text:
\section{Introduction}

In this paper we take the first step towards implementing a program, laid out in \cite{ProcECM}, to find and classify four-dimensional Fano manifolds using mirror symmetry.  We compute quantum periods and quantum differential equations for many known four-dimensional Fano manifolds, using techniques described in~\cite{QC105}.  Our basic reference for the theory of Fano manifolds is the book by Iskovskikh--Prokhorov~\cite{Iskovskikh--Prokhorov}.  Recall that the \emph{index} of a Fano manifold $X$ is the largest integer~$r$ such that ${-K_X} = rH$ for some ample divisor~$H$.  A four-dimensional Fano manifold has index at most~$5$~\cite{Shokurov}.  Four-dimensional Fano manifolds with index $r>1$ have been classified.  In what follows we compute the quantum periods and quantum differential equations for all four-dimensional Fano manifolds of index $r>1$, for all four-dimensional Fano toric manifolds, and for certain other four-dimensional Fano manifolds of index~$1$.

\subsection*{Highlights}  We draw the reader's attention to:
\begin{itemize}
\item \S\ref{sec:double_cover}, where new tools for computing Gromov--Witten invariants (twisted $I$-functions for toric complete intersections~\cite{CCIT:toric_stacks_2} and an improved Quantum Lefschetz theorem~\cite{Coates}) make a big practical difference to the computation of quantum periods.  This should be contrasted with~\cite[\S19]{QC105}, where the new techniques were not available.
\item \S\ref{sec:null_correlation}, which relies on a new construction of  Szurek--Wi\'sniewski's null-correlation bundle~\cite{Szurek--Wisniewski} that may be of independent interest.
\item The tables of regularized quantum period sequences in Appendix~\ref{appendix:periods}.
\item The numerical calculation of quantum differential operators in \S\ref{sec:qdes} and Appendix~\ref{appendix:qdes}.  This suggests in particular that, for each four-dimensional Fano manifold $X$ with Fano index $r>1$, the regularized quantum differential equation of $X$ is either extremal or of low ramification.
\item \S\ref{sec:MW^4_17} and \S\ref{operator:MW^4_17}, which together give an example of a product such that the regularized quantum differential equation for each factor is extremal, but the regularized quantum differential equation for the product itself is not.
\end{itemize}
This paper is accompanied by fully-commented source code, written in the computational algebra system Magma~\cite{Magma}. This will allow the reader to verify the calculations presented here, or to perform similar computations.

\section{Methodology}
\label{sec:methodology}

The quantum period $G_X$  of a Fano manifold $X$ is a generating function
\begin{align}
  \label{eq:quantum_period}
  G_X(t) = 1 + \sum_{d=1}^\infty c_d t^d && \text{$t \in \mathbb{C}$}
\end{align}
for certain genus-zero Gromov--Witten invariants~$c_d$ of $X$.  A precise definition can be found in \cite[\S B]{QC105}, but roughly speaking $c_d$ is the `virtual number' of degree-$d$ rational curves $C$ in $X$ that pass through a given point and satisfy certain constraints on their complex structure.  (The degree of a curve $C$ here is the quantity $\langle {-K_X}, C\rangle$.)  The quantum period is discussed in detail in \cite{ProcECM,QC105}; one property that will be important in what follows is that the regularized quantum period
\begin{align}
  \label{eq:regularized_quantum_period}
  \widehat{G}_X(t) = 1 + \sum_{d=1}^\infty d! c_d t^d && \text{$t \in \mathbb{C}$, $|t| \ll \infty$}
\end{align}
satisfies a differential equation called the \emph{regularized quantum differential equation} of $X$:
\begin{align}
  \label{eq:regularized_QDE}
  L_X \widehat{G}_X \equiv 0 && L_X = \sum_{m=0}^{m=N} p_m(t) D^m
\end{align}
where the $p_m$ are polynomials and $D = t \frac{d}{dt}$.  It is expected that the regularized quantum differential equation for a Fano manifold $X$ is \emph{extremal} or \emph{of low ramification}, as described in \S\ref{sec:qdes} below.  This is a strong constraint on the Gromov--Witten invariants $c_d$ of $X$.

Quantum periods for a broad class of toric complete intersections can be computed using Givental's mirror theorem~\cite{Givental}:

\begin{theorem}[\!\!\protect{\cite[Corollary~C.2]{QC105}}]
  \label{thm:toric_mirror}
  Let $X$ be a toric Fano manifold and let $D_1,\ldots,D_N \in  H^2(X;\mathbb{Q})$ be the cohomology classes Poincar\'e-dual to the   torus-invariant divisors on $X$.  The quantum period of $X$ is:
  \[
  G_X(t) = \sum_{\substack{
      \beta \in H_2(X;\mathbb{Z}) : \\
      \text{$\langle \beta, D_i \rangle \geq 0$ $\forall i$}
    }}
  \frac{
    t^{\langle \beta,{-K_X}\rangle}
  }{
    \prod_{i=1}^N \langle \beta, D_i \rangle !
  }
  \]
\end{theorem}

\begin{theorem}[\!\!\protect{\cite[Corollary~D.5]{QC105}}]
  \label{thm:toric_ci_mirror}
  Let $Y$ be a toric Fano manifold, and let $D_1,\ldots,D_N \in H^2(Y;\mathbb{Q})$ be the cohomology classes Poincar\'e-dual to the torus-invariant divisors on $Y$.  Let $X$ be the complete intersection in $Y$ defined by a regular section of $E = L_1 \oplus \cdots \oplus L_s$ where each $L_i$ is a nef line bundle, and let $\rho_i = c_1(L_i)$, $1 \leq i \leq s$.  Suppose that the class $c_1(Y)-\Lambda$ is ample on $Y$, where $\Lambda =  c_1(L_1) + \cdots + c_1(L_s)$.  Then $X$ is Fano, and the quantum period of $X$ is:
  \[
  G_X(t) = e^{{-c} t} \sum_{\substack{
      \beta \in H_2(Y;\mathbb{Z}) : \\
      \text{$\langle \beta, D_i \rangle \geq 0$ $\forall i$}
      }}
    t^{\langle \beta,{-K_Y}- \Lambda\rangle}
    \frac{
      \prod_{j=1}^s \langle \beta, \rho_j \rangle !
    }{
      \prod_{i=1}^N \langle \beta, D_i \rangle !
    }
    \]
    where $c$ is the unique rational number such that the right-hand side has the form $1 + O(t^2)$.
\end{theorem}

An analogous mirror theorem holds for certain complete intersections in toric Deligne--Mumford stacks, but we will need only the case where the ambient stack is a weighted projective space:

\begin{theorem}[\!\!\protect{\cite[Proposition~D.9]{QC105}}]
  \label{thm:wps_ci_mirror}
  Let $Y$ be the weighted projective space $\mathbb{P}(w_0,\ldots,w_n)$, let $X$ be a smooth Fano manifold given as a complete intersection in $Y$ defined by a section of $E = \mathcal{O}(d_1) \oplus \cdots \oplus \mathcal{O}(d_m)$, and let ${-k} = w_0 + \cdots + w_n - d_1 - \cdots - d_m$.  Suppose that each $d_i$ is a positive integer, that ${-k}>0$, and that $w_i$ divides $d_j$ for all $i$, $j$ such that $0 \leq i \leq n$ and $1 \leq j \leq m$.  Then the quantum period of $X$ is:
  \[
  G_X(t) = e^{{-c} t} \sum_{d=0}^\infty
  t^{{-k} d}
  \frac{
    \prod_{j=1}^m (d d_j) !
  }{
    \prod_{i=1}^n (d w_i) !
  }
  \]
  where $c$ is the unique rational number such that the right-hand side has the form $1 + O(t^2)$.
\end{theorem}

\noindent The quantum period of a product is the product of the quantum periods:

\begin{theorem}[\!\!\protect{\cite[Corollary~E.4]{QC105}}]
  \label{thm:products}
  Let $X$ and $Y$ be smooth projective complex manifolds.  Then:
  \[
  G_{X \times Y}(t) = G_X(t)\, G_Y(t)
  \]
\end{theorem}

\noindent As we will see below, another powerful tool for computing quantum periods is the Abelian/non-Abelian Correspondence of Ciocan-Fontanine--Kim--Sabbah~\cite{Ciocan-Fontanine--Kim--Sabbah}.  We now proceed to the calculation of quantum periods.

\section{Four-Dimensional Fano Manifolds of Index~$5$}
\label{sec:index_5}

The only example here is $\mathbb{P}^4$ \cite{Kobayashi--Ochiai,Kollar,Serpico}.  This is a toric variety.  Theorem~\ref{thm:toric_mirror} yields:
\begin{align*}
  G_{\mathbb{P}^4}(t)  = \sum_{d=0}^\infty \frac{t^{5d}}{(d!)^5}
  &&
  \text{[\hyperref[table:index_5]{regularized quantum period p.~\pageref*{table:index_5}}, \hyperref[operator:P4]{operator p.~\pageref*{operator:P4}}]}
\end{align*}

\section{Four-Dimensional Fano Manifolds of Index~$4$}
\label{sec:index_4}

The only example here is the quadric $Q^4 \subset \mathbb{P}^5$~\cite{Kollar,Serpico}.  This is a complete intersection in a toric variety.  Theorem~\ref{thm:toric_ci_mirror} yields:
\begin{align*}
  G_{Q^4}(t)  = \sum_{d=0}^\infty \frac{(2d)!}{(d!)^6} t^{4d}
  &&
  \text{[\hyperref[table:index_4]{regularized quantum period p.~\pageref*{table:index_4}}, \hyperref[operator:Q4]{operator p.~\pageref*{operator:Q4}}]}
\end{align*}

\section{Four-Dimensional Fano Manifolds of Index~$3$}
\label{sec:index_3}

There are six examples \cite{Fujita:polarized_1,Fujita:polarized_2,Fujita:polarized_3,Fujita:book,Iskovskikh:Fano_1,Iskovskikh:anticanonical,Iskovskikh--Prokhorov}, which are known as del~Pezzo fourfolds:
\begin{itemize}
\item a sextic hypersurface $\FI{4}{1}$ in the weighted projective space $\mathbb{P}^5(1^4,2,3)$;
\item a quartic hypersurface $\FI{4}{2}$ in the weighted projective space $\mathbb{P}^5(1^5,2)$;
\item a cubic hypersurface $\FI{4}{3} \subset \mathbb{P}^5$;
\item a complete intersection $\FI{4}{4} \subset \mathbb{P}^6$ of type $(2H) \cap (2H)$, where $H = \mathcal{O}_{\mathbb{P}^6}(1)$;
\item a complete intersection $\FI{4}{5} \subset \Gr(2,5)$ of type $H \cap H$, where $H$ is the hyperplane bundle; and
\item $\FI{4}{6} = \mathbb{P}^2 \times \mathbb{P}^2$.
\end{itemize}
The first four examples here are complete intersections in weighted projective spaces.  Theorem~\ref{thm:wps_ci_mirror} yields:
\begin{align*}
  & G_{\FI{4}{1}}(t)  = \sum_{d=0}^\infty \frac{(6d)!}{(3d)!(2d)!(d!)^4} t^{3d} \\
  & G_{\FI{4}{2}}(t)  = \sum_{d=0}^\infty \frac{(4d)!}{(2d)!(d!)^5} t^{3d} \\
  & G_{\FI{4}{3}}(t)  = \sum_{d=0}^\infty \frac{(3d)!}{(d!)^6} t^{3d} \\
  & G_{\FI{4}{4}}(t)  = \sum_{d=0}^\infty \frac{(2d)!(2d)!}{(d!)^7} t^{3d}
\end{align*}
For $\FI{4}{5} \subset \Gr(2,5)$ we use the Abelian/non-Abelian Correspondence, applying Theorem~F.1 in \cite{QC105} with $a=2$, $b=c=d=e=0$.  This yields:
\[
G_{\FI{4}{5}}(t) = \sum_{l=0}^\infty \sum_{m=0}^\infty
(-1)^{l+m}
t^{3l+3m}
\frac
{
  (l+m)! (l+m)!
}
{
  (l!)^5 (m!)^5
}
\big(1-5 (m-l)H_m \big)
\]
where $H_m$ is the $m$th harmonic number.
For $\mathbb{P}^2 \times \mathbb{P}^2$, combining Theorem~\ref{thm:products} with \cite[Example~G.2]{QC105} yields:
\[
G_{\mathbb{P}^2 \times \mathbb{P}^2}(t) =
\sum_{l=0}^\infty \sum_{m=0}^\infty \frac{t^{3l+3m}}{(l!)^3 (m!)^3}
\]

\begin{table*}[h!]
  \centering
  \begin{tabular}{cccp{2ex}ccc} \toprule
    \multicolumn{1}{c}{$X$} &
    \multicolumn{1}{c}{$\widehat{G}_X$} &
    \multicolumn{1}{c}{$L_X$} & &
    \multicolumn{1}{c}{$X$} &
    \multicolumn{1}{c}{$\widehat{G}_X$} &
    \multicolumn{1}{c}{$L_X$} \\
    \cmidrule{1-3} \cmidrule{5-7}
    $\FI{4}{1}$ & \hyperref[table:index_3]{p.~\pageref*{table:index_3}} & \hyperref[operator:FI^4_1]{p.~\pageref*{operator:FI^4_1}} &&
    $\FI{4}{4}$ & \hyperref[table:index_3]{p.~\pageref*{table:index_3}} & \hyperref[operator:FI^4_4]{p.~\pageref*{operator:FI^4_4}} \\
    $\FI{4}{2}$ & \hyperref[table:index_3]{p.~\pageref*{table:index_3}} & \hyperref[operator:FI^4_2]{p.~\pageref*{operator:FI^4_2}} &&
    $\FI{4}{5}$ & \hyperref[table:index_3]{p.~\pageref*{table:index_3}} & \hyperref[operator:FI^4_5]{p.~\pageref*{operator:FI^4_5}} \\
    $\FI{4}{3}$ & \hyperref[table:index_3]{p.~\pageref*{table:index_3}} & \hyperref[operator:FI^4_3]{p.~\pageref*{operator:FI^4_3}} &&
    $\mathbb{P}^2 \times \mathbb{P}^2$ & \hyperref[table:index_3]{p.~\pageref*{table:index_3}} & \hyperref[operator:FI^4_6]{p.~\pageref*{operator:FI^4_6}} \\
    \bottomrule \\
  \end{tabular}
\end{table*}

\section{Four-Dimensional Fano Manifolds of Index~$2$}

Consider now a four-dimensional Fano manifold with index~$r=2$ and Picard rank~$\rho$.

\subsection{The Case $\rho=1$}
\label{sec:index_2_rank_1}

Four-dimensional Fano manifolds with index~$r=2$ and Picard rank~$\rho=1$ have been classified~\cite{Mukai:PNAS,Wilson},~\cite[Chapter~5]{Iskovskikh--Prokhorov}.  Up to deformation, there are 9~examples: the `linear unsections' of smooth three-dimensional Fano manifolds with $\rho=1$, $r=1$, and degree at most $144$.   We compute the quantum periods of these examples using the constructions in \cite[\S\S8--16]{QC105}, writing $V^4_k$ for a four-dimensional Fano manifold with $\rho=1$, $r=2$, and degree~$16k$.

\subsubsection{$V^4_2$} \hfill [\hyperref[table:index_2]{regularized quantum period p.~\pageref*{table:index_2}}, \hyperref[operator:V^4_2]{operator p.~\pageref*{operator:V^4_2}}] \label{sec:V^4_2}

This is a sextic hypersurface in $\mathbb{P}^5(1^5,3)$.  Proposition~D.9 in \cite{QC105} yields:
\[
G_{V^4_2}(t) = \sum_{d=0}^\infty \frac{(6d)!}{(d!)^5(3d)!} t^{2d}
\]

\subsubsection{$V^4_4$} \hfill [\hyperref[table:index_2]{regularized quantum period p.~\pageref*{table:index_2}}, \hyperref[operator:V^4_4]{operator p.~\pageref*{operator:V^4_4}}] \label{sec:V^4_4}

This is a quartic hypersurface in $\mathbb{P}^5$.  Theorem~\ref{thm:toric_ci_mirror} yields:
\[
G_{V^4_4}(t) = \sum_{d=0}^\infty \frac{(4d)!}{(d!)^6} t^{2d}
\]

\subsubsection{$V^4_6$} \hfill [\hyperref[table:index_2]{regularized quantum period p.~\pageref*{table:index_2}}, \hyperref[operator:V^4_6]{operator p.~\pageref*{operator:V^4_6}}] \label{sec:V^4_6}

This is a complete intersection of type $(2H) \cap (3H)$ in $\mathbb{P}^6$, where $H = \mathcal{O}_{\mathbb{P}^6}(1)$.  Theorem~\ref{thm:toric_ci_mirror} yields:
\[
G_{V^4_6}(t) = \sum_{d=0}^\infty \frac{(2d)!(3d)!}{(d!)^7} t^{2d}
\]

\subsubsection{$V^4_8$} \hfill [\hyperref[table:index_2]{regularized quantum period p.~\pageref*{table:index_2}}, \hyperref[operator:V^4_8]{operator p.~\pageref*{operator:V^4_8}}] \label{sec:V^4_8}

This is a complete intersection of type $(2H) \cap (2H) \cap (2H)$ in $\mathbb{P}^7$, where $H = \mathcal{O}_{\mathbb{P}^7}(1)$.  Theorem~\ref{thm:toric_ci_mirror} yields:
\[
G_{V^4_8}(t) = \sum_{d=0}^\infty \frac{\big((2d)!\big)^3}{(d!)^8} t^{2d}
\]

\subsubsection{$V^4_{10}$} \hfill [\hyperref[table:index_2]{regularized quantum period p.~\pageref*{table:index_2}}, \hyperref[operator:V^4_10]{operator p.~\pageref*{operator:V^4_10}}] \label{sec:V^4_10}

This is a complete intersection in $\Gr(2,5)$, cut out by a regular section of $\mathcal{O}(1) \oplus \mathcal{O}(2)$ where $\mathcal{O}(1)$ is the pullback of $\mathcal{O}(1)$ on projective space under the Pl\"ucker embedding.  We apply Theorem~F.1 in \cite{QC105} with $a=b=1$ and $c=d=e=0$.  This yields:
\[
G_{V^4_{10}}(t) = \sum_{l=0}^\infty \sum_{m=0}^\infty
(-1)^{l+m}
t^{2l+2m}
\frac
{
  (l+m)! (2l+2m)!
}
{
  (l!)^5 (m!)^5
}
\big(1-5 (m-l)H_m \big)
\]
where $H_m$ is the $m$th harmonic number.

\subsubsection{$V^4_{12}$} \hfill [\hyperref[table:index_2]{regularized quantum period p.~\pageref*{table:index_2}}, \hyperref[operator:V^4_12]{operator p.~\pageref*{operator:V^4_12}}] \label{sec:V^4_12}

This is the subvariety of $\Gr(2,5)$ cut out by a regular section of $S^\star \otimes \det S^\star$, where $S$ is the universal bundle of subspaces on $\Gr(2,5)$.  We apply Theorem~F.1 in \cite{QC105} with $c = 1$ and $a  = b = d = e = 0$. This yields:
\[
G_{V^4_{12}}(t) = \sum_{l=0}^\infty \sum_{m=0}^\infty
({-1})^{l+m} t^{2l+2m}
\frac
{
  (2l+m)! (l+2m)!
}
{
  (l!)^5 (m!)^5
}
\big(1+(m-l)(H_{2l+m} + 2 H_{l + 2m}-5H_m) \big)
\]

\subsubsection{$V^4_{14}$} \hfill [\hyperref[table:index_2]{regularized quantum period p.~\pageref*{table:index_2}}, \hyperref[operator:V^4_14]{operator p.~\pageref*{operator:V^4_14}}] \label{sec:V^4_14}

This is a complete intersection in $\Gr(2,6)$, cut out by a regular section of $\mathcal{O}(1)^{\oplus 4}$ where $\mathcal{O}(1)$ is the pullback of $\mathcal{O}(1)$ on projective space under the Pl\"ucker embedding.  We apply Theorem~F.1 in \cite{QC105} with $a=4$ and $b=c=d=e=0$.  This yields:
\[
G_{V^4_{14}}(t) = \sum_{l=0}^\infty \sum_{m=0}^\infty
(-1)^{l+m}
t^{2l+2m}
\frac
{
  \big((l+m)!\big)^4
}
{
  (l!)^6 (m!)^6
}
\big(1-6 (m-l)H_m \big)
\]

\subsubsection{$V^4_{16}$} \hfill [\hyperref[table:index_2]{regularized quantum period p.~\pageref*{table:index_2}}, \hyperref[operator:V^4_16]{operator p.~\pageref*{operator:V^4_16}}] \label{sec:V^4_16}

This is the subvariety of $\Gr(3,6)$ cut out by a regular section of $\wedge^2 S^\star \oplus (\det S^\star)^{\oplus 2}$, where $S$ is the universal bundle of subspaces on $\Gr(3,6)$.  We apply
Theorem~F.1 in \cite{QC105} with $a=2$, $b=c=d=0$, and $e=1$.   This shows that the quantum period $G_{V^4_{16}}(t)$ is the coefficient of $(p_2-p_1)(p_3-p_1)(p_3-p_2)$ in the expression:
\begin{multline*}
  \sum_{l_1 = 0}^\infty
  \sum_{l_2 = 0}^\infty
  \sum_{l_3 = 0}^\infty
  t^{2l_1 + 2 l_2 + 2 l_3}
  {
    \prod_{k=1}^{l_1+l_2+l_3} (p_1+p_2+p_3 + k)^2
    \over
    \prod_{j=1}^{j=3}
    \prod_{k=1}^{k=l_j} (p_j + k)^6
  }
  \prod_{1 \leq i < j \leq 3} \prod_{k=1}^{l_i + l_j} (p_i + p_j + k) \\
  \times  \prod_{1 \leq i < j \leq 3} \big(p_j-p_i + (l_j-l_i) \big)
\end{multline*}
(Since this expression is totally antisymmetric in $p_1$,~$p_2$,~$p_3$, it is divisible by $(p_2-p_1)(p_3-p_1)(p_3-p_2)$.)

\subsubsection{$V^4_{18}$} \hfill [\hyperref[table:index_2]{regularized quantum period p.~\pageref*{table:index_2}}, \hyperref[operator:V^4_18]{operator p.~\pageref*{operator:V^4_18}}] \label{sec:V^4_18}

This is the subvariety of $\Gr(5,7)$ cut out by a regular section of $\big( S \otimes \det S^\star\big) \oplus \det S^{\star}$, where $S$ is the universal bundle of subspaces on $\Gr(5,7)$.  We apply Theorem~F.1 in \cite{QC105} with $a=d=1$ and $b=c=e=0$.  This shows that the quantum period $G_{V^4_{18}}(t)$ is the coefficient of $\prod_{1 \leq i < j \leq 5} (p_j-p_i)$ in the expression:
\begin{multline*}
  \sum_{l_1 = 0}^\infty
  \sum_{l_2 = 0}^\infty
  \sum_{l_3 = 0}^\infty
  \sum_{l_4 = 0}^\infty
  \sum_{l_5 = 0}^\infty
  t^{2|l|}
  {
    \prod_{k=1}^{k=|l|} (p_1+p_2+\cdots + p_5 + k)
    \over
    \prod_{j=1}^{j=5}
    \prod_{k=1}^{k=l_j} (p_j + k)^7
  }
  \prod_{j=1}^{j=5}  \prod_{k=1}^{|l| - l_j}  (p_1 + p_2 + \cdots + p_5 - p_j + k) \\
  \times
  \prod_{1 \leq i < j \leq 5} \big(p_j-p_i + (l_j-l_i) \big)
\end{multline*}
where $|l| = l_1 + l_2 + \cdots + l_5$.  (As above, antisymmetry implies that the long formula here is divisible by $\prod_{1 \leq i < j \leq 5} (p_j-p_i)$.)

\subsection{The Case $\rho>1$}
\label{sec:index_2_higher_rank}

Four-dimensional Fano manifolds with $\rho>1$ and $r=2$ have been classified by Mukai~\cite{Mukai:PNAS,Mukai:10} and Wi\'sniewski~\cite{Wisniewski:classification}.  There are 18 deformation families, as follows.  We denote the $k$th such deformation family, as given in \cite[Table~12.7]{Iskovskikh--Prokhorov}, by $\MW{4}{k}$.

\subsubsection{$\MW{4}{1}$} \hfill [\hyperref[table:index_2]{regularized quantum period p.~\pageref*{table:index_2}}, \hyperref[operator:MW^4_1]{operator p.~\pageref*{operator:MW^4_1}}] \label{sec:MW^4_1}

This is the product $\mathbb{P}^1 \times B^3_1$.  Combining Theorem~\ref{thm:products} with \cite[Example~G.1]{QC105} and \cite[\S3]{QC105} yields:
\[
G_{\MW{4}{1}}(t) =
\sum_{l=0}^\infty \sum_{m=0}^\infty \frac{(6m)!}{(l!)^2 (m!)^3 (2m)! (3m)!} t^{2l+2m}
\]

\subsubsection{$\MW{4}{2}$} \hfill [\hyperref[table:index_2]{regularized quantum period p.~\pageref*{table:index_2}}, \hyperref[operator:MW^4_2]{operator p.~\pageref*{operator:MW^4_2}}] \label{sec:MW^4_2}

This is the product $\mathbb{P}^1 \times B^3_2$.  Combining Theorem~\ref{thm:products} with \cite[Example~G.1]{QC105} and \cite[\S4]{QC105} yields:
\[
G_{\MW{4}{2}}(t) =
\sum_{l=0}^\infty \sum_{m=0}^\infty \frac{(4m)!}{(l!)^2 (m!)^4 (2m)!} t^{2l+2m}
\]

\subsubsection{$\MW{4}{3}$} \hfill [\hyperref[table:index_2]{regularized quantum period p.~\pageref*{table:index_2}}, \hyperref[operator:MW^4_3]{operator p.~\pageref*{operator:MW^4_3}}] \label{sec:MW^4_3}

This is the product $\mathbb{P}^1 \times B^3_3$.  Combining Theorem~\ref{thm:products} with \cite[Example~G.1]{QC105} and \cite[\S5]{QC105} yields:
\[
G_{\MW{4}{3}}(t) =
\sum_{l=0}^\infty \sum_{m=0}^\infty \frac{(3m)!}{(l!)^2 (m!)^5} t^{2l+2m}
\]

\subsubsection{$\MW{4}{4}$} \hfill [\hyperref[table:index_2]{regularized quantum period p.~\pageref*{table:index_2}}, \hyperref[operator:MW^4_4]{operator p.~\pageref*{operator:MW^4_4}}] \label{sec:MW^4_4}
\label{sec:double_cover}

This is a double cover of $\mathbb{P}^2 \times \mathbb{P}^2$, branched over a divisor of bidegree $(2,2)$.  Consider  the toric
variety $F$ with weight data:
\[
\begin{array}{rrrrrrrl}
  \multicolumn{1}{c}{x_0} &
  \multicolumn{1}{c}{x_1} &
  \multicolumn{1}{c}{x_2} &
  \multicolumn{1}{c}{y_0} &
  \multicolumn{1}{c}{y_1} &
  \multicolumn{1}{c}{y_2} &
  \multicolumn{1}{c}{w} & \\
  \cmidrule{1-7}
  1 & 1 & 1 & 0 & 0 & 0 & 1  & \hspace{1.5ex} L\\
  0 & 0 & 0 & 1 & 1 & 1 & 1 & \hspace{1.5ex} M \\
\end{array}
\]
and $\Amp F = \langle L, L+M \rangle$.  Let $X$ be a member of the linear system $|2L+2M|$ defined by the equation $w^2=f_{2,2}$, where $f_{2,2}$ is a bihomogeneous polynomial of degrees $2$ in $x_0$,~$x_1$,~$x_2$ and $2$ in $y_0$,~$y_1$,~$y_2$.  Let $p \colon F \dashrightarrow \mathbb{P}^2 \times \mathbb{P}^2$ be the rational map which sends (contravariantly) the homogeneous co-ordinate functions $[x_0,x_1,x_2,y_0,y_1,y_2] $ on $\mathbb{P}^2_{x_0,x_1,x_2} \times \mathbb{P}^2_{y_0,y_1,y_2}$ to $[x_0,x_1,x_2,y_0,y_1,y_2]$.  The restriction of $p$ to $X$ is a morphism, which exhibits $X$ as a double cover of $\mathbb{P}^2 \times \mathbb{P}^2$ branched over the locus $(f_{2,2}=0) \subset \mathbb{P}^2_{x_0,x_1,x_2} \times \mathbb{P}^2_{y_0,y_1,y_2}$.  Thus $X = \MW{4}{4}$.

Recall the definition of the $J$-function $J_X(t,z)$ from \cite[equation~11]{Coates--Givental}.  Recall from \cite{Coates} that there is a Lagrangian cone $\mathcal{L}_X \subset H^\bullet(X;\Lambda_X)\otimes \mathbb{C}(\!(z^{-1})\!)$ that encodes all genus-zero Gromov--Witten invariants of $X$, and a Lagrangian cone $\mathcal{L}_{\boldsymbol{e}} \subset H^\bullet(F;\Lambda_F)\otimes \mathbb{C}(\!(z^{-1})\!) \otimes \mathbb{C}(\lambda)$ that encodes all genus-zero $({\boldsymbol{e}},2L+2M)$-twisted Gromov--Witten invariants of $F$.  Here $\Lambda_X$ and $\Lambda_F$ are certain Novikov rings and ${\boldsymbol{e}}$ is the total Chern class with parameter $\lambda$ (or, equivalently, ${\boldsymbol{e}}$ is the $S^1$-equivariant Euler class with respect to an action of $S^1$ described in \cite{Coates}; in this case one should regard $\lambda$ as the standard generator for the $S^1$-equivariant cohomology algebra of a point). The $J$-function $J_X$ is characterised by the fact that $J_X(t,{-z})$ is the unique point on $\mathcal{L}_X$ of the form ${-z} + t + O(z^{-1})$.

Let $p_1$,~$p_2 \in H^2(F;\mathbb{Q})$ denote the first Chern class of $L$,~$L+M$ respectively and let $P_1$,~$P_2 \in H^2(X;\mathbb{Q})$ denote the pullbacks of $p_1$,~$p_2$ along the inclusion map $i \colon X \to F$.  Let $Q_1$,~$Q_2$ denote the elements of the Novikov ring $\Lambda_X$ that are dual respectively to $P_1$,~$P_2$, and note that $\Lambda_X$ and $\Lambda_F$ are canonically isomorphic (via $i_\star$). Theorem~21 in \cite{CCIT:toric_stacks_2} implies that:
\[
I(t_1,t_2,\lambda,z) = z e^{t_1 p_1/z} e^{t_2 p_2/z}
\sum_{l=0}^\infty \sum_{m=0}^\infty
\frac{
  Q_1^l Q_2^m e^{l t_1} e^{m t_2}
  \prod_{k=1}^{k=2m} (\lambda + 2p_2 + k z)
}{
  \prod_{k=1}^{k=l} (p_1 + k z)^3
  \prod_{k=1}^{k=m} (p_2 + k z)
}
\frac{
  \prod_{k={-\infty}}^{k=0} (p_2-p_1+kz)^3
}{
  \prod_{k={-\infty}}^{k=m-l} (p_2-p_1+kz)^3
}
\]
satisfies $I(t_1,t_2,\lambda,{-z}) \in \mathcal{L}_{\boldsymbol{e}}$.  Theorem~1.1 in \cite{Coates} gives that $i^\star \mathcal{L}_{\boldsymbol{e}} \big|_{\lambda = 0} \subset \mathcal{L}_X$, and therefore that:
\[
i^\star I(t_1,t_2,0,{-z}) \in \mathcal{L}_X
\]
Since the hypersurface $X$ misses the locus $y_1 = y_2 = y_3 = 0$ in $F$, we have that $i^\star (p_2-p_1)^3 = 0$.  Thus:
\[
i^\star I(t_1,t_2,0,z) =
z e^{t_1 P_1/z} e^{t_2 P_2/z}
\sum_{l=0}^\infty \sum_{m=l}^\infty
\frac{
  Q_1^l Q_2^m e^{l t_1} e^{m t_2}
  \prod_{k=1}^{k=2m} (2P_2 + k z)
}{
  \prod_{k=1}^{k=l} (P_1 + k z)^3
  \prod_{k=1}^{k=m} (P_2 + k z)
}
\frac{
  1
}{
  \prod_{k=1}^{k=m-l} (P_2-P_1+kz)^3
}
\]
In particular, $i^\star I(t_1,t_2,0,{-z})$ has the form ${-z} + t_1 P_1 + t_2 P_2 + O(z^{-1})$ and, from the characterisation of $J_X$ discussed above, we conclude that $J_X(t_1P_1+t_2P_2,{-z}) = i^\star I(t_1,t_2,0,{-z})$.

To extract the quantum period $G_X$ from the $J$-function $J_X(t_1P_1+t_2P_2,z)$ we take the component along the unit class $1 \in H^\bullet(X;\mathbb{Q})$, set $z=1$, set $t_1 = t_2 = 0$, and set $Q_1 =1$,~$ Q_2 =t^2$, obtaining:
\[
G_{\MW{4}{4}}(t) = \sum_{l=0}^\infty \sum_{m=l}^\infty \frac{(2m)!}{(l!)^3 m! ((m-l)!)^3} t^{2m}
\]
\subsubsection{$\MW{4}{5}$} \hfill [\hyperref[table:index_2]{regularized quantum period p.~\pageref*{table:index_2}}, \hyperref[operator:MW^4_5]{operator p.~\pageref*{operator:MW^4_5}}] \label{sec:MW^4_5}

This is a divisor on $\mathbb{P}^2 \times \mathbb{P}^3$ of bidegree $(1,2)$.  Theorem~\ref{thm:toric_ci_mirror} yields:
\[
G_{\MW{4}{5}}(t) = \sum_{l=0}^\infty \sum_{m=0}^\infty \frac{(l+2m)!}{(l!)^3 (m!)^4} t^{2l+2m}
\]

\subsubsection{$\MW{4}{6}$} \hfill [\hyperref[table:index_2]{regularized quantum period p.~\pageref*{table:index_2}}, \hyperref[operator:MW^4_6]{operator p.~\pageref*{operator:MW^4_6}}] \label{sec:MW^4_6}

This is the product $\mathbb{P}^1 \times B^3_4$.  Combining Theorem~\ref{thm:products} with \cite[Example~G.1]{QC105} and \cite[\S6]{QC105} yields:
\[
G_{\MW{4}{6}}(t) =
\sum_{l=0}^\infty \sum_{m=0}^\infty \frac{(2m)!(2m)!}{(l!)^2 (m!)^6} t^{2l+2m}
\]

\subsubsection{$\MW{4}{7}$} \hfill [\hyperref[table:index_2]{regularized quantum period p.~\pageref*{table:index_2}}, \hyperref[operator:MW^4_7]{operator p.~\pageref*{operator:MW^4_7}}] \label{sec:MW^4_7}

This is a complete intersection of two divisors in $\mathbb{P}^3 \times \mathbb{P}^3$, each of bidegree $(1,1)$.  Theorem~\ref{thm:toric_ci_mirror} yields:
\[
G_{\MW{4}{7}}(t) = \sum_{l=0}^\infty \sum_{m=0}^\infty \frac{(l+m)!(l+m)!}{(l!)^4 (m!)^4} t^{2l+2m}
\]

\subsubsection{$\MW{4}{8}$} \hfill [\hyperref[table:index_2]{regularized quantum period p.~\pageref*{table:index_2}}, \hyperref[operator:MW^4_8]{operator p.~\pageref*{operator:MW^4_8}}] \label{sec:MW^4_8}

This is a divisor on $\mathbb{P}^2 \times Q^3$ of bidegree $(1,1)$.  Theorem~\ref{thm:toric_ci_mirror} yields:
\[
G_{\MW{4}{8}}(t) = \sum_{l=0}^\infty \sum_{m=0}^\infty \frac{(l+m)!(2m)!}{(l!)^3 (m!)^5} t^{2l+2m}
\]

\subsubsection{$\MW{4}{9}$} \hfill [\hyperref[table:index_2]{regularized quantum period p.~\pageref*{table:index_2}}, \hyperref[operator:MW^4_9]{operator p.~\pageref*{operator:MW^4_9}}] \label{sec:MW^4_9}

This is the product $\mathbb{P}^1 \times B^3_5$.  Combining Theorem~\ref{thm:products} with \cite[Example~G.1]{QC105} and \cite[\S7]{QC105} yields:
\[
G_{\MW{4}{9}}(t) = \sum_{l=0}^\infty \sum_{m=0}^\infty \sum_{n=0}^\infty
(-1)^{m+n}
t^{2l+2m+2n}
\frac
{
  \big((m+n)!\big)^3
}
{
  (l!)^2 (m!)^5 (n!)^5
}
\big(1-5 (n-m)H_n \big)
\]

\subsubsection{$\MW{4}{10}$} \hfill [\hyperref[table:index_2]{regularized quantum period p.~\pageref*{table:index_2}}, \hyperref[operator:MW^4_10]{operator p.~\pageref*{operator:MW^4_10}}] \label{sec:MW^4_10}

This is the blow-up of the quadric $Q^4$ along a conic that is not contained in a plane lying in $Q^4$.  Consider the toric variety $F$ with weight data:
\[
\begin{array}{rrrrrrrl}
  \multicolumn{1}{c}{s_0} &
  \multicolumn{1}{c}{s_1} &
  \multicolumn{1}{c}{s_2} &
  \multicolumn{1}{c}{x} &
  \multicolumn{1}{c}{x_3} &
  \multicolumn{1}{c}{x_4} &
  \multicolumn{1}{c}{x_5} & \\
  \cmidrule{1-7}
  1 & 1 & 1 & -1 & 0 & 0 & 0 & \hspace{1.5ex} L\\
  0 & 0 & 0 & 1 & 1  & 1 & 1 & \hspace{1.5ex} M \\
\end{array}
\]
and $\Amp F = \langle L, M\rangle$.  The morphism $F \to \mathbb{P}^5$ that sends (contravariantly) the homogeneous co-ordinate functions $[x_0,x_1,\dots,x_5]$ to $[xs_0, xs_1, x s_2, x_3, x_4, x_5]$ blows up the plane $\Pi = (x_0=x_1=x_2=0)$ in $\mathbb{P}^5$.  Thus a general member of $|2M|$ on $F$ is the blow-up of $Q^4$ with centre a conic on $\Pi$.  In other words, a general member of $|2M|$ on $F$ is $\MW{4}{10}$.  We have:
\begin{itemize}
\item $-K_F=2L+4M$ is ample, so that $F$ is a Fano variety;
\item $\MW{4}{10}\sim 2M$ is ample;
\item $-(K_F+2M)\sim 2L+2M$ is ample.
\end{itemize}
Theorem~\ref{thm:toric_ci_mirror} yields:
\[
G_{\MW{4}{10}}(t) = \sum_{l=0}^\infty \sum_{m=l}^\infty \frac{(2m)!}{(l!)^3(m-l)!(m!)^3} t^{2l+2m}
\]

\subsubsection{$\MW{4}{11}$} \hfill [\hyperref[table:index_2]{regularized quantum period p.~\pageref*{table:index_2}}, \hyperref[operator:MW^4_11]{operator p.~\pageref*{operator:MW^4_11}}] \label{sec:MW^4_11}
\label{sec:null_correlation}

This is the projective bundle $\mathbb{P}_{\mathbb{P}^3}(\mathcal{E}^\vee)$, where $\mathcal{E} \to \mathbb{P}^3$ is the null-correlation bundle of Szurek--Wi\'sniewski~\cite{Szurek--Wisniewski}.

\begin{remark}
  \label{rem:slice}
  For us $\mathbb{P}(E)$ denotes the projective bundle of lines in $E$, whereas in Szurek--Wi\'sniewski and Iskovskikh--Prokhorov, $\mathbb{P}(E)$ denotes the projective bundle of one-dimensional quotients.  With our conventions, if $\pi \colon \mathbb{P}(E) \to X$ is a projective bundle then $E^\star = \pi_\star \mathcal{O}_{\mathbb{P}(E)}(1)$, and so a regular section $s \in \Gamma\big(\mathbb{P}(E),\mathcal{O}_{\mathbb{P}(E)}(1)\big)$ vanishes on $\mathbb{P}(F^\star) \subset \mathbb{P}(E)$, where the vector bundle $F \to X$ is the cokernel of $s \colon \mathcal{O}_{\mathbb{P}(E)} \to E^\star$.
\end{remark}

\begin{proposition}
  \label{pro:better_model}
  Let $V = \mathbb{C}^4$, so that $\mathbb{P}(V) = \mathbb{P}^3$.  Consider the partial flag manifold $\Fl_{1,2}(V)$ and the natural projections
  \begin{equation}
    \label{eq:flag_diagram}
    \begin{aligned}
      \xymatrix{ & \Fl_{1,2}(V) \ar[dl]_{p_1} \ar[dr]^{p_2} \\ \mathbb{P}(V) && \Gr(2,V) }
    \end{aligned}
  \end{equation}
  Let $|L|$ denote the linear system defined by $\mathcal{O}(1)$ for the projective bundle $p_1$.  Then a general element of $|L|$ is $\mathbb{P}(\mathcal{E}^\vee)$, where $\mathcal{E} \to \mathbb{P}(V)$ is the null-correlation bundle.
\end{proposition}

\begin{proof}
  The null-correlation bundle has rank~$2$, and so the perfect pairing $\mathcal{E} \otimes \mathcal{E} \to \det \mathcal{E}$ gives canonical isomorphisms $\mathcal{E}^\vee \cong \mathcal{E} \otimes (\det \mathcal{E})^{-1}$ and $\mathbb{P}(\mathcal{E}^\vee) \cong \mathbb{P}(\mathcal{E})$.
  There is an exact sequence:
  \[
  \xymatrix{0 \ar[r] & \mathcal{E}(-1) \ar[r] & T_{\mathbb{P}(V)}(-2) \ar[r] & \mathcal{O}_{\mathbb{P}(V)} \ar[r] & 0}
  \]
  and the map $s^\star \colon T_{\mathbb{P}(V)}(-2) \to \mathcal{O}_{\mathbb{P}(V)}$ therein defines a section $s \in \Gamma\big(\mathbb{P}(T_{\mathbb{P}(V)}(-2)),\mathcal{O}_{\mathbb{P}(T_{\mathbb{P}(V)}(-2))}(1)\big)$.  The construction in Remark~\ref{rem:slice} now exhibits $\mathbb{P}\big(\mathcal{E}(-1)\big) \cong \mathbb{P}(\mathcal{E})$ as the locus $(s=0)$ in $\mathbb{P}(T_{\mathbb{P}(V)}(-2))$.  We will identify $\mathbb{P}(T_{\mathbb{P}(V)}(-2))$ with the partial flag manifold $\Fl_{1,2}(V)$.

  For a vector bundle $\mathcal{F} \to X$ of rank~$3$, the perfect pairing $\mathcal{F} \otimes \wedge^2 \mathcal{F} \to \det \mathcal{F}$ gives a canonical isomorphism $\mathcal{F}^\star \cong (\wedge^2 \mathcal{F}) \otimes (\det \mathcal{F})^{-1}$.  Applying this with $\mathcal{F} \to X$ equal to $\Omega_{\mathbb{P}(V)}(2) \to \mathbb{P}(V)$ gives:
  \[
  T_{\mathbb{P}(V)}(-2) \cong \Omega_{\mathbb{P}(V)}^2(2)
  \]
  where $\Omega_{\mathbb{P}(V)}^2 := \wedge^2 \Omega_{\mathbb{P}(V)}$.  We thus need to identify $\mathbb{P}\big(\Omega^2_{\mathbb{P}(V)}(2)\big)$ with $\Fl_{1,2}(V)$.

  The Pl\"ucker embedding $\Gr(2,V) \to \mathbb{P}\big(\wedge^2 V\big)$ maps a subspace $W \in \Gr(2,V)$ to the antisymmetric linear map $L_W \colon V^\star \to V$, well-defined up to scale, given by:
  \[
  L_W(f) = f(w_1) w_2 - f(w_2) w_1
  \]
  where $\{w_1,w_2\}$ is a basis for $W$.  The kernel of $L_W$ is the annihilator $W^\perp \subset V^\star$.  If $f \not \in W^\perp$ then $\langle L_W(f) \rangle = \ker f \cap W$; this implies in particular that $\rk L_W = 2$.  Thus the image of the Pl\"ucker embedding consists of (the lines spanned by) antisymmetric linear maps $L_W \colon V^\star \to V$ of rank~$2$, and one can recover $W \in \Gr(2,V)$ from its image $\langle L_W \rangle$ by taking the annihilator of the kernel:
  \[
  W = \big(\ker L_W \big)^\perp
  \]
  There is a canonical isomorphism $\Ann \colon \Gr(2,V) \to \Gr(2,V^\star)$ which maps $W \in \Gr(2,V)$ to $W^\perp$.

  Recall that our goal is to identify $\mathbb{P}\big(\Omega^2_{\mathbb{P}(V)}(2)\big)$ with $\Fl_{1,2}(V)$.  Let $q_1 \colon \mathbb{P}\big(\Omega^2_{\mathbb{P}(V)}(2)\big) \to \mathbb{P}(V)$ denote the projection.  The Euler sequence:
  \[
  \xymatrix{ 0 \ar[r] & \Omega_{\mathbb{P}(V)} \ar[r] & \pi^\star V^\star (-1) \ar[r] & \mathcal{O}_{\mathbb{P}(V)} \ar[r] & 0}
  \]
  gives, via \cite[II, Exercise 5.16]{Hartshorne}:
  \[
  \xymatrix{ 0 \ar[r] & \Omega^2_{\mathbb{P}(V)} \ar[r] & \pi^\star \big(\wedge^2 V^\star \big) (-2) \ar[r] & \Omega_{\mathbb{P}(V)} \ar[r] & 0}
  \]
  and thus:
  \begin{equation}
    \label{eq:Omega_2_sequence}
    \xymatrix{ 0 \ar[r] & \Omega^2_{\mathbb{P}(V)}(2) \ar[r] & \pi^\star \big(\wedge^2 V^\star\big) \ar[r] & \pi^\star V^\star (1) \ar[r] & \mathcal{O}_{\mathbb{P}(V)}(2) \ar[r] & 0}
  \end{equation}
  This defines a map $f \colon \mathbb{P}\big(\Omega^2_{\mathbb{P}(V)}(2)\big) \to \mathbb{P}\big(\wedge^2 V^\star\big)$.  Consider the fiber of the sequence \eqref{eq:Omega_2_sequence} over $[v] \in \mathbb{P}(V)$.  The map $\pi^\star \big(\wedge^2 V^\star \big) \to \pi^\star V^\star (1)$ here is given by contraction with $v$, and so non-zero elements of the kernel are antisymmetric linear maps $V \to V^\star$ of rank~$2$.  (They are antisymmetric, hence have rank $0$,~$2$, or~$4$; they are non-zero, hence are not of rank~$0$; and they have the non-zero element $v$ in their kernel, hence are not of rank~$4$.)\phantom{.}  In particular, we see that the image of $f$ lies in $\Gr(2,V^\star)  \subset \mathbb{P}\big(\wedge^2 V^\star\big)$.  Given $[x] \in \mathbb{P}\big(\Omega^2_{\mathbb{P}(V)}(2)\big)$, write $W_{[x]} \subset V^\star$ for the linear subspace defined by $f([x])$.  Suppose that $[x] \in \mathbb{P}\big(\Omega^2_{\mathbb{P}(V)}(2)\big)$ lies over $[v] \in \mathbb{P}(V)$.  Then, applying the discussion in the previous paragraph but with $V$ there replaced by $V^\star$, we see that $v \in W_{[x]}^\perp$.  Thus, writing $q_2 \colon \mathbb{P}\big(\Omega^2_{\mathbb{P}(V)}(2)\big) \to \Gr(2,V)$ for the composition
  \[
  \xymatrix{ \mathbb{P}\big(\Omega^2_{\mathbb{P}(V)}(2)\big) \ar[r]^f & \Gr(2,V^\star) \ar[r]^\Ann & \Gr(2,V) }
  \]
  we have that $q_1([x]) \subset q_2([x])$, i.e., that the diagram:
  \[
  \xymatrix{
    & \mathbb{P}\big(\Omega^2_{\mathbb{P}(V)}(2)\big) \ar[dl]_{q_1} \ar[rd]^{q_2} \\
    \mathbb{P}(V) && \Gr(2,V)}
  \]
  coincides with the diagram~\eqref{eq:flag_diagram}.  This identifies $\mathbb{P}\big(\Omega^2_{\mathbb{P}(V)}(2)\big)$ with the partial flag manifold $\Fl_{1,2}(V)$, and exhibits $\mathbb{P}(\mathcal{E}^\vee)$ as an element of the linear system $|L|$ as claimed.
\end{proof}

\subsection*{Abelianization:}

To compute the quantum period, we use the Abelian/non-Abelian Correspondence of Ciocan-Fontanine--Kim--Sabbah, as in \cite[\S39]{QC105}.  Consider the situation as in \S3.1 of \cite{Ciocan-Fontanine--Kim--Sabbah} with:
\begin{itemize}
\item $X = \mathbb{C}^{10}$, regarded as the space of pairs:
  \[
  \{(v,w) : \text{$v \in \mathbb{C}^2$ is a row vector, $w$ is a $2 \times 4$ complex matrix}\}
  \]
\item $G = \CC^\times \times \GL_2(\mathbb{C})$, acting on $X$ as:
  \[
  (\lambda, g) \colon (v, w) \mapsto (\lambda v g^{-1}, gw)
  \]
\item $T = (\CC^\times)^3$, the diagonal subtorus in $G$;
\item the group that is denoted by $S$ in \cite{Ciocan-Fontanine--Kim--Sabbah} set equal to the trivial group;
\item $\mathcal{V}$ equal to the representation of $G$ given by the determinant of the standard  representation of the second factor $\GL_2(\mathbb{C})$.
\end{itemize}
Then $X /\!\!/ G$ is the partial flag manifold $\Fl=\Fl_{1,2}(\mathbb{C}^4)$, whereas $X /\!\!/ T$ is the toric variety with weight data:
\[
\begin{array}{rrrrrrrrrrl}
1 & 1 & 1  & 1 & 0 & 0 & 0 & 0 & -1 & 0  & \hspace{1.5ex}  L_1\\
0 & 0 & 0  & 0 & 1 & 1 & 1 & 1 & 0   & -1& \hspace{1.5ex} L_2 \\
0 & 0 & 0  & 0 & 0 & 0 & 0 & 0 & 1   & 1  & \hspace{1.5ex} H
\end{array}
\]
and $\Amp =\langle L_1, L_2 , H \rangle$; that is, $X /\!\!/ T$ is the projective bundle $\mathbb{P}(\mathcal{O}(-1,0)\oplus \mathcal{O}(0,-1))$ over $\mathbb{P}^3\times \mathbb{P}^3$. The non-trivial element of the Weyl group $W=\mathbb{Z}/2\mathbb{Z}$ exchanges the two factors of $\mathbb{P}^3\times \mathbb{P}^3$. The representation $\mathcal{V}$ induces the line bundle $\mathcal{V}_G = L$ over $X /\!\!/ G = \Fl$, where $L$ was defined in the statement of Proposition~\ref{pro:better_model}, whereas the representation $\mathcal{V}$ induces the line bundle $\mathcal{V}_T = L_1+L_2$ over $X /\!\!/ T$.

\subsection*{The Abelian/non-Abelian Correspondence:}

Let $p_1$, $p_2$, and $p_3 \in H^2(X/\!\!/ T;\mathbb{Q})$ denote the first Chern classes of the line bundles $L_1$, $L_2$, and $H$ respectively. We fix a lift of $H^\bullet(X /\!\!/ G;\mathbb{Q})$ to $H^\bullet(X /\!\!/ T,\mathbb{Q})^W$ in the sense of \cite[\S3]{Ciocan-Fontanine--Kim--Sabbah}; there are many possible choices for such a lift, and the precise choice made will be unimportant in what follows.  The lift allows us to regard $H^\bullet(X /\!\!/ G;\mathbb{Q})$ as a subspace of $H^\bullet(X /\!\!/ T,\mathbb{Q})^W$, which maps isomorphically to the Weyl-anti-invariant part $H^\bullet(X /\!\!/ T,\mathbb{Q})^a$ of $H^\bullet(X /\!\!/ T,\mathbb{Q})$ via:
\[
\xymatrix{
  H^\bullet(X /\!\!/ T,\mathbb{Q})^W \ar[rr]^{\cup(p_2-p_1)} &&
  H^\bullet(X /\!\!/ T,\mathbb{Q})^a}
\]
We compute the quantum period of $\MW{4}{11} \subset X /\!\!/ G$ by computing the $J$-function of $\Fl = X /\!\!/ G$ twisted, in the sense of \cite{Coates--Givental}, by the Euler class and the bundle $\mathcal{V}_G$, using the Abelian/non-Abelian Correspondence.

Our first step is to compute the $J$-function of $X /\!\!/ T$ twisted by the Euler class and the bundle $\mathcal{V}_T$.  As in
\cite[\S D.1]{QC105} and as in \cite{Ciocan-Fontanine--Kim--Sabbah}, consider the bundles $\mathcal{V}_T$ and $\mathcal{V}_G$ equipped with the canonical $\CC^\times$-action that rotates fibers and acts trivially on the base.  Recall the definition of the twisted $J$-function $J_{{\boldsymbol{e}},\mathcal{V}_T}$ of $X /\!\!/ T$ from \cite[\S D.1]{QC105}.  We will compute $J_{{\boldsymbol{e}},\mathcal{V}_T}$ using the Quantum Lefschetz theorem; $J_{{\boldsymbol{e}},\mathcal{V}_T}$ is the restriction to the locus $\tau \in H^0(X /\!\!/ T) \oplus H^2(X /\!\!/ T)$ of what was denoted by $J^{S \times \CC^\times}_{\mathcal{V}_T}(\tau)$ in \cite{Ciocan-Fontanine--Kim--Sabbah}.  The toric variety $X /\!\!/ T$ is Fano, so Theorem~C.1 in \cite{QC105} gives:
\[
J_{X /\!\!/ T}(\tau) = e^{\tau/z} \sum_{l, m, n \geq 0} {
  Q_1^l Q_2^m Q_3^n
  e^{l \tau_1} e^{m \tau_2} e^{m \tau_3}
  \over
  \prod_{k=1}^{k=l} (p_1 + k z)^4
  \prod_{k=1}^{k=m} (p_2 + k z)^4
}
{
\prod_{k =-\infty}^{k=0} p_3-p_1 + k z
\over
\prod_{k=-\infty}^{k=n-l} p_3-p_1 + k z
}
{
\prod_{k = -\infty}^{k=0} p_3-p_2 + k z
\over
\prod_{k=-\infty}^{k= n-m} p_3-p_2 + k z
}
\]
where $\tau = \tau_1 p_1 + \tau_2 p_2 + \tau_3 p_3$ and we have identified the group ring $\mathbb{Q}[H_2(X /\!\!/ T;\mathbb{Z})]$ with $\mathbb{Q}[Q_1,Q_2,Q_3]$ via the $\mathbb{Q}$-linear map that sends $Q^\beta$ to $Q_1^{\langle \beta, p_1 \rangle} Q_2^{\langle \beta, p_2\rangle} Q_3^{\langle \beta, p_3 \rangle}$. The line bundles $L_1$, $L_2$, and $H$ are nef, and $c_1(X /\!\!/ T) - c_1(\mathcal{V}_T)$ is ample, so Theorem~D.3 in \cite{QC105} gives:
\begin{multline*}
  J_{{\boldsymbol{e}},\mathcal{V}_T}(\tau) =
  e^{\tau/z}
  \sum_{l=0}^\infty   \sum_{m=0}^\infty   \sum_{n=0}^\infty
  Q_1^l Q_2^m Q_3^n e^{l \tau_1} e^{m \tau_2} e^{m \tau_3}
  {
    \prod_{k=1}^{k=l+m} (\lambda + p_1 + p_2 + k z)
    \over
    \prod_{k=1}^{k=l} (p_1 + k z)^4
    \prod_{k=1}^{k=m} (p_2 + k z)^4
  } \times \\
  {
    \prod_{k =-\infty}^{k=0} p_3-p_1 + k z
    \over
    \prod_{k=-\infty}^{k=n-l} p_3-p_1 + k z
  }
  {
    \prod_{k = -\infty}^{k=0} p_3-p_2 + k z
    \over
    \prod_{k=-\infty}^{k= n-m} p_3-p_2 + k z
  }
\end{multline*}

Consider now $\Fl = X /\!\!/ G$ and a point $t \in H^\bullet(\Fl)$.  Recall that $\Fl=\mathbb{P}(S)$ is the projectivization of the universal bundle $S$ of subspaces on $\Gr := \Gr(2,4)$. Let $\epsilon_1 \in H^2(\Fl;\mathbb{Q})$ be the pullback to $\Fl$ (under the projection map $p_2\colon\Fl\to \Gr$) of the ample generator of $H^2(\Gr)$, and let $\epsilon_2 \in H^2(\Fl;\mathbb{Q})$ be the first Chern class of $\mathcal{O}_{\mathbb{P}(S)}(1)$.  Identify the group ring $\mathbb{Q}[H_2(\Fl;\mathbb{Z})]$ with $\mathbb{Q}[q_1,q_2]$ via the $\mathbb{Q}$-linear map which sends $Q^\beta$ to $q_1^{\langle \beta,\epsilon_1 \rangle} q_2^{\langle \beta,\epsilon_2 \rangle}$.  In \cite[\S 6.1]{Ciocan-Fontanine--Kim--Sabbah} the authors consider the lift $\tilde{J}^{S \times \CC^\times}_{\mathcal{V}_G}(t)$ of their twisted $J$-function $J^{S \times \CC^\times}_{\mathcal{V}_G}(t)$ determined by a choice of lift $H^\bullet(X /\!\!/ G;\mathbb{Q}) \to H^\bullet(X /\!\!/ T,\mathbb{Q})^W$.  We restrict to the locus $t \in H^0(X /\!\!/ G;\mathbb{Q}) \oplus H^2(X /\!\!/ G;\mathbb{Q})$, considering the lift:
\begin{align*}
  \tilde{J}_{{\boldsymbol{e}},\mathcal{V}_G}(t) := \tilde{J}^{S \times \CC^\times}_{\mathcal{V}_G}(t)   && t \in H^0(X /\!\!/ G;\mathbb{Q}) \oplus H^2(X /\!\!/ G;\mathbb{Q})
\end{align*}
of our twisted $J$-function $J_{{\boldsymbol{e}},\mathcal{V}_G}$ determined by our choice of lift $H^\bullet(X /\!\!/ G;\mathbb{Q}) \to H^\bullet(X /\!\!/ T,\mathbb{Q})^W$. Theorems~4.1.1 and~6.1.2 in \cite{Ciocan-Fontanine--Kim--Sabbah} imply that:
\[
\tilde{J}_{{\boldsymbol{e}},\mathcal{V}_G}\big(\varphi(t)\big) \cup (p_2 - p_1) = \Big[
\textstyle
\big(z {\partial \over \partial \tau_2}
-
z {\partial \over \partial \tau_1}
\big) J_{{\boldsymbol{e}},\mathcal{V}_T}(\tau) \Big]_{\tau=t, Q_1 = Q_2 = -q_1, Q_3=q_2}
\]
for some function $\varphi:H^2(X /\!\!/ G;\mathbb{Q}) \to H^\bullet(X /\!\!/ G; \Lambda_G)$. Setting $t = 0$ gives:
\begin{multline*}
  \tilde{J}_{{\boldsymbol{e}},\mathcal{V}_G}\big(\varphi(0)\big) \cup (p_2 - p_1) =
  \sum_{l=0}^\infty   \sum_{m=0}^\infty   \sum_{n=0}^\infty
  (-1)^{l+m} q_1^{l+m} q_2^n
  {
    \prod_{k=1}^{k=l+m} (\lambda + p_1 + p_2 + k z)
    \over
    \prod_{k=1}^{k=l} (p_1 + k z)^4
    \prod_{k=1}^{k=m} (p_2 + k z)^4
  } \times \\
  {
    \prod_{k =-\infty}^{k=0} p_3-p_1 + k z
    \over
    \prod_{k=-\infty}^{k=n-l} p_3-p_1 + k z
  }
  {
    \prod_{k = -\infty}^{k=0} p_3-p_2 + k z
    \over
    \prod_{k=-\infty}^{k= n-m} p_3-p_2 + k z
  }
  \big(p_2-p_1 + (m-l) z \big)
\end{multline*}
For symmetry reasons the right-hand side here is divisible by $p_2-p_1$; it takes the form:
\[
(p_2-p_1)\Big(1 + O(z^{-2})\Big)
\]
whereas:
\[
\tilde{J}_{{\boldsymbol{e}},\mathcal{V}_G}\big(\varphi(0)\big) \cup (p_2 - p_1) =
(p_2-p_1) \Big(1 + \varphi(0) z^{-1} + O(z^{-2}) \Big)
\]
We conclude that $\varphi(0) = 0$.  Thus:
\begin{multline}
  \label{eq:almost_there}
  \tilde{J}_{{\boldsymbol{e}},\mathcal{V}_G}(0) \cup (p_2 - p_1) =
  \sum_{l=0}^\infty   \sum_{m=0}^\infty   \sum_{n=0}^\infty
  (-1)^{l+m} q_1^{l+m} q_2^n
  {
    \prod_{k=1}^{k=l+m} (\lambda + p_1 + p_2 + k z)
    \over
    \prod_{k=1}^{k=l} (p_1 + k z)^4
    \prod_{k=1}^{k=m} (p_2 + k z)^4
  } \times \\
  {
    \prod_{k =-\infty}^{k=0} p_3-p_1 + k z
    \over
    \prod_{k=-\infty}^{k=n-l} p_3-p_1 + k z
  }
  {
    \prod_{k = -\infty}^{k=0} p_3-p_2 + k z
    \over
    \prod_{k=-\infty}^{k= n-m} p_3-p_2 + k z
  }
  \big(p_2-p_1 + (m-l) z \big)
\end{multline}

To extract the quantum period $G_{\MW{4}{11}}$ from the twisted $J$-function $J_{{\boldsymbol{e}},\mathcal{V}_G}(0)$, we proceed as in \cite[Example~D.8]{QC105}: we take the non-equivariant limit, extract the component along the unit class $1 \in H^\bullet(X /\!\!/ G;\mathbb{Q})$, set $z=1$, and set $Q^\beta = t^{\langle \beta, {-K} \rangle}$ where $K = K_{\MW{4}{11}}$.  Thus we consider the right-hand side of \eqref{eq:almost_there}, take the non-equivariant limit, extract the coefficient of $p_2-p_1$, set $z=1$, and set $q_1 = q_2 =2t$, obtaining:
\begin{multline*}
  G_{\MW{4}{11}}(t) =
  \sum_{l=0}^\infty \sum_{m=0}^\infty \sum_{n=\max(l,m)}^\infty
  (-1)^{l+m}
  t^{2l+2m+2n}
  \frac
  {
    (l+m)!
  }
  {
    (l!)^4 (m!)^4 (n-l)! (n-m)!
  }
\Big(1 + (m-l)(H_{n-m}-4H_m)\Big)
  \\
  +
\sum_{l=0}^\infty \sum_{m=l+1}^\infty \sum_{n=l}^{m-1}
  (-1)^{l+n}
  t^{2l+2m+2n}
  \frac
  {
    (l+m)!
    (m-n-1)!
  }
  {
    (l!)^4 (m!)^4 (n-l)!
  }
  (m-l)
\end{multline*}

\begin{remark}
  The quantum period of $\mathbb{P}\big(\Omega^2_{\mathbb{P}(V)}(2)\big)$ can also be computed using Strangeway's reconstruction theorem for the quantum cohomology of Fano bundles~\cite[Theorem~1]{Strangeway}.  Thus the quantum period of $\MW{4}{11}$ can be derived from this result together with the Quantum Lefschetz theorem.  The Gromov--Witten invariants required as input to the reconstruction theorem can be computed via \cite[Lemma~1]{Strangeway}, using Schubert calculus on $\Gr(2,4)$ and intersection numbers in $\mathbb{P}^3$.
\end{remark}

\subsubsection{$\MW{4}{12}$} \hfill [\hyperref[table:index_2]{regularized quantum period p.~\pageref*{table:index_2}}, \hyperref[operator:MW^4_12]{operator p.~\pageref*{operator:MW^4_12}}] \label{sec:MW^4_12}

This is the blow-up of the quadric $Q^4$ along a line.  Consider the toric variety $F$ with weight data:
\[
\begin{array}{rrrrrrrl}
  \multicolumn{1}{c}{s_0} &
  \multicolumn{1}{c}{s_1} &
  \multicolumn{1}{c}{s_2} &
  \multicolumn{1}{c}{s_3} &
  \multicolumn{1}{c}{x} &
  \multicolumn{1}{c}{x_4} &
  \multicolumn{1}{c}{x_5} & \\
  \cmidrule{1-7}
  1 & 1 & 1 & 1 & -1 & 0 & 0 & \hspace{1.5ex} L\\
  0 & 0 & 0 & 0 & 1  & 1 & 1 & \hspace{1.5ex} M \\
\end{array}
\]
and $\Amp F = \langle L, M\rangle$.  The morphism $F \to \mathbb{P}^5$ that sends (contravariantly) the homogeneous co-ordinate functions $[x_0,x_1,\dots,x_5]$ to $[xs_0, xs_1, x s_2, x s_3, x_4, x_5]$ blows up the line $(x_0=x_1=x_2=x_3=0)$ in $\mathbb{P}^5$, and $\MW{4}{12}$ is the proper transform of a quadric containing this line.  Thus $\MW{4}{12}$ is a member of $|L+M|$ in the toric variety $F$.  We have:
\begin{itemize}
\item $-K_F=3L+3M$ is ample, so that $F$ is a Fano variety;
\item $\MW{4}{12}\sim L+M$ is ample;
\item $-(K_F+L+M)\sim 2L+2M$ is ample.
\end{itemize}
Theorem~\ref{thm:toric_ci_mirror} yields:
\[
G_{\MW{4}{12}}(t) = \sum_{l=0}^\infty \sum_{m=l}^\infty \frac{(l+m)!}{(l!)^4(m-l)!(m!)^2} t^{2l+2m}
\]

\subsubsection{$\MW{4}{13}$} \hfill [\hyperref[table:index_2]{regularized quantum period p.~\pageref*{table:index_2}}, \hyperref[operator:MW^4_13]{operator p.~\pageref*{operator:MW^4_13}}] \label{sec:MW^4_13}

This is the projective bundle $\mathbb{P}_{Q^3}\big(\mathcal{O}(1) \oplus \mathcal{O}\big)$ or, equivalently, a member of $|2L|$ in the toric variety $F$ with weight data:
\[
\begin{array}{rrrrrrrl}
  \multicolumn{1}{c}{x_0} &
  \multicolumn{1}{c}{x_1} &
  \multicolumn{1}{c}{x_2} &
  \multicolumn{1}{c}{x_3} &
  \multicolumn{1}{c}{x_4} &
  \multicolumn{1}{c}{u} &
  \multicolumn{1}{c}{v} & \\
  \cmidrule{1-7}
  1 & 1 & 1 & 1 & 1 & 0 & -1 & \hspace{1.5ex} L\\
  0 & 0 & 0 & 0 & 0 & 1 & 1 & \hspace{1.5ex} M \\
\end{array}
\]
and $\Amp F = \langle L, M\rangle$.  We have:
\begin{itemize}
\item $-K_F=4L+2M$ is ample, that is $F$ is a Fano variety;
\item $\MW{4}{13} \sim 2L$ is nef;
\item $-(K_F+2L)\sim 2L+2M$ is ample.
\end{itemize}
The projection $[x_0:x_1:x_2:x_3:x_4:x_5:u:v] \mapsto [x_0:x_1:x_2:x_3:x_4:x_5]$ exhibits $F$ as the scroll $\mathbb{P}_{\mathbb{P}^4}\big(\mathcal{O}(1) \oplus \mathcal{O}\big)$ over $\mathbb{P}^4$, and passing to a member of $|2L|$ restricts this scroll to $Q^3 \subset \mathbb{P}^4$.  Theorem~\ref{thm:toric_ci_mirror} yields:
\[
G_{\MW{4}{13}}(t) = \sum_{l=0}^\infty \sum_{m=l}^\infty \frac{(2l)!}{(l!)^5 m! (m-l)!} t^{2l+2m}
\]

\subsubsection{$\MW{4}{14}$} \hfill [\hyperref[table:index_2]{regularized quantum period p.~\pageref*{table:index_2}}, \hyperref[operator:MW^4_14]{operator p.~\pageref*{operator:MW^4_14}}] \label{sec:MW^4_14}

This is the product $\mathbb{P}^1 \times \mathbb{P}^3$.  Combining Theorem~\ref{thm:products} with \cite[Example~G.1]{QC105} and \cite[\S1]{QC105} yields:
\[
G_{\MW{4}{14}}(t) = \sum_{l=0}^\infty \sum_{m=0}^\infty \frac{t^{2l+4m}}{(l!)^2 (m!)^4}
\]

\subsubsection{$\MW{4}{15}$} \hfill [\hyperref[table:index_2]{regularized quantum period p.~\pageref*{table:index_2}}, \hyperref[operator:MW^4_15]{operator p.~\pageref*{operator:MW^4_15}}] \label{sec:MW^4_15}

This is the projective bundle $\mathbb{P}_{\mathbb{P}^3}\big(\mathcal{O}(1) \oplus \mathcal{O}(-1)\big)$, or in other words, the toric variety with weight data:
\[
\begin{array}{rrrrrrl}
  \multicolumn{1}{c}{x_0} &
  \multicolumn{1}{c}{x_1} &
  \multicolumn{1}{c}{x_2} &
  \multicolumn{1}{c}{x_3} &
  \multicolumn{1}{c}{u} &
  \multicolumn{1}{c}{v} \\
  \cmidrule{1-6}
  1 & 1 & 1 & 1 & 0 & -2 & \hspace{1.5ex} L\\
  0 & 0 & 0 & 0 & 1 & 1 & \hspace{1.5ex} M\\
\end{array}
\]
and $\Amp F = \langle L, M\rangle$.  Theorem~\ref{thm:toric_mirror} yields:
\[
G_{\MW{4}{15}}(t) = \sum_{l=0}^\infty \sum_{m=2l}^\infty \frac{t^{2l+2m}}{(l!)^4 m! (m-2l)!}
\]

\subsubsection{$\MW{4}{16}$} \hfill [\hyperref[table:index_2]{regularized quantum period p.~\pageref*{table:index_2}}, \hyperref[operator:MW^4_16]{operator p.~\pageref*{operator:MW^4_16}}] \label{sec:MW^4_16}

This is the product $\mathbb{P}^1 \times W^3$, where $W^3 \subset \mathbb{P}^2 \times \mathbb{P}^2$ is a divisor of bidegree $(1,1)$.  Theorem~\ref{thm:toric_ci_mirror} yields:
\[
G_{\MW{4}{16}}(t) = \sum_{l=0}^\infty \sum_{m=0}^\infty \sum_{n=0}^\infty \frac{(m+n)!}{(l!)^2 (m!)^3 (n!)^3} t^{2l+2m+2n}
\]

\subsubsection{$\MW{4}{17}$} \hfill [\hyperref[table:index_2]{regularized quantum period p.~\pageref*{table:index_2}}, \hyperref[operator:MW^4_17]{operator p.~\pageref*{operator:MW^4_17}}] \label{sec:MW^4_17}

This is the product $\mathbb{P}^1 \times B^3_7$, where $B^3_7$ is the blow-up of $\mathbb{P}^3$ at a point.  Note that $B^3_7$ is the projective bundle $\mathbb{P}_{\mathbb{P}^2}(\mathcal{O} \oplus \mathcal{O}(-1))$.  It follows that $\MW{4}{17}$ is the toric variety with weight data:
\[
\begin{array}{rrrrrrrl}
  \multicolumn{1}{c}{x_0} &
  \multicolumn{1}{c}{x_1} &
  \multicolumn{1}{c}{y_0} &
  \multicolumn{1}{c}{y_1} &
  \multicolumn{1}{c}{y_2} &
  \multicolumn{1}{c}{u} &
  \multicolumn{1}{c}{v} \\
  \cmidrule{1-7}
  1 & 1 & 0 & 0 & 0 & 0 & 0 & \hspace{1.5ex} L\\
  0 & 0 & 1 & 1 & 1 & 0 & -1 & \hspace{1.5ex} M\\
  0 & 0 & 0 & 0 & 0 & 1 & 1 & \hspace{1.5ex} N\\
\end{array}
\]
and $\Amp F = \langle L, M, N\rangle$.  Theorem~\ref{thm:toric_mirror} yields:
\[
G_{\MW{4}{17}}(t) = \sum_{l=0}^\infty \sum_{m=0}^\infty \sum_{n=m}^\infty \frac{t^{2l+2m+2n}}{(l!)^2 (m!)^3 n! (n-m)!}
\]

\subsubsection{$\MW{4}{18}$} \hfill [\hyperref[table:index_2]{regularized quantum period p.~\pageref*{table:index_2}}, \hyperref[operator:MW^4_18]{operator p.~\pageref*{operator:MW^4_18}}] \label{sec:MW^4_18}

This is the product $\mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^1$.  Combining Theorem~\ref{thm:products} with \cite[Example~G.1]{QC105} yields:
\[
G_{\MW{4}{18}}(t) = \sum_{k=0}^\infty \sum_{l=0}^\infty \sum_{m=0}^\infty \sum_{n=0}^\infty \frac{t^{2k+2l+2m+2n}}{(k!)^2(l!)^2(m!)^2(n!)^2}
\]

\section{Four-Dimensional Fano Toric Manifolds}
\label{sec:toric}

Four-dimensional Fano toric manifolds were classified by Batyrev~\cite{Batyrev} and Sato~\cite{Sato}.  {\O}bro classified Fano toric manifolds in dimensions 2--8~\cite{Obro} and, to standardise notation, we will write $\Obro{4}{k}$ for the $k$th four-dimensional Fano toric manifold in {\O}bro's list.  $\Obro{4}{k}$ is the $(23+k)$th Fano toric manifold in the Graded Ring Database~\cite{GRDB}, as the list there is the concatenation of {\O}bro's lists in dimensions 2--8. We can compute the quantum periods of the $\Obro{4}{k}$ using Theorem~\ref{thm:toric_mirror};  the first few Taylor coefficients of their regularized quantum periods can be found in the tables in the Appendix.

\section{Product Manifolds and Other Index $1$ Examples}
\label{sec:other_index_1}

Quantum periods for one-, two- and three-dimensional Fano manifolds were computed in~\cite{QC105}.  Combining these results with Theorem~\ref{thm:products} allows us to compute the quantum period of any four-dimensional Fano manifold that is a product of lower-dimensional manifolds.  Many of these examples have Fano index $r=1$.

In his thesis~\cite{Strangeway:thesis}, Strangeway determined the quantum periods of two four-dimensional Fano manifolds of index $r=1$ that have not yet been discussed.  These manifolds arise as complete intersections in the $9$-dimensional projective bundle $F = \mathbb{P}\big(\Omega^2_{\mathbb{P}^4}(2) \big)$.  Let $\pi \colon F \to \mathbb{P}^4$ denote the canonical projection, let $p \in H^2(F)$ be the first Chern class of $\pi^\star \mathcal{O}_{\mathbb{P}^4}(1)$, and let $\xi \in H^2(F)$ be the first Chern class of the tautological bundle $\mathcal{O}_F(1)$.
The manifold $F$ is Fano of Picard rank~$2$, with nef cone generated by $\{\xi,p\}$ and ${-K_F} = 6\xi + 2p$.  Let:
\begin{align*}
  & \text{$\mathrm{Str}_1 \subset F$ denote a complete intersection of five divisors of type $\xi$} \\
  & \text{$\mathrm{Str}_2 \subset F$ denote a complete intersection of four divisors of type $\xi$ and a divisor of type $p$} \\
  \intertext{We consider also:}
  & \text{$\mathrm{Str}_3 \subset F$, a complete intersection of four divisors of type $\xi$ and a divisor of type $\xi + p$}
\end{align*}
which was unaccountably omitted from \cite{Strangeway:thesis}.

The manifolds $\mathrm{Str}_k$, $k \in \{1,2,3\}$, each have Picard rank two.  To see this, observe that the ambient manifold $F$ is the blow-up of $\mathbb{P}^9$ along $\Gr(2,5)$, where $\Gr(2,5) \to \mathbb{P}^9$ is the Pl\"ucker embedding~\cite{Strangeway}; the blow-up $F \to \mathbb{P}^9$ and the projection $\pi \colon F \to \mathbb{P}^4$ are the extremal contractions corresponding to the two extremal rays in $\overline{\NE}(F)$.  Thus $\mathrm{Str}_1$ is the blow-up of $\mathbb{P}^4$ along an elliptic curve $E_5 \subset \mathbb{P}^4$ of degree~$5$.  Consider the five-dimensional Fano manifold $F_5$ given by the complete intersection of four divisors of type $\xi$ in $F$.  Then $F_5$ is the blow-up of $\mathbb{P}^5$ along a del~Pezzo surface $S_5$ of degree~$5$; in particular, $F_5$ has Picard rank two.  $\mathrm{Str}_3$ is an ample divisor (of type $\xi + p$) in $F_5$, so the Picard rank of $\mathrm{Str}_3$ is also two.  The manifold $\mathrm{Str}_2$ is a divisor in $F_5$ of type~$p$, and $F_5$ arises as the closure of the graph of the map $\mathbb{P}^5 \to \mathbb{P}^4$ given by the $5$-dimensional linear system of quadrics passing through $S_5$. This exhibits $\mathrm{Str}_2$ as the blow-up of a smooth four-dimensional quadric $Q^4$ along $S_5$, which implies that the Picard rank of $\mathrm{Str}_2$ is two.

We can compute the quantum periods of $\mathrm{Str}_k$, $k \in \{1,2,3\}$, by observing that a complete intersection in $F$ of five divisors of type~$\xi$ and one divisor of type~$p$ is a three-dimensional Fano manifold~$\MM{2}{17}$, `unsectioning' to compute the quantum period for $F$, and then applying the quantum Lefschetz theorem to compute the quantum periods for $\mathrm{Str}_1$,~$\mathrm{Str}_2$, and~$\mathrm{Str}_3$.  Recall the definition of the $J$-function $J_X(t,z)$ from \cite[equation~11]{Coates--Givental}.  The identity component of the $J$-function of $\MM{2}{17}$ is:
\begin{equation}
  \label{eq:equate_me_1}
  e^{-q_1 - q_2}
  \sum_{l_1,l_2,l_3\geq 0}
  (-q_1)^{l_1+l_2} q_2^{l_3}
  \frac
  {
    (l_1+l_2)! (l_1+l_3)!(l_2+l_3)! (l_1+l_2+l_3)!
  }
  {
    (l_1!)^4 (l_2!)^4 (l_3!)^4 z^{l_1+l_2+l_3}
  }
  \Big(1 + (l_2-l_1)(H_{l_2+l_3} - 4H_{l_2})\Big)
\end{equation}
where $q_1$,~$q_2$ are generators of the Novikov ring for $\MM{2}{17}$ dual respectively to $\xi$ and $p$; see~\cite[\S 34]{QC105}.   The identity component of the $J$-function of $F$ takes the form:
\[
\sum_{l=0}^\infty \sum_{m=0}^\infty c_{l,m} z^{-6l-2m} q_1^l q_2^m
\]
for some coefficients $c_{l,m} \in \mathbb{Q}$.  The Quantum Lefschetz theorem implies (cf.~\cite[\S D.1]{QC105}) that the identity component of the $J$-function of $\MM{2}{17}$ is equal to:
\begin{equation}
  \label{eq:equate_me_2}
  e^{-c_{1,0} q_1 -c_{0,1} q_2} \sum_{l=0}^\infty \sum_{m=0}^\infty (l!)^5 m! c_{l,m} z^{-l-m} q_1^l q_2^m
\end{equation}
and it is known that $c_{1,0} = 1$ and $c_{0,1} = 0$~\cite[\S5.1]{Strangeway}.
Equating \eqref{eq:equate_me_1} and \eqref{eq:equate_me_2} determines the $c_{l,m}$:
\[
c_{l,m}=
\sum_{i=0}^{l} \sum_{j=0}^{m}
(-1)^{j+l}
\frac
{
  (m+l-i-j)!(i+m-j)! (m+l-j)!
}
{
  ((l-i)!)^4 (i!)^4 ((m-j)!)^4j!m!(l!)^4
}
\Big(1 + (2i-l)(H_{i+m-j} - 4H_{i})\Big)
\]
The Quantum Lefschetz theorem now gives that:
\begin{align*}
  & G_{\mathrm{Str}_1}(t) = e^{-t}
  \sum_{l=0}^\infty
  \sum_{m=0}^\infty
  (l!)^5 c_{l,m} t^{l+2m} \\
  & G_{\mathrm{Str}_2}(t) =
  \sum_{l=0}^\infty
  \sum_{m=0}^\infty
  (l!)^4 m! \, c_{l,m} t^{2l+m} \\
  & G_{\mathrm{Str}_3}(t) =
  e^{-t} \sum_{l=0}^\infty
  \sum_{m=0}^\infty
  (l!)^4 (l+m)! \, c_{l,m} t^{l+m}
\end{align*}

\section{Numerical Calculations of Quantum Differential Operators}
\label{sec:qdes}

As discussed in \S\ref{sec:methodology}, the regularized quantum period $\widehat{G}_X(t)$ of a Fano manifold $X$ satisfies a differential equation:
\begin{align}
  \label{eq:regularized_qde_again}
  L_X \widehat{G}_X \equiv 0 && L_X = \sum_{k=0}^{k=N} p_k(t) D^k
\end{align}
called the regularized quantum differential equation.  Here the $p_m$ are polynomials and $D = t \frac{d}{dt}$.  The regularized quantum differential equation for $X$ coincides with the (unregularized) quantum differential equation for an anticanonical Calabi--Yau manifold $Y \subset X$; the study of the regularized quantum period from this point of view was pioneered by Batyrev--Ciocan-Fontanine--Kim--van~Straten~\cite{BCFKvS:1,BCFKvS:2}.  The differential equation \eqref{eq:regularized_qde_again} is expected to be Fuchsian, and the local system of solutions to $L_X f \equiv 0$ is expected to be of \emph{low ramification} in the following sense.

\begin{definition}[\!\!\protect{\cite{ProcECM}}]
  Let $S \subset \mathbb{P}^1$ a finite set, and $\mathbb{V} \to \mathbb{P}^1 \setminus S$ a local system.  Fix a basepoint $x \in \mathbb{P}^1 \setminus S$.  For $s \in S$, choose a small loop that winds once anticlockwise around $s$ and connect it to $x$ via a path, thereby making a loop $\gamma_s$ about $s$ based at $x$.  Let $T_s \colon \mathbb{V}_x \to \mathbb{V}_x$ denote the monodromy of $\mathbb{V}$  along $\gamma_s$.  The \emph{ramification} of $\mathbb{V}$ is:
  \[
  \rf(\mathbb{V}) := \sum_{s \in S} \dim\Big(\mathbb{V}_x/{\mathbb{V}_x}^{\!\!\!T_s}\Big)
  \]
  The \emph{ramification defect} of $\mathbb{V}$ is the quantity $\rf(\mathbb{V}) - 2\rk(\mathbb{V})$.  Non-trivial irreducible local systems $\mathbb{V} \to \mathbb{P}^1 \setminus S$ have non-negative ramification defect; this gives a lower bound for the ramification of $\mathbb{V}$.  A local system of ramification defect zero is called \emph{extremal}.
\end{definition}

\begin{definition}
  The \emph{ramification} (respectively \emph{ramification defect}) of a differential operator $L_X$ is the ramification (respectively ramification defect) of the local system of solutions $L_X f \equiv 0$.
\end{definition}

\begin{definition}
  The \emph{quantum differential operator} for a Fano manifold $X$ is the operator $L_X \in \mathbb{Q}[t]\langle D \rangle$ such that $L_X \widehat{G}_X \equiv 0$ which is of lowest order in $D$ and, among all such operators of this order, is of lowest degree in $t$.  (This defines $L_X$ only up to an overall scalar factor, but this suffices for our purposes.)
\end{definition}
Suppose that each of the polynomials $p_0,\ldots,p_N$ are of degree at most $r$, and write:
\begin{align*}
  L_X = \sum_{k=0}^{k=N} p_k(t) D^k && p_k(t) = \sum_{l=0}^r a_{kl} t^l
\end{align*}
The differential equation $L_X \widehat{G}_X \equiv 0$ gives a system of linear equations for the coefficients $a_{kl}$ which, given sufficiently many terms of the Taylor expansion of $\widehat{G}_X$, becomes over-determined.  Given \emph{a~priori} bounds on $N$ and $r$, therefore, we could compute the quantum differential operator $L_X$ by calculating sufficiently many terms in the Taylor expansion.  In general we do not have such bounds, but nonetheless by ensuring the linear system for $(a_{kl})$ is highly over-determined we can be reasonably confident that the operator $L_X$ which we compute is correct.  In addition, since $L_X$ is expected to correspond under mirror symmetry to a Picard--Fuchs differential equation for the mirror family, $L_X$ is expected to be of Fuchsian type.  This is an extremely delicate condition on the coefficients $(a_{kl})$, and it can be checked by exact computation.

We computed candidate quantum differential operators $L_X$ for all four-dimensional Fano manifolds of Fano index $r>1$, and checked the Fuchsian condition in each case.  The operators $L_X$, together with their ramification defects and the log-monodromy data $\{\log T_s : s \in S\}$ in Jordan normal form, can be found in Appendix~\ref{appendix:qdes}.  In $24$ cases, the local system of solutions to the regularized quantum differential equation is extremal, and in the remaining $11$ cases it is of ramification defect~$1$.

To compute the ramification of $L_X$, we follow Kedlaya~\cite[\S7.3]{Kedlaya}.  This involves only linear algebra over a splitting field for $p_N(t)$---recall that every singular point of $L_X$ occurs at a root of $p_N(t)$---and thus can be implemented using exact (not numerical) computer algebra.  For this we use Steel's symbolic implementation of $\overline{\mathbb{Q}}$ in the computational algebra system Magma~\cite{Magma,Steel}.

\subsection*{Source Code} This paper is accompanied by full source code, written in Magma.  See the included file {\tt README.txt} for usage instructions.  The source code, but not the text of this paper, is released under a Creative Commons~CC0 license~\cite{CC0}: see the included file {\tt COPYING.txt} for details.    If you make use of the source code in an academic or commercial context, you should acknowledge this by including a reference or citation to this paper.

\subsection*{Acknowledgements}

We thank Alessio Corti for a number of very useful conversations.    This research was supported by a Royal Society University Research Fellowship (TC); ERC Starting Investigator Grant number~240123; the Leverhulme Trust; grant MK-1297.2014.1; AG Laboratory NRU-HSE, RF government grant ag.~11.G34.31.0023; Grant of Leading Scientific Schools (N.Sh.~2998.2014.1); and EPSRC grant EP/I008128/1.

\pagebreak

\subsection{$\mathbb{P}^4$}
\label{operator:P4}

\hfill [\hyperref[sec:index_5]{description p.~\pageref*{sec:index_5}}, \hyperref[table:index_5]{regularized quantum period p.~\pageref*{table:index_5}}]

The quantum differential operator is:
{\small \[
(5t-1)(625t^4+125t^3+25t^2+5t+1) D^{4} +
31250t^5 D^{3} +
109375t^5 D^{2} +
156250t^5 D +
75000t^5
\]}
The local log-monodromies for the quantum local system:
\begin{align*}
& \tiny \begin{pmatrix}
0&1&0&0\\
0&0&1&0\\
0&0&0&1\\
0&0&0&0
\end{pmatrix} && \text{at $\textstyle t=0$} \\& \tiny \begin{pmatrix}
0&0&0&0\\
0&0&0&0\\
0&0&0&1\\
0&0&0&0
\end{pmatrix} && \text{at $\textstyle t=\frac{1}{5}$} \\& \tiny \begin{pmatrix}
0&0&0&0\\
0&0&0&0\\
0&0&0&1\\
0&0&0&0
\end{pmatrix} && \text{at the roots of $\textstyle 625t^4+125t^3+25t^2+5t+1=0$} \\\end{align*}
The operator $L_X$ is extremal.

\subsection{$Q^4$}
\label{operator:Q4}

\hfill [\hyperref[sec:index_4]{description p.~\pageref*{sec:index_4}}, \hyperref[table:index_4]{regularized quantum period p.~\pageref*{table:index_4}}]

The quantum differential operator is:
{\small \[
(32t^2-1)(32t^2+1) D^{4} +
8192t^4 D^{3} +
23552t^4 D^{2} +
28672t^4 D +
12288t^4
\]}
The local log-monodromies for the quantum local system:
\begin{align*}
& \tiny \begin{pmatrix}
0&1&0&0\\
0&0&1&0\\
0&0&0&1\\
0&0&0&0
\end{pmatrix} && \text{at $\textstyle t=0$} \\& \tiny \begin{pmatrix}
0&0&0&0\\
0&0&0&0\\
0&0&0&1\\
0&0&0&0
\end{pmatrix} && \text{at the roots of $\textstyle 32t^2-1=0$} \\& \tiny \begin{pmatrix}
0&0&0&0\\
0&0&0&0\\
0&0&0&1\\
0&0&0&0
\end{pmatrix} && \text{at the roots of $\textstyle 32t^2+1=0$} \\& \tiny \begin{pmatrix}
0&0&0&0\\
0&0&0&0\\
0&0&0&1\\
0&0&0&0
\end{pmatrix} && \text{at $t=\infty$} \\\end{align*}
The operator $L_X$ is extremal.

\pagebreak

\subsection{$\FI{4}{1}$}
\label{operator:FI^4_1}

\hfill [\hyperref[sec:index_3]{description p.~\pageref*{sec:index_3}}, \hyperref[table:index_3]{regularized quantum period p.~\pageref*{table:index_3}}]

The quantum differential operator is:
{\small \[
(11664t^3-1) D^{4} +
69984t^3 D^{3} +
142884t^3 D^{2} +
113724t^3 D +
29160t^3
\]}
The local log-monodromies for the quantum local system:
\begin{align*}
& \tiny \begin{pmatrix}
0&1&0&0\\
0&0&1&0\\
0&0&0&1\\
0&0&0&0
\end{pmatrix} && \text{at $\textstyle t=0$} \\& \tiny \begin{pmatrix}
0&0&0&0\\
0&0&0&0\\
0&0&0&1\\
0&0&0&0
\end{pmatrix} && \text{at the roots of $\textstyle 11664t^3-1=0$} \\& \tiny \begin{pmatrix}
0&0&0&0\\
0&0&0&0\\
0&0&\frac{1}{2}&0\\
0&0&0&\frac{1}{2}
\end{pmatrix} && \text{at $t=\infty$} \\\end{align*}
The operator $L_X$ is extremal.

\subsection{$\FI{4}{2}$}
\label{operator:FI^4_2}

\hfill [\hyperref[sec:index_3]{description p.~\pageref*{sec:index_3}}, \hyperref[table:index_3]{regularized quantum period p.~\pageref*{table:index_3}}]

The quantum differential operator is:
{\small \[
(12t-1)(144t^2+12t+1) D^{4} +
10368t^3 D^{3} +
21924t^3 D^{2} +
19116t^3 D +
5832t^3
\]}
The local log-monodromies for the quantum local system:
\begin{align*}
& \tiny \begin{pmatrix}
0&1&0&0\\
0&0&1&0\\
0&0&0&1\\
0&0&0&0
\end{pmatrix} && \text{at $\textstyle t=0$} \\& \tiny \begin{pmatrix}
0&0&0&0\\
0&0&0&0\\
0&0&0&1\\
0&0&0&0
\end{pmatrix} && \text{at $\textstyle t=\frac{1}{12}$} \\& \tiny \begin{pmatrix}
0&0&0&0\\
0&0&0&0\\
0&0&0&1\\
0&0&0&0
\end{pmatrix} && \text{at the roots of $\textstyle 144t^2+12t+1=0$} \\& \tiny \begin{pmatrix}
0&0&0&0\\
0&0&0&0\\
0&0&\frac{3}{4}&0\\
0&0&0&\frac{1}{4}
\end{pmatrix} && \text{at $t=\infty$} \\\end{align*}
The operator $L_X$ is extremal.

\subsection{$\FI{4}{3}$}
\label{operator:FI^4_3}

\hfill [\hyperref[sec:index_3]{description p.~\pageref*{sec:index_3}}, \hyperref[table:index_3]{regularized quantum period p.~\pageref*{table:index_3}}]

The quantum differential operator is:
{\small \[
(9t-1)(81t^2+9t+1) D^{4} +
4374t^3 D^{3} +
9477t^3 D^{2} +
8748t^3 D +
2916t^3
\]}
The local log-monodromies for the quantum local system:
\begin{align*}
& \tiny \begin{pmatrix}
0&1&0&0\\
0&0&1&0\\
0&0&0&1\\
0&0&0&0
\end{pmatrix} && \text{at $\textstyle t=0$} \\& \tiny \begin{pmatrix}
0&0&0&0\\
0&0&0&0\\
0&0&0&1\\
0&0&0&0
\end{pmatrix} && \text{at $\textstyle t=\frac{1}{9}$} \\& \tiny \begin{pmatrix}
0&0&0&0\\
0&0&0&0\\
0&0&0&1\\
0&0&0&0
\end{pmatrix} && \text{at the roots of $\textstyle 81t^2+9t+1=0$} \\& \tiny \begin{pmatrix}
0&1&0&0\\
0&0&0&0\\
0&0&0&1\\
0&0&0&0
\end{pmatrix} && \text{at $t=\infty$} \\\end{align*}
The operator $L_X$ is extremal.

\subsection{$\FI{4}{4}$}
\label{operator:FI^4_4}

\hfill [\hyperref[sec:index_3]{description p.~\pageref*{sec:index_3}}, \hyperref[table:index_3]{regularized quantum period p.~\pageref*{table:index_3}}]

The quantum differential operator is:
{\small \[
(432t^3-1) D^{4} +
2592t^3 D^{3} +
5724t^3 D^{2} +
5508t^3 D +
1944t^3
\]}
The local log-monodromies for the quantum local system:
\begin{align*}
& \tiny \begin{pmatrix}
0&1&0&0\\
0&0&1&0\\
0&0&0&1\\
0&0&0&0
\end{pmatrix} && \text{at $\textstyle t=0$} \\& \tiny \begin{pmatrix}
0&0&0&0\\
0&0&0&0\\
0&0&0&1\\
0&0&0&0
\end{pmatrix} && \text{at the roots of $\textstyle 432t^3-1=0$} \\& \tiny \begin{pmatrix}
0&0&0&0\\
0&0&0&0\\
0&0&\frac{1}{2}&1\\
0&0&0&\frac{1}{2}
\end{pmatrix} && \text{at $t=\infty$} \\\end{align*}
The operator $L_X$ is extremal.

\subsection{$\FI{4}{5}$}
\label{operator:FI^4_5}

\hfill [\hyperref[sec:index_3]{description p.~\pageref*{sec:index_3}}, \hyperref[table:index_3]{regularized quantum period p.~\pageref*{table:index_3}}]

The quantum differential operator is:
{\small \[
(729t^6+297t^3-1) D^{4} +
162t^3(54t^3+11) D^{3} +
27t^3(1323t^3+148) D^{2} +
81t^3(702t^3+49) D +
1458t^3(20t^3+1)
\]}
The local log-monodromies for the quantum local system:
\begin{align*}
& \tiny \begin{pmatrix}
0&1&0&0\\
0&0&1&0\\
0&0&0&1\\
0&0&0&0
\end{pmatrix} && \text{at $\textstyle t=0$} \\& \tiny \begin{pmatrix}
0&0&0&0\\
0&0&0&0\\
0&0&0&1\\
0&0&0&0
\end{pmatrix} && \text{at the roots of $\textstyle 729t^6+297t^3-1=0$} \\\end{align*}
The ramification defect of $L_X$ is $1$.

\subsection{$\mathbb{P}^2\times \mathbb{P}^2$}
\label{operator:FI^4_6}

\hfill [\hyperref[sec:index_3]{description p.~\pageref*{sec:index_3}}, \hyperref[table:index_3]{regularized quantum period p.~\pageref*{table:index_3}}]

The quantum differential operator is:
{\small
  \begin{multline*}
    (3t+1)(6t-1)(9t^2-3t+1)(36t^2+6t+1) D^{4} +
    162t^3(432t^3+7) D^{3} + \\
    27t^3(10584t^3+95) D^{2} +
    1296t^3(351t^3+2) D +
    972t^3(240t^3+1)
  \end{multline*}}
The local log-monodromies for the quantum local system:
\begin{align*}
& \tiny \begin{pmatrix}
0&1&0&0\\
0&0&1&0\\
0&0&0&1\\
0&0&0&0
\end{pmatrix} && \text{at $\textstyle t=0$} \\& \tiny \begin{pmatrix}
0&0&0&0\\
0&0&0&0\\
0&0&0&1\\
0&0&0&0
\end{pmatrix} && \text{at $\textstyle t=\frac{1}{6}$} \\& \tiny \begin{pmatrix}
0&0&0&0\\
0&0&0&0\\
0&0&0&1\\
0&0&0&0
\end{pmatrix} && \text{at $\textstyle t=-\frac{1}{3}$} \\& \tiny \begin{pmatrix}
0&0&0&0\\
0&0&0&0\\
0&0&0&1\\
0&0&0&0
\end{pmatrix} && \text{at the roots of $\textstyle 9t^2-3t+1=0$} \\& \tiny \begin{pmatrix}
0&0&0&0\\
0&0&0&0\\
0&0&0&1\\
0&0&0&0
\end{pmatrix} && \text{at the roots of $\textstyle 36t^2+6t+1=0$} \\\end{align*}
The ramification defect of $L_X$ is $1$.

\subsection{$\VV{4}{2}$} \label{operator:V^4_2}  \hfill [\hyperref[sec:V^4_2]{description p.~\pageref*{sec:V^4_2}}, \hyperref[table:index_2]{regularized quantum period p.~\pageref*{table:index_2}}]

The quantum differential operator is:
{\small \[
(6912t^2-1) D^{4} +
27648t^2 D^{3} +
38400t^2 D^{2} +
21504t^2 D +
3840t^2
\]}
The local log-monodromies for the quantum local system:
\begin{align*}
& \tiny \begin{pmatrix}
0&1&0&0\\
0&0&1&0\\
0&0&0&1\\
0&0&0&0
\end{pmatrix} && \text{at $\textstyle t=0$} \\& \tiny \begin{pmatrix}
0&0&0&0\\
0&0&0&0\\
0&0&0&1\\
0&0&0&0
\end{pmatrix} && \text{at the roots of $\textstyle 6912t^2-1=0$} \\& \tiny \begin{pmatrix}
0&1&0&0\\
0&0&0&0\\
0&0&\frac{2}{3}&0\\
0&0&0&\frac{1}{3}
\end{pmatrix} && \text{at $t=\infty$} \\\end{align*}
The operator $L_X$ is extremal.

\subsection{$\VV{4}{4}$} \label{operator:V^4_4}  \hfill [\hyperref[sec:V^4_4]{description p.~\pageref*{sec:V^4_4}}, \hyperref[table:index_2]{regularized quantum period p.~\pageref*{table:index_2}}]

The quantum differential operator is:
{\small \[
(32t-1)(32t+1) D^{4} +
4096t^2 D^{3} +
5888t^2 D^{2} +
3584t^2 D +
768t^2
\]}
The local log-monodromies for the quantum local system:
\begin{align*}
& \tiny \begin{pmatrix}
0&1&0&0\\
0&0&1&0\\
0&0&0&1\\
0&0&0&0
\end{pmatrix} && \text{at $\textstyle t=0$} \\& \tiny \begin{pmatrix}
0&0&0&0\\
0&0&0&0\\
0&0&0&1\\
0&0&0&0
\end{pmatrix} && \text{at $\textstyle t=\frac{1}{32}$} \\& \tiny \begin{pmatrix}
0&0&0&0\\
0&0&0&0\\
0&0&0&1\\
0&0&0&0
\end{pmatrix} && \text{at $\textstyle t=-\frac{1}{32}$} \\& \tiny \begin{pmatrix}
0&1&0&0\\
0&0&0&0\\
0&0&\frac{1}{2}&0\\
0&0&0&\frac{1}{2}
\end{pmatrix} && \text{at $t=\infty$} \\\end{align*}
The operator $L_X$ is extremal.

\subsection{$\VV{4}{6}$} \label{operator:V^4_6}  \hfill [\hyperref[sec:V^4_6]{description p.~\pageref*{sec:V^4_6}}, \hyperref[table:index_2]{regularized quantum period p.~\pageref*{table:index_2}}]

The quantum differential operator is:
{\small \[
(432t^2-1) D^{4} +
1728t^2 D^{3} +
2544t^2 D^{2} +
1632t^2 D +
384t^2
\]}
The local log-monodromies for the quantum local system:
\begin{align*}
& \tiny \begin{pmatrix}
0&1&0&0\\
0&0&1&0\\
0&0&0&1\\
0&0&0&0
\end{pmatrix} && \text{at $\textstyle t=0$} \\& \tiny \begin{pmatrix}
0&0&0&0\\
0&0&0&0\\
0&0&0&1\\
0&0&0&0
\end{pmatrix} && \text{at the roots of $\textstyle 432t^2-1=0$} \\& \tiny \begin{pmatrix}
0&1&0&0\\
0&0&0&0\\
0&0&\frac{2}{3}&0\\
0&0&0&\frac{1}{3}
\end{pmatrix} && \text{at $t=\infty$} \\\end{align*}
The operator $L_X$ is extremal.

\pagebreak

\subsection{$\VV{4}{8}$} \label{operator:V^4_8}  \hfill [\hyperref[sec:V^4_8]{description p.~\pageref*{sec:V^4_8}}, \hyperref[table:index_2]{regularized quantum period p.~\pageref*{table:index_2}}]

The quantum differential operator is:
{\small \[
(16t-1)(16t+1) D^{4} +
1024t^2 D^{3} +
1536t^2 D^{2} +
1024t^2 D +
256t^2
\]}
The local log-monodromies for the quantum local system:
\begin{align*}
& \tiny \begin{pmatrix}
0&1&0&0\\
0&0&1&0\\
0&0&0&1\\
0&0&0&0
\end{pmatrix} && \text{at $\textstyle t=0$} \\& \tiny \begin{pmatrix}
0&0&0&0\\
0&0&0&0\\
0&0&0&1\\
0&0&0&0
\end{pmatrix} && \text{at $\textstyle t=\frac{1}{16}$} \\& \tiny \begin{pmatrix}
0&0&0&0\\
0&0&0&0\\
0&0&0&1\\
0&0&0&0
\end{pmatrix} && \text{at $\textstyle t=-\frac{1}{16}$} \\& \tiny \begin{pmatrix}
0&1&0&0\\
0&0&1&0\\
0&0&0&1\\
0&0&0&0
\end{pmatrix} && \text{at $t=\infty$} \\\end{align*}
The operator $L_X$ is extremal.

\subsection{$\VV{4}{10}$} \label{operator:V^4_10}  \hfill [\hyperref[sec:V^4_10]{description p.~\pageref*{sec:V^4_10}}, \hyperref[table:index_2]{regularized quantum period p.~\pageref*{table:index_2}}]

The quantum differential operator is:
{\small \[
(256t^4+176t^2-1) D^{4} +
64t^2(32t^2+11) D^{3} +
16t^2(352t^2+67) D^{2} +
32t^2(192t^2+23) D +
192t^2(12t^2+1)
\]}
The local log-monodromies for the quantum local system:
\begin{align*}
& \tiny \begin{pmatrix}
0&1&0&0\\
0&0&1&0\\
0&0&0&1\\
0&0&0&0
\end{pmatrix} && \text{at $\textstyle t=0$} \\& \tiny \begin{pmatrix}
0&0&0&0\\
0&0&0&0\\
0&0&0&1\\
0&0&0&0
\end{pmatrix} && \text{at the roots of $\textstyle 256t^4+176t^2-1=0$} \\& \tiny \begin{pmatrix}
0&1&0&0\\
0&0&0&0\\
0&0&0&1\\
0&0&0&0
\end{pmatrix} && \text{at $t=\infty$} \\\end{align*}
The ramification defect of $L_X$ is $1$.

\subsection{$\VV{4}{12}$} \label{operator:V^4_12}  \hfill [\hyperref[sec:V^4_12]{description p.~\pageref*{sec:V^4_12}}, \hyperref[table:index_2]{regularized quantum period p.~\pageref*{table:index_2}}]

The quantum differential operator is:
{\small \[
(4t^2-12t+1)(4t^2+12t+1) D^{4} +
32t^2(4t^2-17) D^{3} +
8t^2(46t^2-105) D^{2} +
16t^2(28t^2-37) D +
32t^2(6t^2-5)
\]}
The local log-monodromies for the quantum local system:
\begin{align*}
& \tiny \begin{pmatrix}
0&1&0&0\\
0&0&1&0\\
0&0&0&1\\
0&0&0&0
\end{pmatrix} && \text{at $\textstyle t=0$} \\& \tiny \begin{pmatrix}
0&0&0&0\\
0&0&0&0\\
0&0&0&1\\
0&0&0&0
\end{pmatrix} && \text{at the roots of $\textstyle 4t^2-12t+1=0$} \\& \tiny \begin{pmatrix}
0&0&0&0\\
0&0&0&0\\
0&0&0&1\\
0&0&0&0
\end{pmatrix} && \text{at the roots of $\textstyle 4t^2+12t+1=0$} \\& \tiny \begin{pmatrix}
0&0&0&0\\
0&0&0&0\\
0&0&0&1\\
0&0&0&0
\end{pmatrix} && \text{at $t=\infty$} \\\end{align*}
The operator $L_X$ is extremal.

\subsection{$\VV{4}{14}$} \label{operator:V^4_14}  \hfill [\hyperref[sec:V^4_14]{description p.~\pageref*{sec:V^4_14}}, \hyperref[table:index_2]{regularized quantum period p.~\pageref*{table:index_2}}]

The quantum differential operator is:
{\small \[
(4t^2+1)(108t^2-1) D^{4} +
32t^2(108t^2+13) D^{3} +
24t^2(406t^2+27) D^{2} +
16t^2(708t^2+29) D +
128t^2(36t^2+1)
\]}
The local log-monodromies for the quantum local system:
\begin{align*}
& \tiny \begin{pmatrix}
0&1&0&0\\
0&0&1&0\\
0&0&0&1\\
0&0&0&0
\end{pmatrix} && \text{at $\textstyle t=0$} \\& \tiny \begin{pmatrix}
0&0&0&0\\
0&0&0&0\\
0&0&0&1\\
0&0&0&0
\end{pmatrix} && \text{at the roots of $\textstyle 108t^2-1=0$} \\& \tiny \begin{pmatrix}
0&0&0&0\\
0&0&0&0\\
0&0&0&1\\
0&0&0&0
\end{pmatrix} && \text{at the roots of $\textstyle 4t^2+1=0$} \\& \tiny \begin{pmatrix}
0&0&0&0\\
0&0&0&0\\
0&0&\frac{2}{3}&0\\
0&0&0&\frac{1}{3}
\end{pmatrix} && \text{at $t=\infty$} \\\end{align*}
The ramification defect of $L_X$ is $1$.

\subsection{$\VV{4}{16}$} \label{operator:V^4_16}  \hfill [\hyperref[sec:V^4_16]{description p.~\pageref*{sec:V^4_16}}, \hyperref[table:index_2]{regularized quantum period p.~\pageref*{table:index_2}}]

The quantum differential operator is:
{\small
  \begin{multline*}
    (16t^2-8t-1)(16t^2+8t-1) D^{4} +
    128t^2(16t^2-3) D^{3} +
    32t^2(184t^2-19) D^{2} + \\
    448t^2(4t-1)(4t+1) D +
    128t^2(24t^2-1)
  \end{multline*}}
The local log-monodromies for the quantum local system:
\begin{align*}
& \tiny \begin{pmatrix}
0&1&0&0\\
0&0&1&0\\
0&0&0&1\\
0&0&0&0
\end{pmatrix} && \text{at $\textstyle t=0$} \\& \tiny \begin{pmatrix}
0&0&0&0\\
0&0&0&0\\
0&0&0&1\\
0&0&0&0
\end{pmatrix} && \text{at the roots of $\textstyle 16t^2-8t-1=0$} \\& \tiny \begin{pmatrix}
0&0&0&0\\
0&0&0&0\\
0&0&0&1\\
0&0&0&0
\end{pmatrix} && \text{at the roots of $\textstyle 16t^2+8t-1=0$} \\& \tiny \begin{pmatrix}
0&0&0&0\\
0&0&0&0\\
0&0&0&1\\
0&0&0&0
\end{pmatrix} && \text{at $t=\infty$} \\\end{align*}
The operator $L_X$ is extremal.

\subsection{$\VV{4}{18}$} \label{operator:V^4_18} \hfill [\hyperref[sec:V^4_18]{description p.~\pageref*{sec:V^4_18}}, \hyperref[table:index_2]{regularized quantum period p.~\pageref*{table:index_2}}]

The quantum differential operator is:
{\small \[
(432t^4+72t^2-1) D^{4} +
288t^2(12t^2+1) D^{3} +
24t^2(414t^2+19) D^{2} +
336t^2(36t^2+1) D +
96t^2(54t^2+1)
\]}
The local log-monodromies for the quantum local system:
\begin{align*}
& \tiny \begin{pmatrix}
0&1&0&0\\
0&0&1&0\\
0&0&0&1\\
0&0&0&0
\end{pmatrix} && \text{at $\textstyle t=0$} \\& \tiny \begin{pmatrix}
0&0&0&0\\
0&0&0&0\\
0&0&0&1\\
0&0&0&0
\end{pmatrix} && \text{at the roots of $\textstyle 432t^4+72t^2-1=0$} \\& \tiny \begin{pmatrix}
0&0&0&0\\
0&0&0&0\\
0&0&0&1\\
0&0&0&0
\end{pmatrix} && \text{at $t=\infty$} \\\end{align*}
The operator $L_X$ is extremal.

\subsection{$\MW{4}{1}$} \label{operator:MW^4_1}  \hfill [\hyperref[sec:MW^4_1]{description p.~\pageref*{sec:MW^4_1}}, \hyperref[table:index_2]{regularized quantum period p.~\pageref*{table:index_2}}]

The quantum differential operator is:
{\small
  \begin{multline*}
    (2t-1)^2(2t+1)^2(1724t^2-4t-1)(1724t^2+4t-1)\\
    \shoveright{(1384128950480t^6-34997928616t^4-263676995t^2+9409) D^{6} }\\
    \shoveleft{+ 2(2t-1)(2t+1)(181010493290961157120t^{12}-17468834144875533568t^{10}+374429340495784832t^8+}\\
    \shoveright{3959280486757728t^6-1227737299988t^4-725619617t^2+18818) D^{5}} \\
    \shoveleft{+ 4\left(3126544884116601804800t^{14}-725557954486979610624t^{12}+31166631689741025792t^{10}- \right. }\\
    \shoveright{\left. 440963660134839040t^8-7399870298607304t^6-348759582360t^4-504354223t^2+9409\right) D^{4}} \\
    \shoveleft{+ 8t^2(6746754749935824947200t^{12}-1095289161198143939072t^{10}+33852557194447324800t^8-}\\
    \shoveright{138056179574882528t^6-6267098983057824t^4-620133376448t^2-10735669) D^{3}} \\
    \shoveleft{+ 32t^2(3803277296533945221760t^{12}-460852243532660846400t^{10}+13408867107650109352t^8+}\\
    \shoveright{72880113460188392t^6-1247120973283936t^4-144892007990t^2-573949) D^{2}}\\
    \shoveleft{+ 256t^4(524004808703094940640t^{10}-51043969670376668752t^8+1573801437077923102t^6+}\\
    \shoveright{16338545311012128t^4-54334824441981t^2-5942952755) D} \\
    \shoveleft{+ 1536t^4(35996404915816139200t^{10}-3025511420019920960t^8+}\\
    99559010182515260t^6+1234528802429310t^4-1125770982819t^2-89024854)
\end{multline*}}
The local log-monodromies for the quantum local system:
\begin{align*}
& \tiny \begin{pmatrix}
0&1&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&1&0&0\\
0&0&0&0&1&0\\
0&0&0&0&0&1\\
0&0&0&0&0&0
\end{pmatrix} && \text{at $\textstyle t=0$} \\& \tiny \begin{pmatrix}
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&\frac{5}{6}&0\\
0&0&0&0&0&\frac{1}{6}
\end{pmatrix} && \text{at $\textstyle t=\frac{1}{2}$} \\& \tiny \begin{pmatrix}
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&\frac{5}{6}&0\\
0&0&0&0&0&\frac{1}{6}
\end{pmatrix} && \text{at $\textstyle t=-\frac{1}{2}$} \\& \tiny \begin{pmatrix}
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&1\\
0&0&0&0&0&0
\end{pmatrix} && \text{at the roots of $\textstyle 1724t^2-4t-1=0$} \\& \tiny \begin{pmatrix}
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&1\\
0&0&0&0&0&0
\end{pmatrix} && \text{at the roots of $\textstyle 1724t^2+4t-1=0$} \\\end{align*}
The operator $L_X$ is extremal.

\pagebreak

\subsection{$\MW{4}{2}$} \label{operator:MW^4_2}  \hfill [\hyperref[sec:MW^4_2]{description p.~\pageref*{sec:MW^4_2}}, \hyperref[table:index_2]{regularized quantum period p.~\pageref*{table:index_2}}]

The quantum differential operator is:
{\small
  \begin{multline*}
    (2t-1)^2(2t+1)^2(14t-1)(14t+1)(18t-1)(18t+1)(7544656t^6-3112t^4-6667t^2+1) D^{6} \\
    \shoveleft{+ 2(2t-1)(2t+1)} \\
    \shoveright{(21081096723456t^{12}-1556683668736t^{10}-25006224512t^8+
    2282941792t^6-4014452t^4-18937t^2+2) D^{5}} \\
    \shoveleft{+ 4\left(364128034314240t^{14}-74616093755392t^{12}+1020882362880t^{10}+\right.} \\
    \shoveright{\left. 104092887296t^8-4150928136t^6-1064664t^4-12743t^2+1\right) D^{4}}\\
    \shoveleft{+ 8t^2(785749968783360t^{12}-103037075309056t^{10}-578957540736t^8+}\\
    \shoveright{167061307168t^6-3800819360t^4-2728128t^2-205) D^{3}}\\
    \shoveleft{+ 32t^2(442942589109888t^{12}-38313217780544t^{10}-591376611224t^8+}\\
    \shoveright{79457857384t^6-864235264t^4-697078t^2-13) D^{2}} \\
    \shoveright{+ 256t^4(61027379435232t^{10}-3662113129808t^8-80739092050t^6+9991256448t^4-46797421t^2-33179) D}\\
    + 7680t^4(838452710592t^{10}-37503518528t^8-1048500436t^6+130762470t^4-249799t^2-110)
  \end{multline*}}
The local log-monodromies for the quantum local system:
\begin{align*}
& \tiny \begin{pmatrix}
0&1&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&1&0&0\\
0&0&0&0&1&0\\
0&0&0&0&0&1\\
0&0&0&0&0&0
\end{pmatrix} && \text{at $\textstyle t=0$} \\& \tiny \begin{pmatrix}
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&1&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&1\\
0&0&0&0&0&0
\end{pmatrix} && \text{at $\textstyle t=\frac{1}{2}$} \\& \tiny \begin{pmatrix}
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&1\\
0&0&0&0&0&0
\end{pmatrix} && \text{at $\textstyle t=\frac{1}{14}$} \\& \tiny \begin{pmatrix}
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&1\\
0&0&0&0&0&0
\end{pmatrix} && \text{at $\textstyle t=\frac{1}{18}$} \\& \tiny \begin{pmatrix}
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&1\\
0&0&0&0&0&0
\end{pmatrix} && \text{at $\textstyle t=-\frac{1}{18}$} \\& \tiny \begin{pmatrix}
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&1\\
0&0&0&0&0&0
\end{pmatrix} && \text{at $\textstyle t=-\frac{1}{14}$} \\& \tiny \begin{pmatrix}
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&1&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&1\\
0&0&0&0&0&0
\end{pmatrix} && \text{at $\textstyle t=-\frac{1}{2}$} \\\end{align*}
The operator $L_X$ is extremal.

\pagebreak

\subsection{$\MW{4}{3}$} \label{operator:MW^4_3}  \hfill [\hyperref[sec:MW^4_3]{description p.~\pageref*{sec:MW^4_3}}, \hyperref[table:index_2]{regularized quantum period p.~\pageref*{table:index_2}}]

The quantum differential operator is:
{\small
  \begin{multline*}
    (2t-1)^2(2t+1)^2(104t^2-4t-1)(104t^2+4t-1)(124883200t^6+3445552t^4-190621t^2+50) D^{6} \\
    \shoveleft{+ 2(2t-1)(2t+1)(59432414412800t^{12}-3046657163264t^{10}-307452667136t^8}\\
    \shoveright{+12555781056t^6-36422116t^4-548263t^2+100) D^{5}} \\
    \shoveleft{+ 4\left(1026559885312000t^{14}-181195540611072t^{12}-5355282845184t^{10}+794966941312t^8 \right.}\\
    \shoveright{\left. -20516326820t^6-1544700t^4-364427t^2+50\right) D^{4}} \\
    \shoveleft{+ 8t^2(2215208173568000t^{12}-216948202172416t^{10}-15687219459072t^8} \\
    \shoveright{+1005364344896t^6-20392735992t^4-36932632t^2-5075) D^{3}}\\
    \shoveleft{+ 64t^2(624378035507200t^{12}-30899527376640t^{10}-4029754891664t^8}\\
    \shoveright{+205143060452t^6-2542262335t^4-5022410t^2-175) D^{2}}\\
    \shoveleft{+ 256t^4(172050086041600t^{10}-3462364820096t^8} \\
    \shoveright{-968115899896t^6+47585432988t^4-312087909t^2-517438) D} \\
    + 1536t^4(11818946048000t^{10}-33971342080t^8-59641497680t^6+2994403590t^4-9592347t^2-9140)
  \end{multline*}}
The local log-monodromies for the quantum local system:
\begin{align*}
& \tiny \begin{pmatrix}
0&1&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&1&0&0\\
0&0&0&0&1&0\\
0&0&0&0&0&1\\
0&0&0&0&0&0
\end{pmatrix} && \text{at $\textstyle t=0$} \\& \tiny \begin{pmatrix}
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&\frac{5}{6}&0\\
0&0&0&0&0&\frac{1}{6}
\end{pmatrix} && \text{at $\textstyle t=\frac{1}{2}$} \\& \tiny \begin{pmatrix}
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&\frac{5}{6}&0\\
0&0&0&0&0&\frac{1}{6}
\end{pmatrix} && \text{at $\textstyle t=-\frac{1}{2}$} \\& \tiny \begin{pmatrix}
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&1\\
0&0&0&0&0&0
\end{pmatrix} && \text{at the roots of $\textstyle 104t^2-4t-1=0$} \\& \tiny \begin{pmatrix}
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&1\\
0&0&0&0&0&0
\end{pmatrix} && \text{at the roots of $\textstyle 104t^2+4t-1=0$} \\\end{align*}
The operator $L_X$ is extremal.

\pagebreak

\subsection{$\MW{4}{4}$} \label{operator:MW^4_4}  \hfill [\hyperref[sec:MW^4_4]{description p.~\pageref*{sec:MW^4_4}}, \hyperref[table:index_2]{regularized quantum period p.~\pageref*{table:index_2}}]

The quantum differential operator is:
{\small \[
(16t^2+1)(128t^2-1) D^{4} +
64t^2(256t^2+7) D^{3} +
16t^2(2816t^2+43) D^{2} +
96t^2(512t^2+5) D +
128t^2(144t^2+1)
\]}
The local log-monodromies for the quantum local system:
\begin{align*}
& \tiny \begin{pmatrix}
0&1&0&0\\
0&0&1&0\\
0&0&0&1\\
0&0&0&0
\end{pmatrix} && \text{at $\textstyle t=0$} \\& \tiny \begin{pmatrix}
0&0&0&0\\
0&0&0&0\\
0&0&0&1\\
0&0&0&0
\end{pmatrix} && \text{at the roots of $\textstyle 128t^2-1=0$} \\& \tiny \begin{pmatrix}
0&0&0&0\\
0&0&0&0\\
0&0&0&1\\
0&0&0&0
\end{pmatrix} && \text{at the roots of $\textstyle 16t^2+1=0$} \\& \tiny \begin{pmatrix}
0&1&0&0\\
0&0&0&0\\
0&0&0&1\\
0&0&0&0
\end{pmatrix} && \text{at $t=\infty$} \\\end{align*}
The ramification defect of $L_X$ is $1$.

\subsection{$\MW{4}{5}$} \label{operator:MW^4_5}  \hfill [\hyperref[sec:MW^4_5]{description p.~\pageref*{sec:MW^4_5}}, \hyperref[table:index_2]{regularized quantum period p.~\pageref*{table:index_2}}]

The quantum differential operator is:
{\small
  \begin{multline*}
    (32t^4-144t^3+40t^2-12t+1)(32t^4+144t^3+40t^2+12t+1)\\
    \shoveright{(1379024896t^8-181690112t^6+32203856t^4+160775t^2+136) D^{6}} \\
    \shoveleft{+2\left(14121214935040t^{16}-152437496479744t^{14}+18570092085248t^{12}-3527276827648t^{10}-\right.} \\
    \shoveright{\left.277037824256t^8-556812800t^6-140567296t^4-499733t^2-272\right) D^{5}}\\
    \shoveleft{+ 4\left(54366677499904t^{16}-340827602288640t^{14}+64054093561856t^{12}-13315483720192t^{10}\right.}\\
    \shoveright{\left.-406200526464t^8-986296160t^6+122964452t^4+370629t^2+136\right) D^{4}}\\
    \shoveleft{+ 8t^2(103084869025792t^{14}-369261049675776t^{12}+91720177627136t^{10}-21127821675520t^8}\\
    \shoveright{-305989253824t^6-1259677440t^4-1614768t^2-5831) D^{3}}\\
    \shoveleft{+128t^2(12720125640704t^{14}-26591028115456t^{12}+8287153333632t^{10}}\\
    \shoveright{-2016286904136t^8-15422020074t^6-337096385t^4-533452t^2-102) D^{2}}\\
    \shoveleft{+ 256t^4(6200095932416t^{12}-7989995554816t^{10}+3020985441920t^8}\\
    \shoveright{-739274260600t^6-4109085596t^4-179641501t^2-331640) D} \\
    \shoveleft{+ 1536t^4(386126970880t^{12}-333456404480t^{10}+147420386560t^8}\\
    -34919816144t^6-204213358t^4-10473275t^2-20944)
  \end{multline*}}
The local log-monodromies for the quantum local system:
\begin{align*}
& \tiny \begin{pmatrix}
0&1&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&1&0&0\\
0&0&0&0&1&0\\
0&0&0&0&0&1\\
0&0&0&0&0&0
\end{pmatrix} && \text{at $\textstyle t=0$} \\& \tiny \begin{pmatrix}
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&1\\
0&0&0&0&0&0
\end{pmatrix} && \text{at the roots of $\textstyle 32t^4-144t^3+40t^2-12t+1=0$} \\& \tiny \begin{pmatrix}
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&1\\
0&0&0&0&0&0
\end{pmatrix} && \text{at the roots of $\textstyle 32t^4+144t^3+40t^2+12t+1=0$} \\& \tiny \begin{pmatrix}
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&1\\
0&0&0&0&0&0
\end{pmatrix} && \text{at $t=\infty$} \\\end{align*}
The ramification defect of $L_X$ is $1$.

\subsection{$\MW{4}{6}$} \label{operator:MW^4_6}  \hfill [\hyperref[sec:MW^4_6]{description p.~\pageref*{sec:MW^4_6}}, \hyperref[table:index_2]{regularized quantum period p.~\pageref*{table:index_2}}]

The quantum differential operator is:
{\small
  \begin{multline*}
    (2t-1)^2(2t+1)^2(6t-1)(6t+1)(10t-1)(10t+1)(1433040t^6+80728t^4-2579t^2+1) D^{6} \\
    \shoveleft{+ 2(2t-1)(2t+1)(226993536000t^{12}-6984049920t^{10}}\\
    \shoveright{-2174297216t^8+67707232t^6-106452t^4-7441t^2+2) D^{5}} \\
    \shoveleft{+ 4\left(3920797440000t^{14}-584516290560t^{12}-52264472064t^{10}\right.}\\
    \shoveright{\left.+4812821760t^8-86750536t^6+175464t^4-4911t^2+1\right) D^{4}} \\
    \shoveleft{+ 8t^2(8460668160000t^{12}-550314631680t^{10}-112753881472t^8}\\
    \shoveright{+5245181728t^6-97715360t^4-329920t^2-69) D^{3}} \\
    \shoveleft{+ 32t^2(4769443728000t^{12}-59034734400t^{10}-53369502424t^8}\\
    \shoveright{+1881440232t^6-26802560t^4-94678t^2-5) D^{2}} \\
    \shoveright{+ 768t^4(219040164000t^{10}+4402781840t^8-2035415446t^6+67181120t^4-620103t^2-1737) D} \\
    + 4608t^4(15046920000t^{10}+593723200t^8-120630700t^6+4065994t^4-22137t^2-34)
  \end{multline*}}
The local log-monodromies for the quantum local system:
\begin{align*}
& \tiny \begin{pmatrix}
0&1&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&1&0&0\\
0&0&0&0&1&0\\
0&0&0&0&0&1\\
0&0&0&0&0&0
\end{pmatrix} && \text{at $\textstyle t=0$} \\& \tiny \begin{pmatrix}
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&\frac{1}{2}&1\\
0&0&0&0&0&\frac{1}{2}
\end{pmatrix} && \text{at $\textstyle t=\frac{1}{2}$} \\& \tiny \begin{pmatrix}
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&1\\
0&0&0&0&0&0
\end{pmatrix} && \text{at $\textstyle t=\frac{1}{6}$} \\& \tiny \begin{pmatrix}
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&1\\
0&0&0&0&0&0
\end{pmatrix} && \text{at $\textstyle t=\frac{1}{10}$} \\& \tiny \begin{pmatrix}
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&1\\
0&0&0&0&0&0
\end{pmatrix} && \text{at $\textstyle t=-\frac{1}{10}$} \\& \tiny \begin{pmatrix}
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&1\\
0&0&0&0&0&0
\end{pmatrix} && \text{at $\textstyle t=-\frac{1}{6}$} \\& \tiny \begin{pmatrix}
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&\frac{1}{2}&1\\
0&0&0&0&0&\frac{1}{2}
\end{pmatrix} && \text{at $\textstyle t=-\frac{1}{2}$} \\\end{align*}
The operator $L_X$ is extremal.

\pagebreak

\subsection{$\MW{4}{7}$} \label{operator:MW^4_7}  \hfill [\hyperref[sec:MW^4_7]{description p.~\pageref*{sec:MW^4_7}}, \hyperref[table:index_2]{regularized quantum period p.~\pageref*{table:index_2}}]

The quantum differential operator is:
{\small \[
(8t-1)(8t+1)(16t^2+1) D^{4} +
64t^2(128t^2+3) D^{3} +
16t^2(1456t^2+19) D^{2} +
32t^2(864t^2+7) D +
64t^2(180t^2+1)
\]}
The local log-monodromies for the quantum local system:
\begin{align*}
& \tiny \begin{pmatrix}
0&1&0&0\\
0&0&1&0\\
0&0&0&1\\
0&0&0&0
\end{pmatrix} && \text{at $\textstyle t=0$} \\& \tiny \begin{pmatrix}
0&0&0&0\\
0&0&0&0\\
0&0&0&1\\
0&0&0&0
\end{pmatrix} && \text{at $\textstyle t=\frac{1}{8}$} \\& \tiny \begin{pmatrix}
0&0&0&0\\
0&0&0&0\\
0&0&0&1\\
0&0&0&0
\end{pmatrix} && \text{at $\textstyle t=-\frac{1}{8}$} \\& \tiny \begin{pmatrix}
0&0&0&0\\
0&0&0&0\\
0&0&0&1\\
0&0&0&0
\end{pmatrix} && \text{at the roots of $\textstyle 16t^2+1=0$} \\& \tiny \begin{pmatrix}
0&0&0&0\\
0&0&0&0\\
0&0&\frac{1}{2}&0\\
0&0&0&\frac{1}{2}
\end{pmatrix} && \text{at $t=\infty$} \\\end{align*}
The ramification defect of $L_X$ is $1$.

\subsection{$\MW{4}{8}$} \label{operator:MW^4_8}  \hfill [\hyperref[sec:MW^4_8]{description p.~\pageref*{sec:MW^4_8}}, \hyperref[table:index_2]{regularized quantum period p.~\pageref*{table:index_2}}]

The quantum differential operator is:
{\small
  \begin{multline*}
    (2t-1)(2t+1)(8000t^6+528t^4+60t^2-1)(64404500t^8-1791160t^6+729408t^4+2144t^2+5) D^{6}  \\
    \shoveleft{+ 4\left(10304720000000t^{16}-1452885248000t^{14}+158410065280t^{12}\right.}\\
    \shoveright{\left.-11837396096t^{10}-726357052t^8+2787020t^6-1531744t^4-3536t^2-5\right) D^{5}} \\
    \shoveleft{+ 4\left(79346344000000t^{16}-7715142848000t^{14}+1509213811040t^{12}\right.}\\
    \shoveright{\left.-96608493568t^{10}-1083561820t^8+48291628t^6+2808508t^4+5602t^2+5\right) D^{4}}\\
    \shoveleft{+ 32t^2(37612228000000t^{14}-2748389976000t^{12}+827208001440t^{10}}\\
    \shoveright{-40569448408t^8-339211968t^6+5857560t^4-32366t^2-51) D^{3}}\\
    \shoveleft{+ 32t^2(74258388500000t^{14}-4572988268000t^{12}+1872888803480t^{10}}\\
    \shoveright{-62657669272t^8-239219588t^6-13755238t^4-70290t^2-15) D^{2}}\\
    \shoveleft{+512t^4(4524416125000t^{12}-258077892500t^{10}+128903007200t^8}\\
    \shoveright{-2846763420t^6-5734916t^4-1401952t^2-4225) D} \\
    +3072t^4(281769687500t^{12}-15710668750t^{10}+8827474000t^8-131446365t^6-200152t^4-95918t^2-245)
  \end{multline*}}
The local log-monodromies for the quantum local system:
\begin{align*}
& \tiny \begin{pmatrix}
0&1&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&1&0&0\\
0&0&0&0&1&0\\
0&0&0&0&0&1\\
0&0&0&0&0&0
\end{pmatrix} && \text{at $\textstyle t=0$} \\& \tiny \begin{pmatrix}
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&1\\
0&0&0&0&0&0
\end{pmatrix} && \text{at $\textstyle t=\frac{1}{2}$} \\& \tiny \begin{pmatrix}
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&1\\
0&0&0&0&0&0
\end{pmatrix} && \text{at $\textstyle t=-\frac{1}{2}$} \\& \tiny \begin{pmatrix}
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&1\\
0&0&0&0&0&0
\end{pmatrix} && \text{at the roots of $\textstyle 8000t^6+528t^4+60t^2-1=0$} \\& \tiny \begin{pmatrix}
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&1\\
0&0&0&0&0&0
\end{pmatrix} && \text{at $t=\infty$} \\\end{align*}
The ramification defect of $L_X$ is $1$.

\subsection{$\MW{4}{9}$} \label{operator:MW^4_9}  \hfill [\hyperref[sec:MW^4_9]{description p.~\pageref*{sec:MW^4_9}}, \hyperref[table:index_2]{regularized quantum period p.~\pageref*{table:index_2}}]

The quantum differential operator is:
{\small
  \begin{multline*}
    (176t^4-144t^3+20t^2+8t-1)(176t^4+144t^3+20t^2-8t-1)(2109888t^6+174528t^4-2941t^2+2) D^{6} \\
    \shoveleft{+ 2\left(718914797568t^{14}-137405030400t^{12}-5419619584t^{10}\right.}\\
    \shoveright{\left.+2224774528t^8-45409520t^6-364024t^4+8407t^2-4\right) D^{5}} \\
    \shoveleft{+ 4\left(3104404807680t^{14}-245927227392t^{12}-51116043392t^{10}\right.}\\
    \shoveright{\left.+3834427616t^8-24121292t^6+553996t^4-5471t^2+2\right) D^{4}} \\
    \shoveleft{+ 8t^2(6698978795520t^{12}-43313504256t^{10}-101803764992t^8}\\
    \shoveright{+3311154624t^6-46433864t^4-300504t^2-107) D^{3}} \\
    \shoveright{+  64t^2(1888172529408t^{12}+75804905088t^{10}-23177683476t^8+452012856t^6-8043103t^4-46340t^2-4) D^{2}} \\
    \shoveright{+ 4608t^4(28905231168t^{10}+2010341344t^8-284249759t^6+4423780t^4-79284t^2-316) D} \\
    +
    55296t^4(992819520t^{10}+86769760t^8-8133271t^6+123438t^4-1829t^2-4)
  \end{multline*}}
The local log-monodromies for the quantum local system:
\begin{align*}
& \tiny \begin{pmatrix}
0&1&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&1&0&0\\
0&0&0&0&1&0\\
0&0&0&0&0&1\\
0&0&0&0&0&0
\end{pmatrix} && \text{at $\textstyle t=0$} \\& \tiny \begin{pmatrix}
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&1\\
0&0&0&0&0&0
\end{pmatrix} && \text{at the roots of $\textstyle 176t^4-144t^3+20t^2+8t-1=0$} \\& \tiny \begin{pmatrix}
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&1\\
0&0&0&0&0&0
\end{pmatrix} && \text{at the roots of $\textstyle 176t^4+144t^3+20t^2-8t-1=0$} \\\end{align*}
The operator $L_X$ is extremal.

\pagebreak

\subsection{$\MW{4}{10}$} \label{operator:MW^4_10}  \hfill [\hyperref[sec:MW^4_10]{description p.~\pageref*{sec:MW^4_10}}, \hyperref[table:index_2]{regularized quantum period p.~\pageref*{table:index_2}}]

The quantum differential operator is:
{\small
  \begin{multline*}
    (4t^2+1)(16384t^6+512t^4+44t^2-1)(1473757184t^8+13746176t^6+7592448t^4+46808t^2+55) D^{6} \\
    \shoveleft{+ 4\left(482920754053120t^{16}+86447658893312t^{14}+4886210281472t^{12}+534951231488t^{10}\right.}\\
    \shoveright{\left.+18501230592t^8+119944000t^6+16234576t^4+72412t^2+55\right) D^{5}} \\
    \shoveleft{+ 4\left(3718489806209024t^{16}+396693178679296t^{14}+31496439267328t^{12}+3577124749312t^{10}\right.}\\
    \shoveright{\left.+37524877312t^8-202496928t^6-28001956t^4-105906t^2-55\right) D^{4}}\\
    \shoveleft{+ 16t^2(3525321504587776t^{14}+219939516448768t^{12}+31919813689344t^{10}}\\
    \shoveright{+2870602727424t^8+30098031616t^6+29514768t^4+38264t^2+755) D^{3}}\\
    \shoveleft{+ 64t^2(1740023841947648t^{14}+63218454626304t^{12}+17856939016192t^{10}}\\
    \shoveright{+1163732019200t^8+10148827776t^6+77992960t^4+145739t^2+55) D^{2}} \\
    \shoveleft{+ 256t^4(424064787152896t^{12}+8915881033728t^{10}+4878831042560t^8}\\
    \shoveright{+233990370304t^6+2025100864t^4+24310608t^2+53911) D} \\
    + 30720t^4(1320486436864t^{12}+15835594752t^{10}+16557547520t^8+617585152t^6+5801632t^4+78192t^2+187)
  \end{multline*}}
The local log-monodromies for the quantum local system:
\begin{align*}
& \tiny \begin{pmatrix}
0&1&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&1&0&0\\
0&0&0&0&1&0\\
0&0&0&0&0&1\\
0&0&0&0&0&0
\end{pmatrix} && \text{at $\textstyle t=0$} \\& \tiny \begin{pmatrix}
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&1\\
0&0&0&0&0&0
\end{pmatrix} && \text{at the roots of $\textstyle 4t^2+1=0$} \\& \tiny \begin{pmatrix}
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&1\\
0&0&0&0&0&0
\end{pmatrix} && \text{at the roots of $\textstyle 16384t^6+512t^4+44t^2-1=0$} \\& \tiny \begin{pmatrix}
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&1\\
0&0&0&0&0&0
\end{pmatrix} && \text{at $t=\infty$} \\\end{align*}
The ramification defect of $L_X$ is $1$.

\subsection{$\MW{4}{11}$} \label{operator:MW^4_11}  \hfill [\hyperref[sec:MW^4_11]{description p.~\pageref*{sec:MW^4_11}}, \hyperref[table:index_2]{regularized quantum period p.~\pageref*{table:index_2}}]

The quantum differential operator is:
{\small
  \begin{multline*}
    (176t^4-96t^3+8t^2-4t-1)(176t^4+96t^3+8t^2+4t-1)(46963840t^6-6320080t^4+10817t^2-60) D^{6} \\
    \shoveleft{+ 2\left(16002270986240t^{14}-4453229608960t^{12}+179162526976t^{10}+27549781760t^8\right.}\\
    \shoveright{\left.-288092960t^6+25314336t^4-28611t^2+120\right) D^{5}} \\
    \shoveleft{+ 4\left(69100715622400t^{14}-16714357862400t^{12}+879319798400t^{10}\right.} \\
    \shoveright{\left.+38172435520t^8-54159304t^6-23633284t^4+6819t^2-60\right) D^{4}}\\
    \shoveleft{+ 8t^2(149112070553600t^{12}-33265162726400t^{10}+1526941953280t^8}\\
    \shoveright{+17140032640t^6-188273072t^4+3160464t^2+1615) D^{3}}\\
    \shoveleft{+ 128t^2(21014345918720t^{12}-4511525020640t^{10}+157602014826t^8}\\
    \shoveright{-362882280t^6+22427697t^4+282960t^2+30) D^{2}}\\
    \shoveright{+ 512t^4(5790594508160t^{10}-1225795140240t^8+31283552519t^6-265518840t^4+10109387t^2+49308) D } \\
    + 15360t^4(79556744960t^{10}-16784238240t^8+319629734t^6-3585072t^4+146945t^2+456)
  \end{multline*}}
  The local log-monodromies for the quantum local system:
\begin{align*}
& \tiny \begin{pmatrix}
0&1&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&1&0&0\\
0&0&0&0&1&0\\
0&0&0&0&0&1\\
0&0&0&0&0&0
\end{pmatrix} && \text{at $\textstyle t=0$} \\& \tiny \begin{pmatrix}
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&1\\
0&0&0&0&0&0
\end{pmatrix} && \text{at the roots of $\textstyle 176t^4-96t^3+8t^2-4t-1=0$} \\& \tiny \begin{pmatrix}
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&1\\
0&0&0&0&0&0
\end{pmatrix} && \text{at the roots of $\textstyle 176t^4+96t^3+8t^2+4t-1=0$} \\\end{align*}
The operator $L_X$ is extremal.

\subsection{$\MW{4}{12}$} \label{operator:MW^4_12}  \hfill [\hyperref[sec:MW^4_12]{description p.~\pageref*{sec:MW^4_12}}, \hyperref[table:index_2]{regularized quantum period p.~\pageref*{table:index_2}}]

The quantum differential operator is:
{\small
  \begin{multline*}
    (16t^4+44t^2-1)(432t^4+36t^2+1)(314924112t^8-117964512t^6+14238144t^4+164850t^2+221) D^{6} \\
    \shoveleft{+ 4\left(10883777310720t^{16}+14017882540032t^{14}-7127443839744t^{12}\right.}\\
    \shoveright{\left.+913572411264t^{10}+49215528432t^8+208091040t^6+29910072t^4+249043t^2+221\right) D^{5}}\\
    \shoveleft{+4\left(83805085292544t^{16}+42638961696768t^{14}-38573795723328t^{12}+6607519967520t^{10}\right.}\\
    \shoveright{\left.+197257285008t^8+2651761872t^6-52981980t^4-386208t^2-221\right) D^{4}}\\
    \shoveleft{+ 16t^2(79451574368256t^{14}+656487804672t^{12}-24209536131072t^{10}}\\
    \shoveright{+5352527374416t^8+119397810240t^6+1063433832t^4+1841528t^2+1513) D^{3}}\\
    \shoveleft{+ 32t^2(78431220245376t^{14}-24146078481216t^{12}-14886119891520t^{10}}\\
    \shoveright{+4353594087744t^8+89900843112t^6+822085824t^4+1734954t^2+221) D^{2}}\\
    \shoveleft{+ 2304t^4(1061924105664t^{12}-538846899984t^{10}-114223084416t^8}\\
    \shoveright{+48172072716t^6+1084223724t^4+9219514t^2+22737) D}\\
    + 27648t^4(33067031760t^{12}-20911507560t^{10}-1683208512t^8+1243973394t^6+32605206t^4+248744t^2+663)
  \end{multline*}}
The local log-monodromies for the quantum local system:
\begin{align*}
& \tiny \begin{pmatrix}
0&1&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&1&0&0\\
0&0&0&0&1&0\\
0&0&0&0&0&1\\
0&0&0&0&0&0
\end{pmatrix} && \text{at $\textstyle t=0$} \\& \tiny \begin{pmatrix}
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&1\\
0&0&0&0&0&0
\end{pmatrix} && \text{at the roots of $\textstyle 432t^4+36t^2+1=0$} \\& \tiny \begin{pmatrix}
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&1\\
0&0&0&0&0&0
\end{pmatrix} && \text{at the roots of $\textstyle 16t^4+44t^2-1=0$} \\& \tiny \begin{pmatrix}
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&1\\
0&0&0&0&0&0
\end{pmatrix} && \text{at $t=\infty$} \\\end{align*}
The ramification defect of $L_X$ is $1$.

\pagebreak

\subsection{$\MW{4}{13}$} \label{operator:MW^4_13} \hfill [\hyperref[sec:MW^4_13]{description p.~\pageref*{sec:MW^4_13}}, \hyperref[table:index_2]{regularized quantum period p.~\pageref*{table:index_2}}]

The quantum differential operator is:
{\small
  \begin{multline*}
    (2t-1)(2t+1)(136t^3-20t^2+6t-1)(136t^3+20t^2+6t+1)(66683996t^6-1058780t^4-19394t^2+15) D^{6} \\
    \shoveleft{+ 4\left(27134518180352t^{14}-3636686279168t^{12}-76696829376t^{10}\right.}\\
    \shoveright{\left.+3834603264t^8-107218310t^6+2061400t^4+29091t^2-15\right) D^{5}}\\
    \shoveleft{+ 4\left(234343566103040t^{14}-21804386465792t^{12}+102596245216t^{10}\right.}\\
    \shoveright{\left.+15956069184t^8+542705432t^6-1694520t^4-36762t^2+15\right) D^{4}}\\
    \shoveleft{+ 32t^2(126422186976640t^{12}-8667906294144t^{10}+129634375864t^8}\\
    \shoveright{+2690207848t^6+17651600t^4-204240t^2-49) D^{3}}\\
    \shoveleft{+ 32t^2(285066614292448t^{12}-15534154687040t^{10}+251203935368t^8}\\
    \shoveright{-291393828t^6-1136928t^4-258660t^2-15) D^{2}}\\
    \shoveright{+ 128t^4(78551346664144t^{10}-3663416463632t^8+48257529676t^6-820898270t^4-4423658t^2-37545) D}\\
    + 3072t^4(1349017239080t^{10}-57143052340t^8+533807468t^6-19445050t^4-103234t^2-345)
  \end{multline*}}
The local log-monodromies for the quantum local system:
\begin{align*}
& \tiny \begin{pmatrix}
0&1&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&1&0&0\\
0&0&0&0&1&0\\
0&0&0&0&0&1\\
0&0&0&0&0&0
\end{pmatrix} && \text{at $\textstyle t=0$} \\& \tiny \begin{pmatrix}
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&1\\
0&0&0&0&0&0
\end{pmatrix} && \text{at $\textstyle t=\frac{1}{2}$} \\& \tiny \begin{pmatrix}
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&1\\
0&0&0&0&0&0
\end{pmatrix} && \text{at $\textstyle t=-\frac{1}{2}$} \\& \tiny \begin{pmatrix}
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&1\\
0&0&0&0&0&0
\end{pmatrix} && \text{at the roots of $\textstyle 136t^3-20t^2+6t-1=0$} \\& \tiny \begin{pmatrix}
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&1\\
0&0&0&0&0&0
\end{pmatrix} && \text{at the roots of $\textstyle 136t^3+20t^2+6t+1=0$} \\\end{align*}
The operator $L_X$ is extremal.

\subsection{$\MW{4}{14}$} \label{operator:MW^4_14}  \hfill [\hyperref[sec:MW^4_14]{description p.~\pageref*{sec:MW^4_14}}, \hyperref[table:index_2]{regularized quantum period p.~\pageref*{table:index_2}}]

The quantum differential operator is:
{\small
  \begin{multline*}
    (2t-1)(2t+1)(6t-1)(6t+1)(20t^2-4t+1)(20t^2+4t+1)(127920t^6-45016t^4-293t^2-1) D^{6} \\
    \shoveleft{+ 2\left(81050112000t^{14}-42347458560t^{12}+4138403072t^{10}\right.}\\
    \shoveright{\left.+109225472t^8+95280t^6+187248t^4+911t^2+2\right) D^{5}}\\
    \shoveleft{+ 4\left(349989120000t^{14}-177028331520t^{12}+12857605632t^{10}\right.}\\
    \shoveright{\left.+227619584t^8+1578760t^6-171496t^4-769t^2-1\right) D^{4}}\\
    \shoveleft{+ 8t^2(755239680000t^{12}-383806594560t^{10}+19129972096t^8}\\
    \shoveright{+278237408t^6+1853984t^4+10304t^2+13) D^{3}}\\
    \shoveright{+ 32t^2(425743344000t^{12}-221217604800t^{10}+7159648792t^8+109530520t^6+1224240t^4+4582t^2+1) D^{2}}\\
    \shoveright{+ 768t^4(19552572000t^{10}-10422296720t^8+211812198t^6+4163952t^4+55827t^2+173) D}\\
    + 23040t^4(268632000t^{10}-146189120t^8+1864348t^6+52366t^4+729t^2+2)
  \end{multline*}}
The local log-monodromies for the quantum local system:
\begin{align*}
& \tiny \begin{pmatrix}
0&1&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&1&0&0\\
0&0&0&0&1&0\\
0&0&0&0&0&1\\
0&0&0&0&0&0
\end{pmatrix} && \text{at $\textstyle t=0$} \\& \tiny \begin{pmatrix}
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&1\\
0&0&0&0&0&0
\end{pmatrix} && \text{at $\textstyle t=\frac{1}{2}$} \\& \tiny \begin{pmatrix}
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&1\\
0&0&0&0&0&0
\end{pmatrix} && \text{at $\textstyle t=\frac{1}{6}$} \\& \tiny \begin{pmatrix}
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&1\\
0&0&0&0&0&0
\end{pmatrix} && \text{at $\textstyle t=-\frac{1}{6}$} \\& \tiny \begin{pmatrix}
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&1\\
0&0&0&0&0&0
\end{pmatrix} && \text{at $\textstyle t=-\frac{1}{2}$} \\& \tiny \begin{pmatrix}
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&1\\
0&0&0&0&0&0
\end{pmatrix} && \text{at the roots of $\textstyle 20t^2-4t+1=0$} \\& \tiny \begin{pmatrix}
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&1\\
0&0&0&0&0&0
\end{pmatrix} && \text{at the roots of $\textstyle 20t^2+4t+1=0$} \\\end{align*}
The operator $L_X$ is extremal.

\subsection{$\MW{4}{15}$} \label{operator:MW^4_15}  \hfill [\hyperref[sec:MW^4_15]{description p.~\pageref*{sec:MW^4_15}}, \hyperref[table:index_2]{regularized quantum period p.~\pageref*{table:index_2}}]

The quantum differential operator is:
{\small
  \begin{multline*}
    (416t^4-144t^3+8t^2+4t-1)(416t^4+144t^3+8t^2-4t-1)(119179008t^6+10942640t^4+192779t^2+980) D^{6} \\
    \shoveleft{+ 2\left(226871066492928t^{14}+10977991065600t^{12}-661584438272t^{10}\right.}\\
    \shoveright{\left.-1433003264t^8-363743360t^6-47681568t^4-641057t^2-1960\right) D^{5}}\\
    \shoveleft{+ 4\left(979670514401280t^{14}+74598187089920t^{12}-1635236114944t^{10}\right.}\\
    \shoveright{\left.-46841533568t^8+1268915136t^6+55009348t^4+503977t^2+980\right) D^{4}}\\
    \shoveleft{+ 8t^2(2114025846865920t^{12}+203938713538560t^{10}-1101057117184t^8}\\
    \shoveright{-117410965440t^6-475968000t^4+3889200t^2-12299) D^{3}}\\
    \shoveleft{+ 128t^2(297929404790784t^{12}+33127871570560t^{10}+111493204952t^8}\\
    \shoveright{-15417741170t^6-129552291t^4+148450t^2-245) D^{2}}\\
    \shoveright{+ 256t^4(164191489173504t^{10}+19958501546880t^8+167108388648t^6-7614297740t^4-82910507t^2-24740) D}\\
    + 1536t^4(11279101317120t^{10}+1447829510400t^8+16271407536t^6-474232070t^4-5907869t^2-4760)
  \end{multline*}}
The local log-monodromies for the quantum local system:
\begin{align*}
& \tiny \begin{pmatrix}
0&1&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&1&0&0\\
0&0&0&0&1&0\\
0&0&0&0&0&1\\
0&0&0&0&0&0
\end{pmatrix} && \text{at $\textstyle t=0$} \\& \tiny \begin{pmatrix}
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&1\\
0&0&0&0&0&0
\end{pmatrix} && \text{at the roots of $\textstyle 416t^4-144t^3+8t^2+4t-1=0$} \\& \tiny \begin{pmatrix}
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&1\\
0&0&0&0&0&0
\end{pmatrix} && \text{at the roots of $\textstyle 416t^4+144t^3+8t^2-4t-1=0$} \\\end{align*}
The operator $L_X$ is extremal.

\subsection{$\MW{4}{16}$} \label{operator:MW^4_16}  \hfill [\hyperref[sec:MW^4_16]{description p.~\pageref*{sec:MW^4_16}}, \hyperref[table:index_2]{regularized quantum period p.~\pageref*{table:index_2}}]

The quantum differential operator is:
{\small
  \begin{multline*}
    (8t^2-4t+1)(8t^2+4t+1)(28t^2-4t-1)(28t^2+4t-1)(1245440t^6+159472t^4-353t^2+2) D^{6} \\
    \shoveleft{+ 2\left(687403171840t^{14}+55847092224t^{12}-2940634112t^{10} \right.}\\
    \shoveright{\left.+646645248t^8-7816080t^6-602296t^4+771t^2-4\right) D^{5}}\\
    \shoveleft{+ 4\left(2968331878400t^{14}+342433251328t^{12}-16569335296t^{10}\right.}\\
    \shoveright{\left.+962246144t^8+20691276t^6+609636t^4-327t^2+2\right) D^{4}}\\
    \shoveright{+ 8t^2(6405347737600t^{12}+905030160384t^{10}-30840349696t^8+529133120t^6+4111656t^4-69016t^2-79) D^{3}}\\
    \shoveright{+ 64t^2(1805409751040t^{12}+290148947200t^{10}-6893123664t^8+13678020t^6-1333787t^4-13834t^2-3) D^{2}}\\
    \shoveright{+ 256t^4(497488517120t^{10}+86925519744t^8-1508603672t^6-6132868t^4-554137t^2-2494) D}\\
    + 1536t^4(34174873600t^{10}+6291237120t^8-85724560t^6-530810t^4-38831t^2-116)
  \end{multline*}}
The local log-monodromies for the quantum local system:
\begin{align*}
& \tiny \begin{pmatrix}
0&1&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&1&0&0\\
0&0&0&0&1&0\\
0&0&0&0&0&1\\
0&0&0&0&0&0
\end{pmatrix} && \text{at $\textstyle t=0$} \\& \tiny \begin{pmatrix}
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&1\\
0&0&0&0&0&0
\end{pmatrix} && \text{at the roots of $\textstyle 8t^2-4t+1=0$} \\& \tiny \begin{pmatrix}
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&1\\
0&0&0&0&0&0
\end{pmatrix} && \text{at the roots of $\textstyle 28t^2-4t-1=0$} \\& \tiny \begin{pmatrix}
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&1\\
0&0&0&0&0&0
\end{pmatrix} && \text{at the roots of $\textstyle 28t^2+4t-1=0$} \\& \tiny \begin{pmatrix}
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&1\\
0&0&0&0&0&0
\end{pmatrix} && \text{at the roots of $\textstyle 8t^2+4t+1=0$} \\\end{align*}
The operator $L_X$ is extremal.

\pagebreak

\subsection{$\MW{4}{17}$} \label{operator:MW^4_17}  \hfill [\hyperref[sec:MW^4_17]{description p.~\pageref*{sec:MW^4_17}}, \hyperref[table:index_2]{regularized quantum period p.~\pageref*{table:index_2}}]

The quantum differential operator is:
{\small
  \begin{multline*}
    (2752t^6-1152t^5+224t^4+96t^3-52t^2+12t-1)(2752t^6+1152t^5+224t^4-96t^3-52t^2-12t-1)\\
    \shoveleft{ (4516691026120601600t^{18}+10894175535784019520t^{16}+161057014788668272t^{14}+223186423846901825t^{12}}\\
    \shoveright{+1489656860655194t^{10}-37076036387883t^8-944190030122t^6+1145046509t^4+34077463t^2+3136) D^{8}} \\
    \shoveleft{+ 2\left(718350728614858094700134400t^{30}+1808773177510341584824565760t^{28}\right.}\\
+11029946474586078383243264t^{26}+38551285621538701581713408t^{24}-573876948723528931270656t^{22}\\
-55410907815277159905280t^{20}-10868619181236655325696t^{18}-205717988100675369984t^{16}\\
+54690133059712073568t^{14}-2288987306843185052t^{12}-22309755134355486t^{10}\\
\shoveright{\left.+253350406694334t^8+8580745232122t^6-7798176720t^4-238542241t^2-18816\right) D^{7}}\\
\shoveleft{+ 4\left(6234258109050375607576166400t^{30}+16563148489700147386089144320t^{28}\right.}\\
+153884025675978814793531392t^{26}+396376397592542781690715136t^{24}-244086311073605059692544t^{22}\\
-293272503356977770941696t^{20}-34041797107502106577312t^{18}+652775043341067168068t^{16}\\
-44917328848849757800t^{14}+11574061561240458564t^{12}+61805608644992588t^{10}\\
\shoveright{\left.-1336778298658238t^8-25795200795638t^6+35399715688t^4+581693399t^2+40768\right) D^{6}} \\
\shoveleft{+8\left(29067549125736936474830438400t^{30}+81851870301201769764088381440t^{28}\right.}\\
+939457832323447863369383936t^{26}+2148139133199504182578666496t^{24}+5371023638579317537625088t^{22}\\
-957869720387013465562880t^{20}-5020278555805201268384t^{18}-1791882536502223335404t^{16}\\
-85650713028232429760t^{14}-20843283163258659946t^{12}-101280721448407430t^{10}\\
\shoveright{\left.+1741410204251200t^8+30185303650830t^6-49184868070t^4-583650151t^2-37632\right) D^{5}}\\
\shoveleft{+16\left(79230237058744125391346073600t^{30}+236991057245518887863871979520t^{28}\right.}\\
+3064756009927980485218619392t^{26}+6725558154012574042351982336t^{24}+25714515380315969257756928t^{22}\\
-2214587183336657609276512t^{20}+14905942833336792786992t^{18}+2425296396558205778092t^{16}\\
+70030906648265241400t^{14}+13039662696372133523t^{12}+68594134702584874t^{10}\\
\shoveright{\left.-642583292426727t^8-12336212241784t^6+23535298811t^4+206616410t^2+12544\right) D^{4}}\\
\shoveleft{+64t^2(64244286367595768799677644800t^{28}+203835202251539030078676664320t^{26}}\\
+2826396757245234563525490688t^{24}+6176058394659175094246477056t^{22}+30060979005675148672022400t^{20}\\
-1750848106552043666387424t^{18}-31680726103451488064224t^{16}+328919199593772520366t^{14}\\
+33052797280494574660t^{12}-311682535251628245t^{10}-3943292702734802t^8\\
\shoveright{-5555362326911t^6+105428531260t^4-716272453t^2-28448) D^{3}}\\
\shoveleft{+512t^2(15009134071247813042455347200t^{28}+50222389883894861028705538560t^{26}}\\
+725335071957348016538310016t^{24}+1603230224901344022068592328t^{22}+9154893897382658634810352t^{20}\\
-428192374996045407448412t^{18}-15519523249001954206792t^{16}+52781806444610795042t^{14}\\
-1276462080564820298t^{12}-107386844952242335t^{10}-792211501649686t^8\\
\shoveright{-5580848553730t^6+16358540366t^4-35218463t^2-1568) D^{2}}\\
\shoveleft{+6144t^6(1216860211646830162557747200t^{24}+4254138344852923650551665920t^{22}}\\
+62879395621367262268113088t^{20}+141203555473546515828319348t^{18}+892698338486015205162136t^{16}\\
-36871341336439978899176t^{14}-1637218553379789670928t^{12}+2207989224928555682t^{10}\\
\shoveright{-562698415042789318t^8-9517978064339709t^6-42261757561740t^4-328350172920t^2+368309382) D}\\
\shoveleft{+15482880t^6(183729957560533831884800t^{24}+663265247268250820234880t^{22}}\\
+9929341713271041302112t^{20}+22607862359961003907882t^{18}+151963033359678362364t^{16}\\
-5846359991941694249t^{14}-276152453516510822t^{12}+64100253358393t^{10}\\
-116439515097332t^8-1405235953421t^6-3886412020t^4-21911230t^2-41272)
  \end{multline*}}
The local log-monodromies for the quantum local system:
\begin{align*}
& \tiny \begin{pmatrix}
0&1&0&0&0&0&0&0\\
0&0&0&0&0&0&0&0\\
0&0&0&1&0&0&0&0\\
0&0&0&0&0&0&0&0\\
0&0&0&0&0&1&0&0\\
0&0&0&0&0&0&1&0\\
0&0&0&0&0&0&0&1\\
0&0&0&0&0&0&0&0
\end{pmatrix} && \text{at $\textstyle t=0$} \\& \tiny \begin{pmatrix}
0&0&0&0&0&0&0&0\\
0&0&0&0&0&0&0&0\\
0&0&0&0&0&0&0&0\\
0&0&0&0&0&0&0&0\\
0&0&0&0&0&0&0&0\\
0&0&0&0&0&0&0&0\\
0&0&0&0&0&0&0&1\\
0&0&0&0&0&0&0&0
\end{pmatrix} && \text{at the roots of $\textstyle 2752t^6-1152t^5+224t^4+96t^3-52t^2+12t-1=0$} \\& \tiny \begin{pmatrix}
0&0&0&0&0&0&0&0\\
0&0&0&0&0&0&0&0\\
0&0&0&0&0&0&0&0\\
0&0&0&0&0&0&0&0\\
0&0&0&0&0&0&0&0\\
0&0&0&0&0&0&0&0\\
0&0&0&0&0&0&0&1\\
0&0&0&0&0&0&0&0
\end{pmatrix} && \text{at the roots of $\textstyle 2752t^6+1152t^5+224t^4-96t^3-52t^2-12t-1=0$} \\\end{align*}
The ramification defect of $L_X$ is $1$.

\subsection{$\MW{4}{18}$} \label{operator:MW^4_18}  \hfill [\hyperref[sec:MW^4_18]{description p.~\pageref*{sec:MW^4_18}}, \hyperref[table:index_2]{regularized quantum period p.~\pageref*{table:index_2}}]

The quantum differential operator is:
{\small
  \begin{multline*}
    (4t-1)(4t+1)(8t-1)(8t+1) D^{4} +
    64t^2(128t^2-5) D^{3} +
    16t^2(1472t^2-33) D^{2} + \\
    32t^2(896t^2-13) D +
    128t^2(96t^2-1)
  \end{multline*}}
The local log-monodromies for the quantum local system:
\begin{align*}
& \tiny \begin{pmatrix}
0&1&0&0\\
0&0&1&0\\
0&0&0&1\\
0&0&0&0
\end{pmatrix} && \text{at $\textstyle t=0$} \\& \tiny \begin{pmatrix}
0&0&0&0\\
0&0&0&0\\
0&0&0&1\\
0&0&0&0
\end{pmatrix} && \text{at $\textstyle t=\frac{1}{4}$} \\& \tiny \begin{pmatrix}
0&0&0&0\\
0&0&0&0\\
0&0&0&1\\
0&0&0&0
\end{pmatrix} && \text{at $\textstyle t=\frac{1}{8}$} \\& \tiny \begin{pmatrix}
0&0&0&0\\
0&0&0&0\\
0&0&0&1\\
0&0&0&0
\end{pmatrix} && \text{at $\textstyle t=-\frac{1}{8}$} \\& \tiny \begin{pmatrix}
0&0&0&0\\
0&0&0&0\\
0&0&0&1\\
0&0&0&0
\end{pmatrix} && \text{at $\textstyle t=-\frac{1}{4}$} \\& \tiny \begin{pmatrix}
0&0&0&0\\
0&0&0&0\\
0&0&0&1\\
0&0&0&0
\end{pmatrix} && \text{at $t=\infty$} \\\end{align*}
The operator $L_X$ is extremal.

redpajama_set_name: RedPajamaArXiv