text:
\section{Introduction}
\label{sec-intro}

The topologically twisted $\mathcal{N}=4$ super Yang-Mills \cite{Vafa:1994tf} on a complex surface $S$, also known as Vafa-Witten (VW) theory,
appears to be at the heart of intersection of various theories, problems, approaches and conjectures in
mathematics and physics. On one hand, its partition function captures important topological information
about the surface $S$ encoded in the so called Vafa-Witten invariants, the Euler
numbers of the moduli spaces of semi-stable sheaves on $S$, which are the same as the moduli spaces of instantons
in the topologically twisted gauge theory.
The evaluation of these invariants is an important problem that has many links
to other mathematical subjects. In particular, it turns out that the generating functions of VW invariants exhibit non-trivial
modular properties and provide examples of either modular or mock modular forms \cite{Zwegers-thesis,MR2605321}.
This fact makes them a beautiful playground for number theory that can be used as a source for new ideas and identities.

On the other hand, the same invariants can be interpreted as BPS indices in supersymmetric gauge theory,
while their modular properties are understood as a consequence of S-duality \cite{Montonen:1977sn,Vafa:1994tf}.
Moreover, they can be related to generalized Donaldson-Thomas (DT) invariants of non-compact Calabi-Yau (CY) threefolds
because VW theory appears as an effective theory of M5-branes wrapped on $S\times T^2$ and reduced along the torus
where $S$ is a divisor of a CY.
Regarding these non-compact manifolds as local limits of compact CYs establishes further relations
with physics of BPS black holes, instantons and topological strings (see e.g. \cite{Minahan:1998vr}).

The connection to string theory outlined above furnishes us with a plethora of methods based on dualities and
other physical considerations that can bring new insights into the problem of evaluation of VW invariants
and understanding their modular properties.
In particular, recently an important progress has been achieved due to a complete
characterization of a modular anomaly of the generating functions of certain BPS indices in
string theory compactifications with $N=2$ supersymmetry \cite{Alexandrov:2018lgp,Alexandrov:2019rth}.
More precisely, reducing from M-theory to type IIA string, the above M5-brane system gives rise to a bound state
of D4-D2-D0-branes wrapped on even dimensional cycles of a compact Calabi-Yau threefold $\mathfrak{Y}$, which at strong coupling
corresponds to a BPS black hole with a set of electro-magnetic charges. Hence, it makes sense to consider
a generating function of black hole degeneracies (obtained by summing over D0-brane charge),
whose mathematical interpretation is the generating function of the generalized DT invariants
counting semi-stable coherent sheaves supported on a divisor $\mathcal{D}\subset\mathfrak{Y}$. Whereas for $\mathcal{D}$ {\it irreducible}
the generating function, evaluated in the so-called attractor chamber of the moduli space,
is known to be a modular form under S-duality group $SL(2,\mathds{Z})$
\cite{Maldacena:1997de,Gaiotto:2005gf,deBoer:2006vg},
for {\it reducible} divisors where $\mathcal{D}=\sum_{i=1}^n\mathcal{D}_i$ it exhibits a modular anomaly \cite{Alexandrov:2016tnf}.
Remarkably, the anomaly turns out to be of a very special type and can be completely characterized by constructing
modular completions of the generating functions, i.e. their non-holomorphic deformations which
do transform as modular forms. Using duality constraints of string theory\footnote{These constraints require
the moduli space of compactified theory to carry an isometric action of $SL(2,\mathds{Z})$.
At the same time, the metric on this moduli space receives D-instanton corrections weighted by DT invariants.
The completion was found by combining the isometry condition with the explicit description of D-instantons
in the twistor formalism \cite{Alexandrov:2008gh,Alexandrov:2009zh,Alexandrov:2012bu,Alexandrov:2012au}
(see \cite{Alexandrov:2011va,Alexandrov:2013yva} for reviews).},
in \cite{Alexandrov:2018lgp} a general formula for such completion in terms of the original holomorphic generating functions
has been found for a generic divisor. Its form implies that the generating functions are actually
vector valued {\it higher depth mock} modular forms, objects generalizing the notion of the usual mock modularity
\cite{bringmann2017higher}.

Then in \cite{Alexandrov:2019rth} this construction has been upgraded to include a refinement parameter $y$.
Its logarithm $z=\frac{\log y}{2\pi\mathrm{i}}$ transforms as an elliptic parameter so that the refined generating functions
turn out to be {\it higher depth mock Jacobi} forms.
Amazingly, the refined construction is in fact much simpler than the unrefined one and thus represents a natural framework
for further developments. In particular, taking the local limit on $\mathfrak{Y}$ given by the canonical bundle
over a complex surface $S$ with $b_2^+(S)=1$ and $b_1(S)=0$, it has been used to get a formula for the modular completion
$\widehat h_{N,\mu}(\tau,z)$ of the generating functions $h_{N,\mu}(\tau,z)$
of refined VW invariants of $S$ with $U(N)$ gauge group \cite{Alexandrov:2019rth}.
The point is that for this class of surfaces such generating functions are known to have a modular anomaly \cite{Vafa:1994tf},
which can be traced back to the existence of non-holomorphic contributions to the partition function
of VW theory generated by Q-exact terms and coming from boundaries of the moduli space \cite{Dabholkar:2020fde}.
They ensure that the partition function is a true modular form given by the non-holomorphic
completion of the holomorphic generating functions.

Note that the formula of \cite{Alexandrov:2019rth} only expresses $\widehat h_{N,\mu}$ in terms of $h_{N_i,\mu_i}$ with $N_i\le N$,
which remained unknown at that stage. Nevertheless, it allows to trade the modular anomaly of the
generating functions for the holomorphic anomaly of their completions.
Combined with the requirement to have a well-defined unrefined limit $z\to 0$,
it can then be used to actually find the generating functions themselves,
similarly to solution of the topological string on elliptic CY threefolds \cite{Huang:2015sta,Gu:2017ccq}.
This program has been realized for Hirzebruch and del Pezzo surfaces, ${\mathbb F}_m$ and ${\mathbb B}_m$, in \cite{Alexandrov:2020bwg}
which produced a closed formula for both $h_{N,\mu}$ and $\widehat h_{N,\mu}$ for {\it all} ranks $N$. (For earlier results
on VW invariants of these surfaces, see
\cite{Yoshioka:1995,yoshioka1999euler,Manschot:2011dj,Manschot:2011ym,Klemm:2012sx,Beaujard:2020sgs}.)

This result however has two omissions.
First, as indicated above, the generating functions $h_{N,\mu}$ count the invariants only in a special chamber of the moduli space,
parametrized by a choice of polarization of $S$ provided by the K\"ahler form $J\in H^2(S,\mathds{R})$.
This chamber corresponds to the attractor chamber of the CY geometry and,
in the terminology of \cite{Beaujard:2020sgs}, is called the {\it canonical} chamber, specified by $J=-K_S=c_1(S)$
where $K_S$ is the canonical class and $c_1(S)$ is the first Chern class of the surface.
In other chambers the invariants may have different values due to the wall-crossing phenomenon.
Although there are well established wall-crossing formulae which allow to compute the change of invariants from one chamber to another
\cite{ks,Joyce:2008pc,Joyce:2009xv}, it is certainly desirable to have an explicit formula for the generating functions
in arbitrary chamber.

Second, the construction of \cite{Alexandrov:2020bwg} relied on the existence of a null vector belonging to the integer homology lattice
$\Lambda_S=H_2(S,\mathds{Z})$. Clearly, such vector does not exist in the case $b_2(S)=1$ corresponding to $S=\mathds{P}^2$.
Thus, this case remained uncaptured by the that construction.

In fact, the VW invariants of $\mathds{P}^2$ are determined by those of ${\mathbb F}_1$ by means of the
blow-up formula \cite{Yoshioka:1996,0961.14022,Li:1998nv}.
However this formula does not apply directly to $h_{N,\mu}$. Instead, one has to go to a different chamber
of the moduli space and pass through the so-called stack invariants \cite{joyce2008configurations}.
In principle, this is not a big deal and this procedure has been realized for any $N$ in \cite{Manschot:2014cca},
generalizing previous results in \cite{Yoshioka:1994,Manschot:2010nc,Manschot:2011ym} and expressing the generating functions
as combinations of generalized Appell functions.
However, the corresponding result for the completion, which is important not only as a characterization
of the modular properties of the  VW invariants, but also as a prediction for the partition function
of the physical theory, was missing so far. Of course, it can be obtained by either substituting
the generating functions of \cite{Manschot:2014cca} into the formula for the completion of \cite{Alexandrov:2019rth},
or by applying the general procedure developed in \cite{Alexandrov:2016enp} for constructing modular completions of
indefinite theta series and generalized Appell functions. But the former approach leads to expressions
with obscure modular properties, whereas the latter is very cumbersome and has been realized so far only for $N=3$
\cite{Manschot:2017xcr}.
On the other hand, the construction of \cite{Alexandrov:2020bwg} is designed to produce manifestly modular expressions
for completions, so that it is natural to try whether it can encompass the case of $\mathds{P}^2$ as well.
Besides, this might be useful keeping in mind its possible extension to finding DT invariants of compact CY threefolds
where the existence of a null vector belonging to the lattice is not guarantied either.

In this paper we fill in the above two gaps.
First, we conjecture (see eq. \eqref{genJhN}) an explicit expression for the generating functions of refined VW invariants of
Hirzebruch and del Pezzo surfaces for a two-parameter family of polarizations $J$
(which exhaust all possible $J$ in the case of ${\mathbb F}_m$).
Next, we show how one can avoid the obstruction of the absence of a null vector in $\Lambda_S$:
to this end, the formula for the modular completion $\widehat h_{N,\mu}$ can be rewritten
in terms of a set of functions $\check h_{N,\check\mu}$ defined by an `extended lattice' $\check\Lambda_S=\Lambda_S\oplus \mathds{Z}$
that can already possess such a vector.
Then the construction of \cite{Alexandrov:2020bwg} allows to find both $\check h_{N,\check\mu}$ and their completions $\lefteqn{\widehat\chh}\phantom{h}_{N,\check\mu}$,
while $h_{N,\mu}$ and $\widehat h_{N,\mu}$ follow from them in a trivial way.

It turns out that a particular case of this construction is nothing else but a variant of the blow-up formula.
However, it differs from the standard formula in that it is written directly in terms of the generating functions of VW invariants and
does not require the use of stack invariants. It can also be written in a way which is manifestly modular with respect
to the full duality group $SL(2,\mathds{Z})$ (see eq. \eqref{compl-blowup}),
whereas usually only invariance under its congruence subgroup is apparent.
Thus, in a sense, our derivation provides a proof of consistency of the blow-up formula with modularity
for any rank $N$.

Specializing the above results for the case of $\mathds{P}^2$ allows to obtain explicit expressions for both
$h^{\,\mathds{P}^2}_{N,\mu}$ and $\widehat h^{\,\mathds{P}^2}_{N,\mu}$. The formula \eqref{whhP2} for the modular completion
is the main new result of this work.

The organization of the paper is as follows. In the next section we review the results of \cite{Alexandrov:2020bwg}
for $S={\mathbb F}_m$ and ${\mathbb B}_m$ obtained in the canonical chamber of the moduli space and extend them to other chambers.
In section \ref{sec-latext} we present our construction based on the extended lattice, that allows to
find generating functions in the absence of a null vector in $\Lambda_S$, and relate it to the blow-up formula.
Then in section \ref{sec-P2} we apply it to the case of $\mathds{P}^2$. Section \ref{sec-concl} presents our conclusions.
Appendices contain useful details on theta functions, stack invariants, the blow-up formula and generalized error functions.

\section{Generating functions of refined Vafa-Witten invariants}
\label{sec-compl}

In this section we define the refined VW invariants, their generating functions and
discuss them for Hirzebruch and del Pezzo surfaces. The basic definitions are collected in $\S$\ref{subsec-VWinv}.
Then in $\S$\ref{subsec-results} we present the formula for $h_{N,\mu}$ in the canonical chamber found in \cite{Alexandrov:2020bwg}.
In $\S$\ref{subsec-gencham} it is generalized to a two-parameter family of polarizations.
Finally, in $\S$\ref{subsec-modcompl} we provide explicit results for the modular completions $\widehat h_{N,\mu}$.

\subsection{Refined VW invariants}
\label{subsec-VWinv}

Vafa-Witten invariants of a surface $S$ are defined as Euler numbers of the moduli spaces $\mathcal{M}_{\gamma,J}$
of semi-stable coherent sheaves on $S$ where $\gamma=(N,\mu,{1\over 2}\mu^2-n)$ is the Chern character of the sheaf and
$J$ is a polarization entering the stability condition.\footnote{We refer to \cite{Beaujard:2020sgs}
and references therein for a more rigorous and detailed exposition.}
From physics point of view, $N$ is the rank of the gauge group $U(N)$, $J$ is the K\"ahler form,
whereas $\mu=-c_1(F)\in \Lambda_S$ and $n=\int_S c_2(F)\in \mathds{Z}$ are Chern classes of the gauge bundle.
The moduli space $\mathcal{M}_{\gamma,J}$ parametrizes solutions of hermitian Yang-Mills equations
and depends on $J$ through the self-duality condition on the field strength $F$.
An important property, corresponding to the spectral flow symmetry,
is that the moduli space does not change upon tensoring $F$ with a line bundle $\mathcal{L}$.
This leads to the shift $\mu\to \mu-Nc_1(\mathcal{L})$, but leaves invariant the rank $N$ and
the Bogomolov discriminant
\begin{equation}
\Delta(F):= \frac{1}{N} \left( n - \frac{N-1}{2N} \mu^2 \right),
\label{Bogom}
\end{equation}
where $\mu^2\equiv \int_S\mu^2$.
Due to this, the parameter $\mu$ can be restricted to $\Lambda_S/N\Lambda_S$.

In this paper we will be interested in the refined VW invariants, which are defined
by Poincar\'e polynomials of $\mathcal{M}_{\gamma,J}$. More precisely, they are given by
\begin{equation}
\Omega_J(\gamma,y)=\frac{\sum_{p=0}^{d} y^{2p-d_{\mathds{C}}(\mathcal{M}_{\gamma,J})}\, b_p(\mathcal{M}_{\gamma,J})}{y-y^{-1}}\, ,
\label{intOm}
\end{equation}
where $y=e^{2\pi\mathrm{i} z}$ is the refinement parameter, $d_{\mathds{C}}(\mathcal{M})$ is the complex dimension of $\mathcal{M}$
and $b_p(\mathcal{M})$ is its Betti number.
The parameter $z$ is taken to be complex $z=\alpha-\tau\beta$ with both $\alpha$ and $\beta$ non-vanishing.
To exhibit modular properties of the refined invariants, one has to work actually in terms of their rational counterparts
\cite{Manschot:2010xp,Manschot:2017xcr}
\begin{equation}
\bar\Omega_J(\gamma,y) =  \sum_{m|\gamma} \frac{1}{m}\, \Omega_J(\gamma/m, - (-y)^{m}).
\label{defcref}
\end{equation}
It is these invariants that we use to define the generating functions\footnote{Sometimes we will drop the upper index
indicating the surface when it does not cause a confusion.}
\begin{equation}
\label{defhVWref}
h^S_{N,\mu,J}(\tau,z) =
\sum_{n\geq 0}
\bar\Omega_J(\gamma,y)\,
\mbox{q}^{N\(\Delta(F) - \tfrac{\chi(S)}{24}\)},
\end{equation}
where as usual $\mbox{q}=e^{2\pi\mathrm{i}\tau}$.
As follows from \eqref{intOm}, this function has single poles at $z=0$ and $z={1\over 2}$ with the residues
given by the generating function of the unrefined VW invariants.

For $N=1$, there is no dependence on $\mu$ and $J$, and
the generating function is known for any $S$ \cite{Gottsche:1990}.
For surfaces with $b_1(S)=0$, it is given by
\begin{equation}
\label{h10anySref}
h^S_{1,0}(\tau,z) = \frac{\mathrm{i}}{\theta_1(\tau,2z)\, \eta(\tau)^{b_2(S)-1}},
\end{equation}
where $\theta_1(\tau,z)$ is the Jacobi theta function and $\eta(\tau)$ is the Dedekind eta function.
For higher ranks one obtains already a vector valued function with components labelled by $\mu\in\Lambda_S/N\Lambda_S$.
Regarding dependence on $J$, when $b_2^+(S)>1$ it was found to be absent, but
for $b_2^+(S)=1$ and $b_2(S)>1$ it does affect the generating functions.
Of course, this dependence is piecewise constant and can be captured by the standard wall-crossing formulae
\cite{ks,Joyce:2008pc,Joyce:2009xv}.
In the following, we will be interested in surfaces with $b_2^+(S)=1$ and $b_1(S)=0$
which include, in particular, Hirzebruch ${\mathbb F}_m$, del Pezzo ${\mathbb B}_m$ and $\mathds{P}^2$.
Thus, for all of them except $\mathds{P}^2$ there is a non-trivial wall-crossing.

\subsection{Generating functions for ${\mathbb F}_m$ and ${\mathbb B}_m$: canonical chamber}
\label{subsec-results}

Among all possible choices of polarization $J$ there is one special $J=-K_S$ which corresponds to the canonical chamber in
the moduli space. On one hand, this is the chamber with the richest BPS spectrum \cite{Beaujard:2020sgs}.
On the other hand, after uplifting to  a CY threefold, it corresponds to the attractor chamber of the CY moduli space where
the moduli are fixed by the charges and one expects that no bound states are present \cite{Alexandrov:2018iao}
(except the so-called scaling solutions \cite{Denef:2007vg,Bena:2012hf}).
This is the chamber where the construction of \cite{Alexandrov:2018lgp,Alexandrov:2019rth} applies,
and in \cite{Alexandrov:2020bwg} it was used to get an explicit formula for the generating functions $h^S_{N,\mu}:=h^S_{N,\mu,-K_S}$
for Hirzebruch and del Pezzo surfaces, which we now review.

First, let us recall a few basic facts about these surfaces.
The Hirzebruch surface ${\mathbb F}_m$ is a projectivization of the $\mathcal{O}(m)\oplus \mathcal{O}(0)$ bundle over $\mathds{P}^1$.
It has $b_2(S)=2$ and in the basis given by the curves corresponding to the fiber $[f]$ and the section of the bundle $[s]$,
the intersection matrix and the first Chern class are the following
\begin{equation}
C_{\alpha\beta} = \begin{pmatrix} 0 & 1 \\ 1 & -m \end{pmatrix},
\qquad
c_1({\mathbb F}_m) = (m+2)[f]+2[s].
\label{dataFk}
\end{equation}
The del Pezzo surface ${\mathbb B}_m$ is the blow-up of $\mathds{P}^2$ over $m$ generic points.
It has $b_2(S)=m+1$ and a basis of $\Lambda_{{\mathbb B}_m}$ is given by the hyperplane class of $\mathds{P}^2$ and
the exceptional divisors of the blow-up denoted, respectively, by $D_1$ and $D_2,\dots,D_{m+1}$.
In this basis the intersection matrix and the first Chern class are given by
\begin{equation}
C_{\alpha\beta}= {\rm diag} (1,-1,\dots,-1),
\qquad
c_1({\mathbb B}_m) = 3D_1 -\sum_{\alpha=2}^{m+1}D_\alpha.
\label{dataBk}
\end{equation}
A few comments are in order:
\begin{itemize}
\item
For all above surfaces as well as for $\mathds{P}^2$, the signature of the intersection matrix is $(1,b_2-1)$ and
\begin{equation}
c_1^2(S)=10-b_2(S).
\label{c1b2}
\end{equation}
\item
In fact, ${\mathbb F}_1={\mathbb B}_1$, i.e. it is the blow-up of $\mathds{P}^2$ at one point,
and by changing the basis to
\begin{equation}
D_1=[f]+[s],
\qquad
D_2=[s],
\label{baseF1}
\end{equation}
one brings $C_{\alpha\beta}$ and $c_1({\mathbb F}_1)$ to the form \eqref{dataBk} with $m=1$.

\item
The crucial role in the construction of the generating functions of VW invariants is played by a null vector $v_0\in\Lambda_S$.
For $S={\mathbb F}_m$ and ${\mathbb B}_m$, this vector must be chosen as follows \cite{Alexandrov:2020bwg}
\begin{equation}
v_0({\mathbb F}_m)=[f],
\qquad
v_0({\mathbb B}_m)=D_1-D_2.
\label{nullv}
\end{equation}
\end{itemize}

Next, one has to introduce several notations.
Let $\hat\gamma=( N, q)$ where $q\in \Lambda_S+\frac{N}{2} K_S$ and it is further decomposed
into the part spanning $N\Lambda_S$ and the residue class $\mu\in\Lambda_S/N\Lambda_S$,
that reflects the spectral flow symmetry.
In a basis $D_\alpha$, $\alpha=1,\dots, b_2(S)$, of $H_2(S,\mathds{Z})$ this decomposition is given by
\begin{equation}
\label{quant-q}
q_\alpha =  N \, C_{\alpha\beta} \epsilon^{\beta} + \mu_\alpha- \frac{N}{2}\, C_{\alpha\beta} c_1^\beta,
\qquad
\epsilon^\alpha\in \mathds{Z},
\end{equation}
where $C_{\alpha\beta}=D_\alpha\cap D_\beta$ and $c_1^\alpha$ are the components
of the first Chern class $c_1(S)=-K_S=c_1^\alpha D_\alpha$ of the surface.
Then we introduce a combination of theta series\footnote{In \cite{Alexandrov:2020bwg} the same notation was used
for the normalized combinations differing from \eqref{combTh} by the factor of $(h_{1,0}^S)^N$.}
\begin{equation}
\Theta^S_{N,\mu}(\tau,z;\{\Phi_n\})=\sum_{n=1}^\infty\frac{1}{2^{n-1}}
\sum_{\sum_{i=1}^n \hat\gamma_i=\hat\gamma}
\Phi_n(\{\hat\gamma_i\})
\, \mbox{q}^{{1\over 2} Q_n(\{\hat\gamma_i\})}
\, y^{\sum_{i<j}\gamma_{ij}(c_1(S))}\prod_{i=1}^n H^S_{N_i,\mu_i}(\tau,z),
\label{combTh}
\end{equation}
where the sum goes over all decompositions of the charge $\hat\gamma=(N,\mu- \frac{N}{2}c_1)$,
i.e. with the spectral flow parameter set to zero, and the charges
$q_{i}$ are quantized as in \eqref{quant-q} with $N$ replaced by $N_i$.
The other notations used in \eqref{combTh} are the following:
\begin{itemize}
\item
the quadratic form $Q_n$ given by
\begin{equation}
Q_n(\{\hat\gamma_i\})= \frac{1}{N}\, q^2-\sum_{i=1}^n \frac{1}{N_i} q_{i}^2
=-\sum_{i<j}\frac{(N_i q_j - N_j q_i)^2}{NN_iN_j}\, ,
\label{defQlr}
\end{equation}
where $q^2=C^{\alpha\beta}q_\alpha q_\beta$ and $C^{\alpha\beta}$ is the inverse of $C_{\alpha\beta}$;
for the charges satisfying $\sum_i q_i=q$ with $q$ fixed, the signature of $Q_n$ is $(n-1,(n-1)(b_2-1))$;

\item
anti-symmetrized Dirac product of charges depending on a vector $v\in \Lambda_S$
\begin{equation}
\label{gam12}
\gamma_{ij}(v)=
v^{\alpha} (N_i q_{j,\alpha} -N_j q_{i,\alpha});
\end{equation}

\item
the generating function $H^S_{N,\mu}$ of stack invariants evaluated at $J=v_0$, which is given in \eqref{gfHN},

\item
the kernels $\Phi_n$ depending on $n$ charges that determine the theta series and will be specified below.

\end{itemize}
Note that both \eqref{defQlr} and \eqref{gam12} are invariant under an overall shift of the spectral flow parameters $\epsilon_i^\alpha$.
The same is supposed to be true (and indeed will be) for the kernels $\Phi_n$.
Therefore, the r.h.s. of \eqref{combTh} is invariant under the spectral flow shift of $q$,
which explains why this symmetry is fixed in $\hat\gamma$.

The main result of \cite{Alexandrov:2020bwg} is that the generating functions of refined VW invariants in
the canonical chamber are expressed through the combinations \eqref{combTh}
\begin{equation}
h^S_{N,\mu}=\Theta^S_{N,\mu}(\tau,z;\{\Phi_n(-K_S)\})
\label{whgN}
\end{equation}
with the following kernels
\begin{equation}
\Phi_n(\{\hat\gamma_i\};v)
=\sum_{\mathcal{J}\subseteq \mathscr{Z}_{n-1}}\[e_{|\mathcal{J}|}\,\delta_\mathcal{J}(v)
\prod_{k\in \mathscr{Z}_{n-1}\setminus \mathcal{J}}\Bigl(\mbox{\rm sgn}(\Gamma_k(v))-\mbox{\rm sgn}(\mathscr{B}_k)\Bigr)\],
\label{kerg}
\end{equation}
where $\mathscr{Z}_{n}=\{1,\dots,n\}$, $|\mathcal{J}|$ is the cardinality of the set,
$e_{m-1}$ is the $m$-th Taylor coefficient of ${\rm arctanh}$, namely
\begin{equation}
e_m=\left\{ \begin{array}{cc}
0 & \mbox{ if }m \mbox{ is odd},
\\
\frac{1}{m+1}& \mbox{ if }m \mbox{ is even},
\end{array}\right.
\label{valek}
\end{equation}
$\delta_\mathcal{J}(v)=\prod_{k\in \mathcal{J}}\delta_{\Gamma_k(v)}$ is the product of Kronecker deltas, and
\begin{equation}
\begin{split}
\Gamma_{k}(v)=&\,\sum_{i=1}^k\sum_{j=k+1}^n \gamma_{ij}(v),
\\
\mathscr{B}_k=&\, \gamma_{k,k+1}(v_0)+\beta N_k N_{k+1}(N_k+N_{k+1})\, v_0\cdot c_1(S).
\end{split}
\label{defbk}
\end{equation}
The structure of the kernel \eqref{kerg} is in fact very simple. Generically,
it is given by a product of differences of two sign functions.
This is a typical structure for a kernel of indefinite theta series ensuring their convergence \cite{Alexandrov:2016enp}.
In our case, one set of sign functions is determined by the null vector $v_0$ \eqref{nullv} and the second
by the vector $v$, which according to \eqref{whgN} is equal to the anti-canonical class $-K_S=c_1(S)$.
However, for some values of charges the arguments of the second set
may vanish.\footnote{For $\mathscr{B}_k$ this does not happen for generic $\beta$.} Such situations require a special attention.
The formula \eqref{kerg} is written using convention $\mbox{\rm sgn}(0)=0$, however it turns out that the construction
taking its roots in modularity requires, roughly speaking,
that a product of even number of sign functions with vanishing arguments is to be replaced by
the non-vanishing number $e_m$, i.e. $(\mbox{\rm sgn}(0))^m\to e_m$. This is the origin of the sum over subsets in \eqref{kerg}.

\subsection{Generating functions for ${\mathbb F}_m$ and ${\mathbb B}_m$: generic chamber}
\label{subsec-gencham}

Given the result \eqref{whgN}, it is easy to guess an expression for the generating functions
for a more general polarization $J$:

\begin{conj}
For $J\in\mbox{\rm Span}(c_1(S),v_0(S))^+$, where $v_0(S)$ is specified in \eqref{nullv}
and $+$ indicates the restriction to the K\"ahler cone, one has
\begin{equation}
\fbox{$h^S_{N,\mu,J}=\Theta^S_{N,\mu}(\tau,z;\{\Phi_n(J)\})$}
\label{genJhN}
\end{equation}
\label{conj-J}
\end{conj}

\vspace{-0.2cm}
Note that for $S={\mathbb F}_m$ the restriction on $J$ is empty since $\mbox{\rm Span}(c_1(S),v_0)^+$ coincides with the full K\"ahler cone.
On the other hand, for $S={\mathbb B}_m$ it restricts $J$ to belong to a two-dimensional plane in the $m+1$ dimensional moduli space.
This restriction is necessary because for more general $J$ the formula \eqref{genJhN} appears to not have
a well-defined unrefined limit and to be inconsistent with
the formula for the completion \eqref{exp-whhr} given below (see footnote \ref{foot-J}).

We will not try to prove this conjecture. Instead, we provide several evidences in its favor:
\begin{itemize}
\item
We have checked numerically that it agrees with the formulae for $h^{{\mathbb F}_m}_{N,\mu,J_{k,\ell}}$, $N=2,3$,
where $J_{k,\ell}=k([s]+m[f])+\ell [f]$, $k,\ell\ge 0$, which were given in \cite{Manschot:2011ym},
provided one does not hit a wall of marginal stability what may happen for $J\cdot\mu =0\!\! \mod N$.

\item
In the canonical chamber $J=-K_S$ it reduces to the result \eqref{whgN}.

\item
For $J=v_0$ it reduces to the generating function of stack invariants $H^S_{N,\mu}$ \eqref{gfHN}
plus the so-called `zero mode contributions'
where some of the charges are fixed by the condition that at least one of $\Gamma_k(v_0)$ vanishes.
This is the expected structure since the zero mode contributions account for the difference between
stack and VW invariants.

\item
As we will se below, it nicely agrees with the blow up formula.

\end{itemize}

\subsection{Modular completion}
\label{subsec-modcompl}

As explained in the Introduction, the result \eqref{whgN} has been derived
by requiring a proper behavior under modular transformations
which act on the arguments of the generating functions in the following way
\begin{equation}
\tau\to \frac{a\tau+b}{c\tau+d}\, ,
\qquad
z\to \frac{z}{c\tau+d}\, ,
\qquad
\(\begin{array}{cc}
a & b \\ c & d
\end{array}\)\in SL(2,\mathds{Z}).
\label{transf-tz}
\end{equation}
Identifying $\tau$ with the complexified coupling constant of $\mathcal{N}=4$ super-Yang-Mills, one recognizes
in \eqref{transf-tz} the standard S-duality which leaves this theory invariant. This suggests that
the partition function of VW theory transforms as a modular form. Furthermore, since the dependence on $\bar \tau$
can be shown to be Q-exact \cite{Vafa:1994tf}, the standard arguments imply that the partition function is holomorphic in $\tau$
and is essentially captured (up to a simple theta series) by the generating function $h_{N,\mu}$.
The latter therefore is expected to behave as a (vector valued) Jacobi form \cite{MR781735}.
The transformation properties of such objects are shown in \eqref{Jacobi} and characterized by two numbers: weight and index.
In our case they are given by \cite{Alexandrov:2019rth}
\begin{equation}
\label{indexconjVW}
w_S = -\frac12\, b_2(S),
\qquad
m_S(N) = -\frac16\, K_S^2 (N^3-N)-2N .
\end{equation}

All these expectations turn out to be true for $h^S_{1,0}$ \eqref{h10anySref}.
However, for $N\ge 2$ and $b_2^+(S)=1$ they were found to fail \cite{Vafa:1994tf}.
In this case the non-holomorphic contributions from Q-exact terms can be shown to be non-vanishing
due to boundaries of the moduli space \cite{Dabholkar:2020fde}.
As a result, the full partition function is not holomorphic anymore and does not reduce to $h_{N,\mu}$,
while it is still believed to be modular.
This implies that $h_{N,\mu}$ has a modular anomaly, but at the same time it has a non-holomorphic modular completion
which then gives the partition function. An explicit expression for such completion is one of our main interests.

For $S={\mathbb F}_m$ and ${\mathbb B}_m$, the modular completion was found in \cite{Alexandrov:2020bwg} in the canonical chamber,
but given Conjecture \ref{conj-J} it can easily be generalized to the class of polarizations described there.
Indeed, there is a general recipe \cite{Alexandrov:2016enp,Nazaroglu:2016lmr}
for constructing completions of indefinite theta series with kernels given by combinations of
sign functions, which is explained in appendix \ref{ap-generr}.
Applying it to the generating functions \eqref{genJhN},
one obtains\footnote{We assume that the modular group does not act on the polarization $J$.
In VW theory this is not true and it transforms as $J\to J/|c\tau+d|$. However, a rescaling of $J$ does not affect VW invariants.
Therefore, to take into account the transformation of $J$, it is enough to replace in all equations below $J$ by
$J/\sqrt{\tau_2}$, which does stay invariant. We avoid from doing so
explicitly because then the specialization to the canonical chamber would read $J=\sqrt{\tau_2}\,c_1$.}
\begin{equation}
\fbox{$\widehat h^{\,S}_{N,\mu,J}(\tau,z)=\Theta^S_{N,\mu}(\tau,z;\{\widehat\Phi_n(J)\})$}
\label{complFBJ}
\end{equation}
where the kernels
\begin{equation}
\widehat\Phi_n(\{\hat\gamma_i\};J)=\sum_{\mathcal{J}\subseteq \mathscr{Z}_{n-1}} \Phi_{|\mathcal{J}|}^E(\{ {\boldsymbol v}_{\ell}(J)\}_{\ell\in \mathcal{J}};{\boldsymbol x})
\prod_{k\in \mathscr{Z}_{n-1}\setminus \mathcal{J}}\Bigl(-\mbox{\rm sgn}(\mathscr{B}_k)\Bigr)
\label{kerhg}
\end{equation}
are expressed through the generalized error functions $\Phi_n^E$ \eqref{generrPhiME}
with the vectors ${\boldsymbol x}$ and ${\boldsymbol v}_\ell(J)$ defined in \eqref{vectors}.

We also need another expression for the completion which expresses it through the holomorphic functions $h_{N_i,\mu_i}$
and holds more generally than for $S={\mathbb F}_m$ and ${\mathbb B}_m$.
It has been derived in \cite{Alexandrov:2019rth} for $J=-K_S$ and was the key for deriving \eqref{whgN} in \cite{Alexandrov:2020bwg}.
Here, in accordance with the previous discussion, we generalize it to $J\in\mbox{\rm Span}(c_1,v_0)^+$
in which case it reads\footnote{Inspecting the derivation of \eqref{whgN} from \eqref{exp-whhr} done in \cite{Alexandrov:2020bwg}
for $J=c_1$, it is a easy to see that the only place where it depends on the polarization is the derivation
of the holomorphic modular factor given by the functions $H^S_{N,\mu}$. However, this derivation involves only charges
satisfying $v_0\cdot q_i=0$, while for such charges and $J\in\mbox{\rm Span}(c_1,v_0)^+$,
one has $J\cdot q_i=ac_1\cdot q_i$ with positive $a$. Since the coefficient $a$ is canceled in sign functions, this implies that
the derivation still goes through. This provides a justification for \eqref{exp-whhr} and at the same time demonstrates
the need of the restriction on $J$. \label{foot-J}}
\begin{equation}
\fbox{$\displaystyle
\widehat h^{\,S}_{N,\mu,J}(\tau,z)= \sum_{n=1}^\infty\frac{1}{2^{n-1}}
\sum_{\sum_{i=1}^n \hat\gamma_i=\hat\gamma}
\mathscr{R}_n(\{\hat\gamma_i\},J)
\, \mbox{q}^{{1\over 2} Q_n(\{\hat\gamma_i\})}
\, y^{\sum_{i<j} \gamma_{ij}(c_1) }
\prod_{i=1}^n h^S_{N_i,\mu_i,J}(\tau,z)$}
\label{exp-whhr}
\end{equation}
It involves the same objects which already appeared in \eqref{combTh}.
The only new one is the coefficient $\mathscr{R}_n$. We will not need its explicit expression and
therefore relegate its definition to appendix \ref{ap-Rn}.
The only relevant information for us is that it is expressed through the same generalized error functions
which appeared in \eqref{kerhg}.

\section{Modularity, lattice extension and blow-up}
\label{sec-latext}

In this section we would like to address the problem which arises when the lattice $\Lambda_S$ does not possess
a null vector, which plays the crucial role in the construction of \cite{Alexandrov:2020bwg} and appears explicitly
in the definition of kernels \eqref{kerg} and \eqref{kerhg}. An obvious example of such situation
is $S=\mathds{P}^2$ where the lattice $\Lambda_{\mathds{P}^2}$ is one-dimensional. But we will not restrict ourselves to this
particular case and proceed in full generality since this might find applications in attempts to extend the present approach
to calculation of DT invariants of compact CY threefolds. As we will see, it pays off because
it allows to rederive the blow-up formula, reviewed in appendix \ref{ap-stack},
from the formula for the modular completion \eqref{exp-whhr}
and moreover to give it a slightly new formulation.

\subsection{Generating functions from lattice extension}

Our starting point is the equation \eqref{exp-whhr} expressing the modular completion $\widehat h^S_{N,\mu,J}$
through the holomorphic generating functions $h^S_{N_i,\mu_i,J}$.
Let us multiply this equation by the `blow-up function'
$B_{N,\ell}(\tau,\kappa z)= \theta^{(N)}_{\ell}(\tau,\kappa z)/\eta^{N}(\tau)$ \eqref{defBNk} where $\kappa\in \mathds{Z}$.
The parameter $\kappa$ is introduced here due to the following reason.
On one hand, taking $\kappa=0$ seems to be the simplest possibility which allows to perform the manipulations below.
On the other hand, the choice $\kappa=1$ appears to be the most natural one from the geometric point of view
since it will be shown to produce the blow-up construction.
To keep both possibilities available, we prefer to keep $\kappa$ generic.

Now, in each term on the r.h.s. of the resulting equation we apply to the theta function $\theta^{(N)}_{\ell}$
the identity \eqref{decompThN} with $N_i$ equal to the components of the charges $\hat\gamma_i$.
Then $\eta^{-N}$ and all factors $\theta^{(N_i)}_{\ell_i}$ can be absorbed into a redefinition of the generating functions
and their completions
\begin{equation}
\begin{split}
\check h^S_{N,\check\mu,J}(\tau,z)=&\, B_{N,\ell}(\tau,\kappa z)\, h^S_{N,\mu,J}(\tau,z),
\\
\lefteqn{\widehat\chh}\phantom{h}^{\,S}_{N,\check\mu,J}(\tau,z)=&\, B_{N,\ell}(\tau,\kappa z) \,\widehat h^{\,S}_{N,\mu,J}(\tau,z),
\end{split}
\label{defchh}
\end{equation}
where $\check\mu=(\mu,\ell)$. Given the modular properties of $B_{N,\ell}$, the new functions $\lefteqn{\widehat\chh}\phantom{h}^{\,S}_{N,\check\mu,J}$
must behave as vector valued Jacobi forms with weight and index given by
\begin{equation}
\label{indexconjVWch}
\check w_S = -\frac12\, (b_2(S)+1),
\qquad
\check m_S(N) = \frac16\, (N^3-N)(\kappa^2- K_S^2)-2N .
\end{equation}
In this way one arrives at the following equation
\begin{equation}
\lefteqn{\widehat\chh}\phantom{h}^{\,S}_{N,\check\mu,J}= \sum_{n=1}^\infty\frac{1}{2^{n-1}}
\sum_{\sum_{i=1}^n \hat\gamma_i=\hat\gamma}
\mathscr{R}_n(\{\hat\gamma_i\},J)
\, \mbox{q}^{{1\over 2} Q_n(\{\hat\gamma_i\})}
\, y^{\sum_{i<j} \gamma_{ij}(c_1) }
\sum_{\ell_i=0}^{N_i-1}\theta^{(\vec N)}_{\ell,\vec\ell}(\tau,\kappa z)\prod_{i=1}^n\check h^S_{N_i,\check\mu_i,J}.
\label{exp-whhr-ch}
\end{equation}
The form of the theta series $\theta^{(\vec N)}_{\ell,\vec\ell}(\tau,\kappa z)$ \eqref{defthNll} and the relation
\eqref{sumgam} for the power of the refinement parameter suggest that the theta series can also be absorbed.
To this end, let us extend the lattice $\Lambda_S$ by adding to it the trivial integer lattice,
$\check\Lambda_S=\Lambda_S\oplus \mathds{Z}$, so that the new bilinear form is given by
\begin{equation}
\check C=\(\begin{array}{cc}
C & 0 \\  0 & -1
\end{array}\).
\label{extC}
\end{equation}
Following this idea we also define $\check\gamma=(N,\check q)$ with $\check q\in N\check\Lambda_S+\check\mu-\frac{N}{2}\, \check c_1$
and $\check c_1=(c_1, -\kappa)$. Finally, we take the quadratic form $\check Q_n$ to be the same as in \eqref{defQlr}, but evaluated on
the extended lattice.
It is easy to check that, rewritten in terms of the extended lattice, eq. \eqref{exp-whhr-ch}
takes exactly the same form as the initial equation \eqref{exp-whhr}
\begin{equation}
\lefteqn{\widehat\chh}\phantom{h}^{\,S}_{N,\check\mu,J}(\tau,z)= \sum_{n=1}^\infty\frac{1}{2^{n-1}}
\sum_{\sum_{i=1}^n \check\gamma_i=\check\gamma}
\mathscr{R}_n(\{\check\gamma_i\},J)
\, \mbox{q}^{{1\over 2} \check Q_n(\{\check\gamma_i\})}
\, y^{\check c_1\cdot\sum_{i=1}^n \mathscr{N}_i \check q_i }
\prod_{i=1}^n \check h^S_{N_i,\check\mu_i,J}(\tau,z).
\label{exp-whhr-c}
\end{equation}
Note that we could replace in the coefficient $\mathscr{R}_n$ the charges $\hat\gamma_i$ by the new ones $\check\gamma_i$ because
its dependence on charges and the first Chern class $c_1$ is captured by the scalar products
\eqref{defGam} and the contraction with $J$ ensures that the additional components of $\check\gamma_i$ and $\check c_1$ do not contribute.

Thus, we have found that the functions $\check h^S_{N,\check\mu,J}$ and their completions satisfy the same equations
as the generating functions of refined VW invariants of, may be fictional or may be actual, surface $\check S$
with $b_2(\check S)=b_2(S)+1$, bilinear form given by \eqref{extC} and the first Chern class $c_1(\check S)=\check c_1$.
This allows to conclude that
\begin{equation}
\begin{split}
\check h^S_{N,\check\mu,J}(\tau,z)=&\,  h^{\check S}_{N,\check\mu,\iota(J)}(\tau,z),
\\
\lefteqn{\widehat\chh}\phantom{h}^{\,S}_{N,\check\mu,J}(\tau,z)=&\, \widehat h^{\,\check S}_{N,\check\mu,\iota(J)}(\tau,z),
\end{split}
\label{defchhchS}
\end{equation}
where $\iota(J)=(J,0)$ is the embedding $\iota:\Lambda_S\hookrightarrow \Lambda_{\check S}=\check\Lambda_S $.
The difference with respect to the original problem of finding $h^S_{N,\mu,J}$
and their completions is that the new lattice $\Lambda_{\check S}$ may have a null vector even if the old one $\Lambda_S$
did not have it. If $\Lambda_{\check S}$ still does not have a null vector, one can repeat the construction
by further extending the lattice until it has it.\footnote{Such situation may happen for non-unimodular lattices. For instance, if
$\Lambda_S=\mathds{Z}$ with quadratic form equal to $2k^2$, $k\in\Lambda_S$,
its extension does not have null vectors because the null vectors of quadratic form ${\rm diag}(2,-1)$ are irrational.
In this case one would have to extend $\Lambda_S$ twice.
See \cite{Alexandrov:2017qhn} for an example of using such construction for proving S-duality of
multiple D3-instantons in type IIB string theory on a CY threefold.}
Assuming that such vector does exist, one can apply the approach of \cite{Alexandrov:2020bwg} to get the generating functions and
their completions for $\check S$.\footnote{If $\check S$ is a fictional surface, the existence of a solution is not guaranteed!}
In particular, if $\check S$ coincides with one of Hirzebruch or del Pezzo surfaces, one could immediately
borrow results from \eqref{genJhN} and \eqref{complFBJ}. In any case, the generating functions and their completions for
the initial surface $S$ then follow from \eqref{defchh} and \eqref{defchhchS}, i.e.
\begin{equation}
\begin{split}
h^S_{N,\mu,J}(\tau,z)=&\, \frac{h^{\check S}_{N,(\mu,\ell),\iota(J)}(\tau,z)}{B_{N,\ell}(\tau,\kappa z)}\,,
\qquad
\widehat h^{\,S}_{N,\mu,J}(\tau,z)=\frac{\widehat h^{\,\check S}_{N,(\mu,\ell),\iota(J)}(\tau,z)}{ B_{N,\ell}(\tau,\kappa z)} \,.
\end{split}
\label{relhch}
\end{equation}
Note that the index $\ell$ on the r.h.s. is arbitrary. Therefore, the functions $h^{\check S}_{N,\check\mu,\iota(J)}$
must satisfy the following integrability conditions
\begin{equation}
B_{N,k}(\tau,\kappa z) \, h^{\check S}_{N,(\mu,\ell),\iota(J)}(\tau,z)=B_{N,\ell}(\tau,\kappa z) \, h^{\check S}_{N,(\mu,k),\iota(J)}(\tau,z)
\label{inegrcond}
\end{equation}
to ensure the independence of the ratios \eqref{relhch} on this index.\footnote{Similar conditions
for $\widehat h^{\,\check S}_{N,\check\mu,\iota(J)}$ will then automatically follow due to the modular properties of the blow-up functions.}
Of course, these conditions trivially hold for \eqref{defchh}, but the point is that the construction of the generating functions
of $\check S$ does not know {\it a priori} about them and they can be extremely non-trivial.
In fact, they can be viewed as consistency conditions of the whole construction.

There is however one important drawback of the relations \eqref{relhch}:
they make the modular properties of the completion $\widehat h^{\,S}_{N,\mu,J}$ unobvious.
It is unclear why a ratio of two vector valued modular functions should itself form
a vector representation of the modular group.
Fortunately, one can get an alternative representation with help of the identity \eqref{diagThN}.
Applying it to \eqref{defchh}, one obtains
\begin{equation}
\begin{split}
h^S_{N,\mu,J}(\tau,z)=&\, \frac{\eta(\tau)^N}{\prod_{j=1}^N \theta_1(\tau,\kappa_j z)}
\sum_{\ell=0}^{N-1}\theta_{N,\ell}(\tau,\kappa' z)\,h^{\check S}_{N,(\mu,\ell),\iota(J)}(\tau,z),
\\
\widehat h^{\,S}_{N,\mu,J}(\tau,z)=&\,  \frac{\eta(\tau)^N}{\prod_{j=1}^N \theta_1(\tau,\kappa_j z)}
\sum_{\ell=0}^{N-1}\theta_{N,\ell}(\tau,\kappa' z)\,\widehat h^{\,\check S}_{N,(\mu,\ell),\iota(J)}(\tau,z),
\end{split}
\label{h-chh}
\end{equation}
where $\kappa_j=\kappa'+(2j-N-1) \kappa$ and $\kappa'$ should be chosen such that none of $\kappa_j$ vanishes.
In this representation the numerator is a contraction of two modular vectors and denominator
involves only modular scalars so that the result is manifestly modular.
In particular, the modular properties of theta functions and
$\widehat h^{\,\check S}_{N,\check\mu,\check J}$ ensure that $\widehat h^{\,S}_{N,\mu,J}$
does transform as a Jacobi form of weight and index specified in \eqref{indexconjVW}.

\subsection{Relation to the blow-up formula}

The above construction depends on the integer valued parameter $\kappa$.
It affects the auxiliary surface $\check S$ through its first Chern class $\check c_1=(c_1,-\kappa)$
and appears in the relations \eqref{relhch} and \eqref{h-chh} between generating functions.
It is unlikely that the construction goes through and gives the same result for any $\kappa$.
So what should be the value of this parameter?

The simplest possibility is to set $\kappa=0$. This simplifies various equations including
\eqref{h-chh} where one can take $\kappa_j=\kappa'=1$. However, there is no guarantee that there exists
$\widehat h^{\,\check S}_{N,\check\mu,\check J}$ with all required properties.
For instance, if $S=\mathds{P}^2$, ${\mathbb F}_m$ or ${\mathbb B}_m$, the first Chern class of $\check S$ satisfies
\begin{equation}
\check c_1^2=c_1^2(S)=10-b_2(S)=11-b_2(\check S),
\end{equation}
and thus spoils \eqref{c1b2}, so that $\check S$ is probably fictional and a solution for generating functions may not exist.
And indeed, in the case $S=\mathds{P}^2$ one can show that it is impossible to find an analogue of the functions
$H^{{\mathbb F}_m}_{N,\mu}(\tau,z)$, which would have the same behavior near $z=0$ and $z=1/2$
but transform with index (see eq. \eqref{indexconjVWch})
$\check m_{\mathds{P}^2}(N)=-\frac12\, (3N^3+N)$ instead of $-\frac23\, (2N^3+N)$.

Another natural choice is $\kappa=1$. In this case the multiplication factors in the relations \eqref{relhch}
become identical to the standard blow-up functions, whereas the relations themselves are reminiscent the blow formula
\eqref{blowupf}. This is not an accident since a blow-up of $S$ gives a surface $\check S$ with exactly the same data
as one obtains for $\kappa=1$: the intersection matrix \eqref{extC} and the first Chern class $\check c_1=c_1-D_e$,
where $D_e$ is the exceptional divisor of the blow-up responsible for the factor $\mathds{Z}$ in $\Lambda_{\check S}=\Lambda_S\oplus\mathds{Z}$.
Thus, in this case our construction reproduces the standard blow-up and shows its consistency with
the modular properties of the completions.
In particular, choosing in \eqref{h-chh} $\kappa'=N$, one arrives at the following manifestly modular relation
equivalent to the blow-up formula
\begin{equation}
\fbox{$\displaystyle\widehat h^{\,S}_{N,\mu,J}(\tau,z)=  \frac{\eta(\tau)^N}{\prod_{j=1}^N \theta_1(\tau,(2j-1) z)}
\sum_{\ell=0}^{N-1}\theta_{N,\ell}(\tau,N z)\,\widehat h^{\,\check S}_{N,\iota(\mu)+\ell D_e,\iota(J)}(\tau,z)$}
\label{compl-blowup}
\end{equation}

There is however an important difference between \eqref{blowupf} and \eqref{relhch} for $\kappa=1$:
whereas the former is formulated in terms of the generating functions of stack invariants, the latter
is written directly in terms of the generating functions of VW invariants.
We recall that the reason for the use of stack invariants was that they are, in contrast to VW invariants,
well defined directly on walls of marginal stability, while,
as we will see below, a blow-up of $\mathds{P}^2$ leads to a polarization precisely corresponding
to one of such walls. Our results indicate that the generating functions $h^{\,S}_{N,\mu,J}$
and their completions $\widehat h^{\,S}_{N,\mu,J}$ are in fact also well defined on the walls!
In particular, the functions \eqref{genJhN} and \eqref{complFBJ} can be evaluated for any polarization
from the allowed two-parameter family $J\in\mbox{\rm Span}(c_1,v_0)^+$ and for any residue class $\mu$.
This is because the modular completions are actually smooth across the walls and provide
an unambiguous definition of the holomorphic generating functions everywhere including the walls.
In our case this is realized by a prescription for the contributions
with vanishing arguments of sign functions in the kernel \eqref{kerg}.

We emphasize that this does not mean that the VW invariants are defined on the walls!
In fact, one can check (see the next section for an example) that the rational invariants extracted from $h^{\,S}_{N,\mu,J}$
on a wall do {\it not} lead to {\it integer} invariants
after application of the inverse of the formula \eqref{defcref}.
These are interesting questions whether these rational numbers have any geometrical meaning and
whether they can be converted to integer numbers.

\section{Generating functions for $\mathds{P}^2$ and their modular completion}
\label{sec-P2}

Now we are ready to turn to the main motivation of this work and to provide explicit expressions for
the generating functions of refined VW invariants of $\mathds{P}^2$ and their modular completions.
Using the construction of the previous section they can be related to the corresponding functions for the blow-up of $\mathds{P}^2$
which is known to be the Hirzebruch surface ${\mathbb F}_1$. Indeed, the bilinear form \eqref{extC} and the first Chern class $\check c_1$
are given in this case by
\begin{equation}
\check C=\(\begin{array}{cc}
1 & 0 \\  0 & -1
\end{array}\),
\qquad
\check c_1=(3,-1)
\end{equation}
and coincide with the intersection form and the first Chern class of ${\mathbb F}_1$ in the basis \eqref{baseF1}.
Thus, $D_2=[s]$ is the exceptional divisor $D_e$ of the blow-up, whereas $\iota(J)=D_1=[f]+[s]$.\footnote{In this basis
the null vector is $v_0({\mathbb F}_1)=D_1-D_2$.}
As a result, we arrive at the following representation for the generating functions
\begin{equation}
\fbox{$\displaystyle
h^{\mathds{P}^2}_{N,\mu}(\tau,z)=\frac{\Theta^{{\mathbb F}_1}_{N,\mu D_1+\ell D_2}(\tau,z;\{\Phi_n(D_1)\})}{B_{N,\ell}(\tau, z)}
$}
\label{hP2}
\end{equation}

For $N=2$ and 3 we have checked that this expression does reproduce the generating functions computed in
\cite{Yoshioka:1994,Manschot:2010nc}. The integrability conditions \eqref{inegrcond}, which the generating functions of ${\mathbb F}_1$
should satisfy, are known to follow at $N=2$ from the periodicity property for the classical Appell function
and at $N=3$ from its generalization proven in \cite{Bringmann:2015}.
For higher ranks they remain unexplored.

The modular completion of the generating functions \eqref{hP2}, which was known until now only up to $N=3$,
can be given in the following two forms
\begin{equation}
\fbox{$\displaystyle
\begin{split}
\widehat h^{\,\mathds{P}^2}_{N,\mu}(\tau,z)=&\,\frac{\Theta^{{\mathbb F}_1}_{N,\mu D_1+\ell D_2}(\tau,z;\{\widehat\Phi_n(D_1)\})}{ B_{N,\ell}(\tau,z)}
\\
= &\, \frac{\eta(\tau)^N}{\prod_{j=1}^N \theta_1(\tau,(2j-1) z)}
\sum_{\ell=0}^{N-1}\theta_{N,\ell}(\tau,N z)\,\Theta^{{\mathbb F}_1}_{N,\mu D_1+\ell D_2}(\tau,z;\{\widehat\Phi_n(D_1)\}
\end{split}$}
\label{whhP2}
\end{equation}
Whereas the first representation is simpler to evaluate, the second provides a direct access to the modular properties
of $\widehat h^{\,\mathds{P}^2}_{N,\mu}$.

Finally, we note that the polarization $J=D_1$ for ${\mathbb F}_1$ is a wall of marginal stability for
(some of) the VW invariants with $\check\mu=\check\mu^2 D_2$.
Nevertheless, our version of the blow-up formula \eqref{hP2} perfectly works for all $\check\mu$.
In the example below we demonstrate this for $N=3$ and $\mu=0$
as well as the fact mentioned in the end of the previous section
that a naive calculation of the refined VW invariants of ${\mathbb F}_1$ for this polarization
leads to rational numbers.

\subsubsection*{Example: $N=3$, $\mu=0$}

For these values of parameters the generating function of refined VW invariants of $\mathds{P}^2$ is known
and can be found, for instance, in \cite[Eq.(6.22))]{Manschot:2017xcr}. In our notations it reads
\begin{eqnarray}
h^{\mathds{P}^2}_{3,0}(\tau,z) &= & -\frac{\mathrm{i}}{\theta_1(\tau,2z)^3\,\theta^{(3)}_{0}(\tau,z) }\[
\frac13\,\theta^{(3)}_{0}(\tau,z) +  y^4\sum_{k_1,k_2\in \mathds{Z}}
\frac{y^{-2k_1-4k_2}q^{k_1^2+k_2^2+k_1k_2+2k_1+k_2}}{(1-y^4q^{2k_1+k_2})(1-y^4q^{k_2-k_1})}
\right.
\nonumber\\
&& +\frac{2\mathrm{i}
  \eta(\tau)^3}{\theta_1(\tau,4z)}
  \( \sum_{k\in   \mathds{Z}}\frac{y^{-6k}q^{3k^2}}{1-y^6q^{3k}} -\frac12\, \theta_3(6\tau,6z) \)
\left.
-\frac{\eta(\tau)^6\,\theta_1(\tau,2z)}{\theta_1(\tau,4z)^2\,\theta_1(\tau,6z)}\].
\end{eqnarray}
Its first few terms of the expansion at small $q$ are given by
\begin{equation}
h^{\mathds{P}^2}_{3,0}=\textstyle q^{-3/8}\[\frac{1}{3 (y^3-y^{-3})}
+\frac{y^{12}+2 y^{10}+4y^8 +\frac{16}{3}\,y^6+6y^4+6y^2 +\frac{19}{3}+6y^{-2}+6y^{-4}+\frac{16}{3}\,y^{-6}
+4y^{-8}+2 y^{-10}+y^{-12}}{y^3-y^{-3}}\,q^3
+O(q^4)\].
\label{exphP2N3}
\end{equation}
This expansion allows to read off the rational invariants
$\bar\Omega(\gamma,y)$ for small second Chern classes of the sheaf.
Inverting the formula \eqref{defcref}, one then extracts the integer valued invariants which encode
the Betti numbers of moduli spaces $\mathcal{M}_{\gamma,J}$:
\begin{equation}
h^{\mathds{P}^2}_{3,0}-\frac13\,h^{\mathds{P}^2}_{1,0}(3\tau,3z)=
\frac{q^{-3/8}}{y-y^{-1}}\Bigl[(y^2+1 + y^{-2}) (y^8 + y^4 +1+ y^{-4} + y^{-8})\, q^3 +O(q^4)\Bigr],
\label{inthP2N3}
\end{equation}
in agreement with \cite[Eq.(A.39)]{Beaujard:2020sgs}.

Let us compare this result with \eqref{hP2}. The expansion of the numerator and denominator gives
\begin{eqnarray}
h^{{\mathbb F}_1}_{3,0,D_1}(\tau,z)&=& q^{-1/2}\[\textstyle \frac{1}{3 (y^3-y^{-3})}
+\frac{y^2+1+y^{-2}}{3 (y-y^{-1})}\,q
+\frac{y^2+1+y^{-2}}{y-y^{-1}}\,q^2
\right.
\\
&+&\left.
\textstyle\frac{y^{12}+2 y^{10}+4y^8 +6y^6+9y^4+12y^2 +\frac{43}{3}+12y^{-2}+9y^{-4}+6y^{-6}+4y^{-8}+2 y^{-10}+y^{-12}}{y^3-y^{-3}}\,q^3
+O(q^4)\],
\nonumber
\\
B_{3,0}& =& q^{-1/8}\Bigl[1 + (y^2 +1+ y^{-2})^2 q + 3(y^2 +1+ y^{-2})^2 q^2
\Bigr.
\\
&&\Bigl.
+ \(2 y^6+ 9 y^4+ 18 y^2 +24 + 18y^{-2} + 9y^{-4} + 2y^{-6}\) q^3
+O(q^4) \Bigr].
\nonumber
\end{eqnarray}
It is immediate to check that the ratio of these two expansions indeed reproduces \eqref{exphP2N3}.
Thus, our blow-up formula \eqref{hP2} allows to bypass stack invariants even
when the polarization induced on the blow-up surface corresponds to a wall of marginal stability.
On the other hand, applying the same inverse formula as in \eqref{inthP2N3}, one obtains
\begin{eqnarray}
h^{{\mathbb F}_1}_{3,0,D_1}(\tau,z)&-&\frac13\,h^{{\mathbb F}_1}_{1,0}(3\tau,3z)=\frac{q^{-1/2}}{y-y^{-1}}\Bigl[
\textstyle \frac{1}{3}\,(y^2+1+y^{-2})\,q
+(y^2+1+y^{-2})\,q^2
\\
&+&
\textstyle\(y^{-10}+y^{-8}+2y^{-6}+\frac{8}{3}\, y^{-4}+\frac{13}{3}\,y^{-2}+5+\frac{13}{3}\,y^2+\frac{8}{3}\,y^4
+2y^6+y^8+y^{10}\)q^3
+O(q^4)\Bigr].
\nonumber
\end{eqnarray}
Although this procedure does produce symmetric Poincar\'e polynomials as in \eqref{intOm}, their coefficients are not all integer.
Thus, the resulting numbers cannot be interpreted as Betti numbers of some moduli spaces and their interpretation remains open.

\section{Conclusions}
\label{sec-concl}

In this paper we studied generating functions of refined VW invariants
for $U(N)$ Vafa-Witten theory on a complex surface $S$
with $b_2^+(S)=1$ and $b_1(S)=0$. Such generating functions are higher depth
mock modular forms and possess non-holomorphic modular completions.
Our approach originates in the equation satisfied by these completions in the canonical chamber
of the moduli space \cite{Alexandrov:2019rth}
which, combined with constraints from the unrefined limit,
turns out be sufficiently restrictive to find the generating functions explicitly.
In this way, we obtained the following results, which all hold for arbitrary rank $N$:
\begin{itemize}
\item
Generalizing the results for Hirzebruch and del Pezzo surfaces found in \cite{Alexandrov:2020bwg} in the canonical chamber,
we extended them to the chambers corresponding to a two-parameter family of polarizations, which for Hirzebruch surfaces
cover the whole K\"ahler cone. In particular, in \eqref{genJhN} we provided explicit expressions for
the generating functions and in \eqref{complFBJ} for their completions.

\item
We also proposed the extension \eqref{exp-whhr} of the equation on the completion mentioned above to arbitrary polarization.

\item
We showed that this equation implies a version of the blow-up formula \eqref{relhch} for the generating functions of VW invariants,
which in contrast to the usual approach does not require the use of stack invariants.
A similar formula holds for the modular completions and it can nicely be rewritten in a manifestly modular way \eqref{compl-blowup}.

\item
Specializing to $S=\mathds{P}^2$, we arrived at the closed formula \eqref{hP2} for the generating functions $h^{\mathds{P}^2}_{N,\mu}$,
which have been computed before in a different form \cite{Manschot:2014cca},
and the new formula \eqref{whhP2} for their completions $\widehat h^{\,\mathds{P}^2}_{N,\mu}$.

\end{itemize}

Probably, the most unexpected result is that the generating functions \eqref{genJhN} which we proposed make sense
everywhere in the moduli space, including walls of marginal stability. This is implied by the fact
that they can be used in the blow-up formula and produce correct VW invariants on all surfaces related by
the blow-up or blow-down construction
(see Fig.1 in \cite{Beaujard:2020sgs} for a nice representation of such relations between different surfaces).
This fact was the key for avoiding stack invariants.
However, this leaves us with the open question of geometric interpretation of the numbers extracted from
the generating functions evaluated on a wall. We observed that inverting the formula \eqref{defcref},
one does get symmetric Poincar\'e polynomials, but their coefficients are not generically integer.
May be the first problem to consider is whether there is a general prescription which converts these rational numbers
into integer.

Finally, we would like to note that one of the motivations of this work was to develop methods which
might be useful in extending the approach based on the equation for the modular completion to compact CY threefolds,
the domain where it was originally derived. A serious complication arising for such extension is that the integer parameter $N$
is replaced by a vector $p^a$ valued in (dual to) the lattice of electric charges $\Lambda=H_2(\mathfrak{Y},\mathds{Z})$.
A way to avoid this complication is to consider a CY with $b_2(\mathfrak{Y})=1$, the case analogous to $S=\mathds{P}^2$ for VW theory.
Thus, we expect that the results obtained here may find a direct application for the computation of DT invariants
of compact CYs with one K\"ahler modulus.

\section*{Acknowledgements}

The author is grateful to Jan Manschot and Boris Pioline for valuable discussions.

redpajama_set_name: RedPajamaArXiv


After over 50 years of operation, NASA is about to permanently enter the world of instant (or near instant) information exchange with the general public in a new way that largely symbolizes one of the changing ways the premiere US space agency will interact with the public and disseminate information about science operations and real-time orbital operations on the International Space Station.
While NASA already has a fairly good track record of releasing operational and scientific data sets from its various missions, the decision has been made to proceed to the next step and take the ISS into the live data streaming environment of the 21st century through webpage and mobile device app browsing.
The change, however, is not just a change in the live streaming of information between the ISS and MCC (Mission Control Center) Houston and Moscow. It is also a change in how NASA shares real-time data with the public at large.
This means that household and personal computing and communication devices can now access the streamable information from the International Space Station and MCC Houston.
But that ability will require an increase in the need for "Mobile Device usage" including computers with internet connection, smart phones, tablets, and the creation of new apps for some of these devices to link up with the soon-to-be up-and-running ISS Live program.
The Open Government initiative for NASA and the International Space Station, which had its US core segment designated a US National Lab in the last couple of years, has kick started a Mission Operations Directorate plan call the Public Data Services Initiative.
At its core, the ISSLive! mobile application (which will be available over the internet and for purchase in app stores for various smart phone and tablet devices) will provide users with "simplified" console information readouts on public telemetry and timeline data for the International Space Station as well as current live tracking ability of the International Space Station, ISS MCC Houston "over the shoulder" viewing opportunities, ISS crew timelines (broken up into daily increments), MCC console displays, and Acquisition of Signal (AOS) and Loss of Signal (LOS) indicators/predictors.
"Aggressive schedule for development. Must focus on ISSLive! Core Concept" which includes "public telemetry and timeline data, education review of technical content, export control for technical content, approvals for space organization icons and flight controller icons/patches, and above all ISSLive! Team and Product integration," notes the ISSLive! presentation.
Specifically, the public telemetry and timeline data will use a "LightStreamer Server (Port 80) and a public API for telemetry." A timeline XML is currently under evaluation.
Web pages will also be used, as will HTML content as part of this new data streaming initiative.
For this new application, considerations to sensitive and restricted information had to be taken into account, as well as larger issues of the Open Government initiative with respect to Human Spaceflight on the ISS.
Specifically, ITAR requirements have to be met to ensure the protection of restricted technology data – a holdover from the Cold War Rocket and Tech Protection programs.
Furthermore, crew privacy and proprietary information concerns had to be addressed and reviewed to make sure that ISS crews' rights to privacy would be maintained throughout the program.
Lastly, agreements and approvals for this application had to be gained from the International Space Station partner family.
In order to ensure a timely and smooth rollout of the ISSLive! application, a demo for stakeholders was rolled out on August 31 via the web browser Firefox (as bugs were still being worked out on other browsers) to review the telemetry and timeline part of the app.
Input from those involved was solicited to identify any issues with information accuracy. "Need everyone to provide feedback on content accuracy for all wording and parameters," noted the ISSLive! Demo presentation – also available for download on L2.
The demo lasted from August 31 to September 7, 2011.
After this, a thorough review was initiated with a target rollout date for the application of Oct. 1, 2011 – the start of the new fiscal year for the US government.
Based on the review, a prioritized list of work to be completed by Oct. 1 was created. The list included the following: working links between Link/Navigation; proper rendering of all pages; preventing a freeze on the active console on 3D Unity view; making sure that Station rendered properly and completely; ensuring proper connection of ISS model to real-time tracking and orientation data; ensuring proper telemetry readouts.
It is hoped the application will be available as of Tuesday of this week.
(Images via NASA and L2 content).


text:
SULT1E1 () – білок, який кодується однойменним геном, розташованим у людей на короткому плечі 4-ї хромосоми.  Довжина поліпептидного ланцюга білка становить 294 амінокислот, а молекулярна маса — 35 126.

Кодований геном білок за функцією належить до трансфераз.
Задіяний у такому біологічному процесі як поліморфізм.
Білок має сайт для зв'язування з ліпідами, стероїдами.
Локалізований у цитоплазмі.

Література

Примітки

Див. також
 Хромосома 4

Некатегоризовані білки

redpajama_set_name: RedPajamaWikipedia


text:
The Caribbean Immigrant Victims Of COVID-19 – 2020
Celebrities Who Like Us
Caribbean and Latin America Daily News
New Oil Discovery Made In This Caribbean Community Nation
3 Factors You Need To Consider Before Buying A Business
Preparing For A Global Career: A Guide
Tips For Caribbean Nationals Planning On Relocating To Kuwait
Can't Decide Where to Invest? Consider Investing In the Caribbean
Home Sports Cricket Caribbean Sporting Events Canned As COVID-19 Cases Uptick
Caribbean Sporting Events Canned As COVID-19 Cases Uptick
newsamericas
FLASH BACK: A hotel staff checks West Indies' cricketer Sunil Ambris' body temperature upon arriving at a hotel in Colombo on February 10, 2020 where the West Indies played three one day international (ODI) and two T20 cricket matches during their tour of Sri Lanka. (Photo by ISHARA S. KODIKARA/AFP via Getty Images)
News Americas, NEW YORK, NY, Fri. March 20, 2020: AS COVID-19 cases across the region continue to uptick, sporting events across the Caribbean and in its Diaspora are also being canned.
The 35th annual 2020 CARIFTA has been postponed due the novel coronavirus Pandemic. The Games was scheduled to take place in Bermuda this April10-13, 2020. The news came as the Cayman Islands joined Antigua and Barbuda, and the Bahamas in withdrawing from the competition.
"We would have never predicted an occurrence of this magnitude a week ago. We were deep in preparations making our final adjustments," The Bermuda National Athletics Association said in a statement. "We were ready to host the 2020 CARIFTA Games and ready to welcome the 27 countries to our shores."
Bermuda has remained coronavirus free.
Cricket West Indies (CWI) has also taken the decision to suspend all CWI Tournaments and face-to-face group meetings for a minimum of 30 days, beginning on Monday March 16th.
Suspended are the:
Final two rounds of the West indies Championship – 26th to 29th March and 2nd – 5th April
The Women's CMI Super 50 Cup, Guyana – 27th March to 12th April
Regional Under 15s Boys Championship, Antigua – 9th to 20th April
Regional Under 19s Girls Championship, Trinidad – 6th to 12th April
CWI has also advised all territorial boards and local cricket associations to follow the advice of their respective ministries of health.
The Jamaican government as well as ISSA and Sponsor, GraceKennedy have also canceled this year's staging of the Inter-secondary Schools Sports Association (ISSA)/GraceKennedy Boys' and Girls' Athletics Championships in Jamaica and all participation at the Penn Relays since last week.
Then on Friday, March 13th, Team Jamaica Bickle announced it was also pulling out of the Penn Relays. On Monday, March 16th, The Division of Recreation and Intercollegiate Athletics at the University of Pennsylvania announced that the 126th running of the Penn Relays will be canceled for the first time in its history this year.
The Penn Relays, which has been contested uninterrupted since 1895 and celebrated its 125th consecutive running last year. It was set for April 23rd through 25th.
The Grenada Athletic Invitational has also been cancelled.
And the first three meets of the 2020 Wanda Diamond League season, scheduled for April 17 in Qatar and May 9 and 16, 2020 in China, are also canned.
Meanwhile, as uncertainty swirls around the Summer Olympics in Japan, President of the Jamaica Olympic Association (JOA) Christopher Samuda said the JOA is willing to withdraw Jamaica's athletes from the 2020 Tokyo Olympics if, in light of the evolving Coronavirus pandemic, Japan was not able to guarantee their health and safety.
Concacaf has also postponed the following competitions scheduled to take place over the next 30 days:
2020 Scotiabank Concacaf Champions League
2020 Concacaf Men's Olympic Qualifiers
First Round of the Concacaf 2021 Gold Cup Qualifiers
2020 Concacaf Caribbean Club Shield
Globally, the Football Association has agreed to extend the 2019-2020 season indefinitely, while prolonging the suspension of all its leagues until at least April 30th.
Last week, the FA halted the Premier League, the English Football League, domestic cup competitions and the women's professional game in response to the coronavirus pandemic wreaking havoc on the sporting world.
#Caribbeansports
#Caribbeansportsingeventscancelled
#COVID19cancellations
#sportingeventscancelled
#WestIndiescricket
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U.S. VP-Elect To Deliver Remarks At Caribbean Inauguration Celebration In Her...
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redpajama_set_name: RedPajamaCommonCrawl


Bamboom is a brand with the best in nature for your baby, realized in organic bamboo. Bamboom is an exclusive collection of baby items, manufactured in Italy. Bamboo offers comfort and safety for your baby and it is practical to use. A trendy line, combined with super-soft natural materials.
Wonderfully warm footmuff from Bamboom specially for car sea..


Senior Corey Lewis battles for possession against Brandon.
Two talented teams met on the pitch at Clinton High School Thursday night when Clinton hosted the Brandon Bulldogs.
The Lady Arrows (5-1-2) dropped their first match of the season.
Clinton and Brandon remained scoreless at the end of the first half, but Brandon took the lead in the middle of the second half.
Lady Arrow keeper Kennedi Richardson made an athletic save in a sure-goal, but the Lady Bulldogs recovered the rebound and scored, sending the Lady Arrows down by one.
The Arrows (3-4-1) battled the Bulldogs well all throughout the first half as both teams went into the halftime break scoreless.
Clinton looks to rebound after losses when they host the Madison Central Jaguars on Tuesday, Dec. 4 at Arrow Field.
This game will be live streamed and can be watched at ArrowAthletics.org.
The Lady Arrows will start at 5:30 P.M., and the Arrows are scheduled for 7:15 P.M.


text:
Q: A fluid full-height bootstrap layout I'm trying to make a game layout which fills the entire browser window, but doesn't scroll. The layout is something like this:
 +----------------------+
 +       Header         +
 +------+---------------+
 +      +               +
 +      +               +
 + Side +    Main       +
 +      +  Fill Height  +
 +      +               +
 +      +               +
 +----------------------+
 +       Footer         +
 +------+---------------+

The sidebar should scroll if necessary, but the main content should not. The layout should be responsive: if the browser is too narrow the side goes under "header" and switches the way it is presented (likely Footer disappears as well).
I'd like to use Bootstrap but I can not find any details on how such a layout might be possible. Does it make sense to even use the bootstrap layouts in this display? Or should I rather just code the CSS with absolute positions (% based) and used bootstrap widgets within the sections?

A: I think you can create this type of layout by putting the sidebar and main content in a 100% height/width wrapper.
See this example template at Bootply: http://bootply.com/102032
EDIT: Pad the sidebar_wrapper according to the height of the header and footer:
  #sidebar-wrapper {
      height: 100%;
      padding: 50px 0 50px 0; /* pad top and bottom by height of header and footer */
      position: fixed;
      border-right: 1px solid gray;
  }

redpajama_set_name: RedPajamaStackExchange


text:
<?xml version="1.0" encoding="utf-8"?>
<RelativeLayout xmlns:android="http://schemas.android.com/apk/res/android"
    xmlns:tools="http://schemas.android.com/tools"
    android:id="@+id/activity_celebrity_profile"
    android:layout_width="match_parent"
    android:layout_height="match_parent"
    android:background="@color/myColorOneBody"
    tools:context="com.vumobile.celeb.ui.CelebrityProfileActivity">

    <include
        android:id="@+id/toolbar_celeb"
        layout="@layout/toolbar_celeb" />

    <FrameLayout
        android:id="@+id/frame"
        android:layout_width="150dp"
        android:layout_height="160dp"
        android:layout_below="@+id/toolbar_celeb"
        android:layout_centerHorizontal="true"
        android:layout_gravity="center"
        android:layout_marginTop="10dp"
        android:descendantFocusability="blocksDescendants">

            <com.vumobile.celeb.Utils.RoundedImageView
                android:id="@+id/imgProfilePic"
                android:layout_width="132dp"
                android:layout_height="132dp"
                android:padding="9dp"
                android:layout_gravity="center"
                android:scaleType="fitXY"
                android:background="@drawable/logo" />

            <ImageView
                android:layout_width="150dp"
                android:layout_height="150dp"
                android:layout_gravity="center"
                android:src="@drawable/b" />

    </FrameLayout>

    <TextView
        android:id="@+id/txtStatus"
        android:layout_width="wrap_content"
        android:layout_height="wrap_content"
        android:layout_below="@+id/frame"
        android:layout_centerHorizontal="true"
        android:layout_marginTop="20dp"
        android:text="" />

    <Button
        android:id="@+id/btnGoLive"
        android:layout_width="wrap_content"
        android:layout_height="wrap_content"
        android:layout_below="@+id/login_button"
        android:layout_centerHorizontal="true"
        android:text="Go Live"
        android:visibility="gone" />

    <TextView
        android:id="@+id/txtConfirmationMessage"
        android:layout_width="150sp"
        android:layout_height="wrap_content"
        android:layout_below="@+id/login_button"
        android:layout_centerHorizontal="true"
        android:layout_marginTop="80dp"
        android:gravity="center"
        android:text="You will get confirmation notification soon"
        android:visibility="gone" />

    <com.facebook.login.widget.LoginButton
        android:id="@+id/login_button"
        android:layout_width="200dp"
        android:layout_height="wrap_content"
        android:layout_below="@+id/txtStatus"
        android:layout_centerHorizontal="true"
        android:layout_marginTop="58dp"
        android:gravity="center"
        android:paddingBottom="12dp"
        android:paddingTop="12dp" />

</RelativeLayout>

redpajama_set_name: RedPajamaGithub


My father and I were in the kitchen of the house where I grew up one morning in 1987, having a nice conversation. This was not a typical event because I'd never had a substantial conversation with my father before that, nothing more than one-word answers or brief sentences.
I don't remember exactly what we talked about, but I think it was just chit chat, something about the weather, maybe a news story or something. Winter morning sun streamed through the windows over the table and in the back door and diffused through the white sheers on the dining room window in the tiny ranch house as we stood there exchanging words. He spoke with some animation, responding to what I'd said, asking questions of what I thought about something.
I was 25 and don't remember ever hearing his natural conversational voice aside from the subdued, minimal answers he gave to questions and occasional brief comments. Just a year before he had injured himself in the small bakery where he worked, a very unusual circumstance for all the years since his childhood working in the family bakery, through service in WWII and then in other family and otherwise small bakeries after the war to that point. He was treated for the injury to his hand but the wise emergency room doctors and nurses had noticed some respiratory and cognitive issues. He was diagnosed with lung cancer just the day before the Challenger shuttle broke up over the watching nation of students and teachers and citizens, and I think I cried hard for that tragedy in large part because the shock of the diagnosis had just begun to wear off. The mass was right at the point where his lungs separated from his trachea and impacted both lungs. The surgery was long and difficult, but when he was healing well physically and was not returning to a full mental state they realized his cognitive issues may not have been entirely due to the cancer.
After tests and trials of a few medications doctors determined he had parkinsonism or Parkinson Syndrome, evidencing the symptoms of Parkinson's disease like the fixed, unblinking gaze, shuffling gait, rigidity and slow movements. The cause is usually unknown but can be induced by certain drugs or environmental toxins, and sometimes from head trauma or brain damage. The doctors traced it back to a nearly fatal malarial fever my father had had while serving in India in the Asian-Pacific theater of WWII that likely caused mild brain damage at the time, but may have unknowingly caused enough damage in the right areas that he slowly produced progressively less dopamine and more symptoms over a period of decades. This concurred with what others had seen. By the time I remember him he was the silent, stone-faced person at the dinner table, or driving the car, or sleeping on the couch before he went to work in the bakery in the middle of the night, the person holding me in the photo.
Apparently the current medications and treatments were somewhat effective considering his comparative ease of movement and his conversational ability. The conversation continued for a while and then he left the kitchen to talk to my mother who was in my old bedroom at her vanity getting ready to leave the house.
I didn't hear my mother's response.
My mother was getting ready to leave the house because this was the day my father was going to the hospital for probably the last time before he entered a nursing home, and I was there to drive them and help them through admissions. The medications and treatments had indeed loosened up his body enabling him to move and speak as he hadn't in years, but it wasn't consistent. His mind was quickly becoming fragmented and he was developing frequent pneumonia and had also had prostate surgery. Because he hadn't driven since his surgery I had arrived at the house in the morning ready to take them to many appointments during the previous year. My father would not return from this one. After an extended hospital stay it was determined that skilled nursing care was necessary to manage all his conditions.
I was the youngest. I was the first family member my father "forgot" as his mind marched backward in time through the next four years and he seemed to relive his life in reverse, slipped deeper into dementia, and lost speech entirely along with any control over his body.
Even though my father had no idea who I was, the person he was in that moment found that I was a nice person. I'll take that. I'd discovered with my mother under medical circumstances that the person underneath the mask of lifelong undiagnosed issues sometimes surfaces with great clarity at odd moments. I got to see for just a moment who that relaxed, broadly smiling person was in a photo I found in my brother's baby book taken just four years earlier than the still-faced photo of him holding me. Happy Father's Day, dad, either way.
Read an essay about the photo of that smiling man, Father's Day.


The city wasdivided in two urban areas after the catastrophic earthquake in 1693. Ibla, the old town was built in the same place as the former one, while the first group of buildings of "Ragusa Superiore" (upper Ragusa) rose on the hill.
Ibla looks like a beautiful, large crib when you approach it from the street above, especially in the night, while when you walk along the streets downtown it looks like an elegant party, because of all nice clubs and cultural events. Among the places to visit we mention "Duomo di San Giorgio", "Portale di San Giorgio" a magnificent arch from fifteenth century and "Chiesa di Santa Maria dell'Itria".
Downtown in "Ragusa Superiore" is interesting too, full with things to see and do, it spreads around the "Cattedrale di San Giovanni" from eighteenth century.