text:
\begin{document}

\maketitle
\begin{abstract}
A sheaf quantization is a sheaf associated to a Lagrangian brane. By using the ideas of exact WKB analysis, spectral networks, and scattering diagrams, we sheaf-quantize spectral curves over the Novikov ring under some assumptions on the behavior of Stokes curves.

For Schr\"odinger equations, we prove that the local system associated to the sheaf quantization (microlocalization a.k.a. abelianization) over the spectral curve can be identified with the Voros--Iwaki--Nakanishi coordinate and the Fock--Goncharov coordinate.

We expect that these sheaf quantizations are the object-level realizations of the $\hbar$-enhanced Riemann--Hilbert correspondence. In the subsequent work, we confirm this for the case of Schr\"odinger equations.

\end{abstract}
\setcounter{tocdepth}{1}
\tableofcontents
\section{Introduction}
A quantization of a Lagrangian submanifold $L$ usually means a module supported on $L$ over a quantized symplectic manifold. For example, a $\cD$-module on a complex manifold is a quantization of its characteristic variety. Via the Riemann--Hilbert correspondence, one can introduce a notion of quantization with more topological flavor: {\em sheaf quantization}. A sheaf is a sheaf quantization of its microsupport $\SS$~\cite{KS}.

Characteristic varieties and microsupports are always conic. To quantize non-conic Lagrangians, it is classical to use $\cD_\hbar$-modules. A $\cD_\hbar$-module $\cE$ is a quantization of $\supp(\cE/\hbar\cE)$. Tamarkin introduced an analogue of this trick on the Riemann--Hilbert dual side~\cite{Tam}. Namely, for a real manifold $M$, consider the product $T^*M\times T^*_{\tau>0}\bR_t$ where $\bR_t$ is the real line coordinatized by $t$ and $\tau$ is the dual cotangent coordinate of $t$. Let $\rho$ be a map defined by $\rho\colon T^*M\times T^*_{\tau>0}\bR_t\rightarrow T^*M; (x, \xi, t,\tau)\mapsto (x, \xi/\tau)$ where $x\in M$ and $\xi\in T^*_xM$. We say that a sheaf $\cE$ on $M\times \bR$ is a {\em sheaf quantization} of $\rho(\SS(\cE)\cap (T^*M\times T^*_{\tau>0}\bR_t))$. This idea is quite useful to study symplectic topology. For example, Tamarkin proved non-displacement results~\cite{Tam} and Guillermou proved nearby Lagrangian results~\cite{Guillermou}. The statements obtained by this method are parallel to those obtained by Floer-theoretic methods. In fact, one of Tamarkin's first motivations to introduce $\bR_t$ is to realize the Novikov ring $\Lambda_0$ in sheaf theory. Also, Jin--Treumann~\cite{JT} proved a sheaf quantization is associated to a Lagrangian brane, which is nothing but a necessary condition to be an object of Fukaya category.

In the first version of quantization (using $\cD$-modules), via the Riemann--Hilbert correspondence, a sheaf quantization over a complex manifold is equivalent to a quantization by a regular holonomic $\cD$-module. However, in the second version (using $\cD_\hbar$-modules), there is no such Riemann--Hilbert correspondence, so the relation is not clear.

One of the purpose of this paper is to reveal a part of the relationship explicitly, using exact WKB analysis. We will explore it more abstractly by constructing a functor relating $\cD^\hbar$-modules and sheaf quantizations in the subsequent publication~\cite{hRH}

Naively speaking, the Riemann--Hilbert functor is about solving differential equations. Similarly, in our case, we have to solve $\cD_\hbar$-modules. The appropriate method is {\em exact WKB analysis}~\cite{Voros, DDP, KawaiTakei}. Classically, the WKB approximation means a semi-classical approximation of quantum mechanics obtained by considering the Planck constant very small. Exact WKB analysis is a method to solve differential equations with $\hbar$ along this line. We solve a given differential equation with special formal power series (WKB solutions) in $\hbar$ and try to sum up the series by Borel's method.

Exact WKB analysis is quite successful for the second order differential equations over complex 1-dimensional spaces. The result is, roughly speaking, for each $\hbarr\in \bC^\times_\hbar$, we have a chamber decomposition of the base space, solutions over each chamber, and connection formulas between solutions in different chambers. For us, this data give a gluing of sheaves.

\begin{theorem}
Let $C$ be a Riemann surface and $\cM$ be a rank 2 flat meromorphic $\hbar$-connection which is weakly GMN (Definition~\ref{weaklyGMN}) and satisfies Assumption~\ref{WKBregular}. For a sufficiently small $\hbarr \in \bC^{\times}_\hbar$, there exist two sheaf quantizations $S_{\cM}^{\shbarr, \pm}$ of the spectral curve of $\cM$ as an object of $\mu(T^*C, \Re(\lambda/\hbarr), \sqrt{-1}\Im(\lambda/\hbarr),\Lambda_0)$.
\end{theorem}

We do not explain the definition of $\mu(T^*X, \Re(\lambda/\hbarr), \sqrt{-1}\Im(\lambda/\hbarr),\Lambda_0)$ here, but we would like to make some comments. In the previous works~\cite{Guillermou, JT}, to sheaf-quantize a Lagrangian, we have to assume that the Lagrangian is exact. Since spectral curves are not exact, we will use $\bR$-equivariant sheaves. This equivariance is compatible with the Novikov ring realization. We moreover incorporate {\em B-fields} in sheaf quantization. The category $\mu(T^*X, \Re(\lambda/\hbarr), \sqrt{-1}\Im(\lambda/\hbarr),\Lambda_0)$ is a combination of such ideas. The construction is partially motivated by the author's category of $\Lambda$-modules~\cite{Kuw}. Also, a similar construction will be used in our forthcoming joint work with Shende to construct Fukaya categories of compact surfaces using microlocal sheaves.

In the case of higher order differential equations, there are two major problems in exact WKB analysis:
\begin{enumerate}
    \item The summability of WKB solutions is not well-established.
    \item We cannot obtain a chamber decomposition in a similar way as in the quadratic case, since there exist collisions of Stokes curves (chamber walls)~\cite{BNR} and the collisions give anomalous monodromies.
\end{enumerate}
 In this paper, we will only address the second problem. To cancel the anomalous monodromies, we have to introduce new Stokes curves emanating from the collisions~\cite{BNR}. Then the new Stokes curves may have more collisions. Hence, in general, we have to add infinitely many new Stokes curves. This phenomenon is recently rediscovered in physics in the work of Gaiotto--Moore--Neitzke as {\em spectral network}~\cite{GMN}.

 We will introduce the notion of {\em inductive tameness}, to have a tame behavior of new Stokes curves in each step. Then we can organize it into an inductive sequence of data. This is an analogue of the notion of {\em scattering diagram}~\cite{Affinestr, GS} and we call it a {\em spectral scattering diagram}. We can mimic the inductive construction of the scattering diagram and obtain the following:
\begin{theorem}
Let $C$ be a Riemann surface and $\cM$ be a flat meromorphic $\hbar$-connection which is inductively $\hbarr$-tame for $\hbarr\in \bC^\times$, there exists a sheaf quantization of the spectral curve as an object of $\mu(T^*C, \Re(\lambda/\hbarr), \sqrt{-1}\Im(\lambda/\hbarr),\Lambda^{\bC((\hbar))}_0)$.
\end{theorem}
The reason why the base field is $\bC((\hbar))$ is that we do not address the summability. We also discuss a conjectural story of the summability in 9.7.

\begin{remark}
In the literature of exact WKB analysis, there is another approach to collisions of Stokes curves using the notion of {\em virtual turning points}~\cite{AKT}.
\end{remark}

As we have already mentioned, we expect our construction is a kind of Riemann--Hilbert correspondence, {\em $\hbar$-Riemann--Hilbert correspondence}. In the subsequent work~\cite{hRH}, we construct a functor from the category of $\cD_\hbar$-modules to the category of sheaf quantizations, although the coefficients we take is enlarged to obtain the Novikov rings on the side of $\cD_\hbar$-modules. For the second order case, our sheaf quantization here can be comparable to the sheaf quantization obtained via the $\hbar$-RH functor. For the higher order case, we expect that a similar coincidence can be obtained after exact WKB analysis is settled.

\begin{comment}
Namely, we expect the above theorem is a part of the object-level realization of the following conjecture:
\begin{conjecture}
Let $X$ be a complex manifold. Fix $\theta\in \bR/\bZ$, there exists
\begin{enumerate}
    \item an exhausting filtration $\{\cD_{\theta, |\hbarr|}\}_{|\hbarr|\in \bR_{>0}}$ of $D^b_{reg}(\cD_\hbar(X))$ where $reg$ means WKB-regularity,
    \item a family of categories $\{\cS_{\theta, |\hbarr|}\}_{|\hbarr|\in \bR_{>0}}$ defined over the convergent Novikov ring $\Lambda_0^{conv}$ with an inclusion $\cS_{\theta, |\hbarr|}\otimes_{\Lambda_0^{conv}}\Lambda_0\hookrightarrow \mu(T^*X, \Re(\lambda/\hbarr), \sqrt{-1}\Im(\lambda/\hbarr),\Lambda_0)$,
    \item
    a family of fully faithful functors
    $\cD_{\theta, |\hbarr|}\otimes_{\bC[\hbar]}\bC\hookrightarrow \cS_{\theta, |\hbarr|} \otimes_{\Lambda_0^{conv}}\bC$ where the tensor of the LHS is defined by setting $\hbar=\hbarr:=|\hbarr|e^{2\pi\sqrt{-1}\theta}$, the tensor of RHS is defined by setting $T^c=e^{-c}$.
    \end{enumerate}
\end{conjecture}
\end{comment}

The full statement will be stated in the body of the paper.
One can also ask relations to other realizations of irregular Riemann--Hilbert correspondence. We discuss relationships to (1) D'Agnolo--Kashiwara's holonomic Riemann--Hilbert correspondence~\cite{DK}, and (2) Shende--Treumann--Williams--Zaslow's Legendrian knot description~\cite{STWZ}.

As an application of the above comparisons, we consider cluster coordinates of character varieties. For a quadratic differential on $C$, there is an associated Schr\"odinger equation. By choosing a spin structure, we can lift the equation to a meromorphic flat $\hbar$-connection. Then we can apply our main theorem to the connection. Then we can obtain the following statement.
\begin{theorem}
For a quadratic differential $\phi$, the microlocalization of the sheaf quantization $S^{\shbarr, \pm}_{\phi, \sqrt{K}}$ can be identified with
\begin{enumerate}
\item the Voros--Iwaki--Nakanishi coordinate and
\item the Fock--Goncharov coordinate (under the condition that the order of any pole of $\phi$ is greater than $3$).
\end{enumerate}
\end{theorem}
The coincidence of the latter two coordinates was known by Allegretti by a different method~\cite{Allegretti} (see also Iwaki~\cite{Iwaki} for an earlier discussion). In our proof, the second statement can be considered as a comparison of two Lagrangian submanifolds (and their sheaf quantizations) arising from one differential equation.

\begin{remark}
Microlocal sheaf theory has already met exact WKB analysis in \cite{GetmanekoTamarkin}. In contrast to our treatment, they applied microlocal sheaf theory after the Laplace transform. Hence their solution sheaf corresponds to Borel-transformed solutions. It may be interesting to know the relationship between our sheaves and their sheaves.
\end{remark}

\begin{remark}
Recently, Kontsevich--Soibelman~\cite{KonTalk} announced a formulation of Riemann--Hilbert correspondence for deformation quantization. From this point of view, our construction (or our theorem in \cite{hRH}) can be considered as an intermediate step. The other step to prove Kontsevich--Soibelman's conjecture should be a version of Nadler--Zaslow equivalence~\cite{NZ}.
\end{remark}

\subsection*{Acknowledgment}
I would like to thank Kohei Iwaki, Tsukasa Ishibashi, Hiroshi Ohta, and Vivek Shende for many comments and discussions. Especially, Iwaki-san asked me to find relationship between exact WKB analysis and microlocal sheaf theory for some years. I am very happy to write this note as my "homework report" of his lectures. I also would like to thank the members of GHKK reading seminar (Ishibashi, Kano, Mizuno, Oya) where I learned the notion of scattering diagram.
This work was partially supported by World Premier International Research Center Initiative (WPI), MEXT, Japan
and JSPS KAKENHI Grant Number JP18K13405.

\subsection*{Notation}
Our base field $\bK$ is $\bC$ or the formal Laurent series ring $\bC((\hbar))$.

Let $R$ be a commutative ring. Consider $\bR_{\geq 0}$ as a semigroup and we denote the group ring by $R[\bR_{\geq0}]$. We denote the indeterminate of $R[\bR_{\geq 0}]$ by $T^a$ for $a\in \bR_{\geq 0}$. The Novikov ring $\Lambda^R_0$ is the projective limit $\lim_{a\rightarrow \infty}R[\bR_{\geq 0}]/T^aR[\bR_{\geq 0}]$. We denote the fraction field by $\Lambda^R$ and call it the Novikov field. If the context is clear, we omit the superscript $R$.

For $R=\bC$, we say an element $\sum_{c\geq c_0}a_cT^c\in \Lambda^\bC$ is convergent if the series $\sum_{c\geq c_0}a_ce^{-c}$ is absolutely convergent. We denote the subring of $\Lambda^\bC_0$ consisting of convergent elements by $\Lambda_0^{\bC, conv}$.

A category usually means a dg-category in this paper.

We will use $\hbar$ for a general element or the coordinate function of $\bC^\times_\hbar$ and $\hbarr$ for some fixed element.

\section{Microlocal category}
Here we will introduce our category which is the living place of our sheaf quantizations. This category is a refined version of the category introduced by Tamarkin~\cite{Tam}.

Let $M$ be a differentiable manifold. We denote the cotangent bundle of $M$ by $T^*M$. Let $\bR_t$ be the real line coordinatized by $t$. We denote the cotangent coordinate of $T^*\bR_t$ by $\tau.$ We denote the subset of $T^*M\times T^*\bR_t$ defined by $\tau>0$ by $T^*_{\tau>0}(M\times \bR_t)$.

We consider the translation action of $\bR_t$ on $M\times \bR_t$ as a discontinuous action. Let $\Sh^\bR_{\heartsuit}(M\times \bR)$ be the abelian category of $\bR_t$-equivariant sheaves and $\Sh^\bR(M\times \bR_t)$ be the derived dg-category of $\Sh^\bR_\heartsuit(M\times \bR_t)$. Note that the equivariant derived category in the sense of Bernstein--Lunts~\cite{BernsteinLunts} coincide with the naive derived category in this case, since $\bR_t$ is a discrete group. Since $\Sh^\bR_{\heartsuit}(M\times \bR)$ is a Grothendieck abelian category, one can equip $\Sh^\bR(M\times \bR_t)$ with the injective model structure as in \cite{Hovey}. We denote a fibrant replacement functor by $Q$.

We denote the translation along $\bR_t$ by $c\in \bR$ by $T_c$. An object of $\Sh^\bR(M\times \bR)$ is a complex of $\bR_t$-equivariant sheaves. Hence an isomorphism $\cE\cong T_a\cE$ is specified for $\cE\in \Sh^\bR(M\times \bR)$. Then $Q(\cE)\cong Q(T_a(\cE))$. On the other hand, $Q(\cE)$ itself is equipped with an equivariant structure which gives $Q(\cE)\cong T_aQ(\cE)$. Hence we have a strict isomorphism $T_aQ(\cE)\cong Q(T_a(\cE))$.

Let $\star$ be the convolution product defined on $\Sh(M\times \bR_t)$. For the constant sheaf $\bK_{[a,\infty)}$, we have the induced functor $\star\bK_{[a,\infty)}\colon \Sh^\bR(M\times \bR_t)\rightarrow \Sh^\bR(M\times \bR_t)$. We denote the essential image of $(-)\star\bK_{[0, \infty)}$ by $\Sh^\bR_{>0}(M\times \bR_t)$. We define the non-equivariant version $\Sh_{>0}(M\times \bR_t)$ similarly. The fibrant replacement can be restricted to $\Sh^\bR_{>0}(M\times \bR_t)$.

We denote the subcategory of equivariant modules with microsupports in $\{\tau\leq 0\}$ by $\Sh_{\leq 0}(M\times \bR_t)$.
It is known that the quotient $\Sh(M\times \bR_t)/\Sh_{\leq 0}(M\times \bR_t)$ is quasi-equivalent to $\Sh_{>0}(M\times\bR_t)$ (\cite{Tam}).

\begin{definition}[locally trivial equivariant structure]
An object $\cE\in \Sh_{>0}^\bR(M\times \bR_t)$ is said to be locally trivially equivariant if the following holds: for any point in $x\in M$, there exists a neighborhood $U$ of $x$ and a constructible object $\cE'\in \Sh_{>0}(M\times \bR_t)$ such that $\cE|_U\cong \bigoplus_{c\in \bR}T_c\cE'$ where the right hand side is equipped with the obvious equivariant structure. We denote the subcategory of $\Sh^\bR_{>0}(M\times \bR_t)$ spanned by locally trivially equivariant objects by $\Sh^\bR(M)$.
\end{definition}

The following is a key example.
\begin{example}
We consider $\bigoplus_ c\bK_{M\times \{t\geq c\}}$ as a trivially equivariant object in $\Sh^\bR_{>0}(M\times \bR_t)$. We would like to calculate the space of endomorphisms of $\bigoplus_ c\bK_{M\times \{t\geq c\}}$. Then
\begin{equation}
    \End_{\Sh^{\bR}(M)}(\bigoplus_ c\bK_{M\times \{t\geq c\}})\cong \Hom_{\Sh(M\times \bR_t)}(\bK_{M\times \{t\geq 0\}}, \bigoplus_ c\bK_{M\times \{t\geq c\}})\cong \Lambda_0\otimes_\bK H^*(M, \bK).
\end{equation}
\end{example}

An object $\cE\in \Sh^\bR_{>0}(M\times\bR_t)$ is represented by
\begin{equation}
    \cE'\star \bK_{[0,\infty)}:=m_!(Q(\cE'\boxtimes \bK_{[0,\infty)})).
\end{equation}
where $m\colon M\times\bR_t\times\bR_t\rightarrow M\times\bR_t$ is the addition of $\bR_t$. The canonical map $T^a\colon \bK_{[b, \infty)}\rightarrow \bK_{[a+b, \infty)}$ ($a\in \bR_{\geq 0}$) satisfies $T^aT^b=T^{a+b}$ for $a,b\in \bR_{\geq 0}$ which are strict equalities. The $\bR_t$-equivariant structure of $\cE'\star \bK_{[a+b, \infty)}$ implies that
\begin{equation}
    \cE'\star \bK_{[a, \infty)}=m_!(Q(\cE'\boxtimes T_a\bK_{[0, \infty)}))\cong T_a(\cE'\star \bK_{[0, \infty)})\cong \cE'\star \bK_{[0,\infty)}.
\end{equation}
Together with $T^a$, it induces a canonical map $T^a\colon \cE\rightarrow \cE$. Again we get a strict equality $T^aT^b=T^{a+b}$. This means we have a homomorphism $\bC[\bR_{\geq 0}]\rightarrow \End(\cE)$. This makes
\begin{equation}
    \Hom_{\Sh^{\bR_t}_{>0}(M\times\bR_t)}(\cF, \cE)
\end{equation}
into a complex of $\bC[\bR_{\geq 0}]$-modules. It is easy to see that the $\bC[\bR_{\geq 0}]$-action on $\Hom_{\Sh^{\bR_t}_{>0}(M\times\bR_t)}(\cF, \cE)$ induced by $\bC[\bR_{\geq 0}]\rightarrow \End(\cF)$ coincides with the above action on the cohomological level.

Consider the completion
\begin{equation}\label{completion}
    \widehat{\Hom}(\cE,\cF):=\lim_{a\rightarrow \infty}\Hom(\cE,\cF)/T^a\Hom(\cE,\cF).
\end{equation}
This is a $\Lambda_0$-module and the composition morphism is still well-defined.

\begin{remark}
The canonical map $\Hom(\cE, \cF)\rightarrow \widehat{\Hom}(\cE, \cF)$ often becomes a quasi-isomorphism.
\end{remark}

We define the new category $\Sh^{\Lambda_0}(M)$ to be the category with the same objects as of $\Sh^\bR(M)$, but the Hom-spaces are defined by (\ref{completion}). We denote the category of dg $\Lambda_0$-modules over $\Sh^{\Lambda_0}(M)$ by $\Mod(\Sh^{\Lambda_0}(M))$. We also denote the category of dg $\Lambda_0/T^c\Lambda_0$-modules by $\Mod^{\Lambda_0/T^c\Lambda_0}(\Sh^{\Lambda_0}(M))$. Since any $\Lambda_0/T^c\Lambda_0$-module can be considered as a $\Lambda_0$-module, we have a fully faithful embedding $\Mod^{\Lambda_0/T^c\Lambda_0}(\Sh^{\Lambda_0}(M))\hookrightarrow \Mod(\Sh^{\Lambda_0}(M))$.

We are interested in the following class of objects.
\begin{definition}[Locally Yoneda module]
\begin{enumerate}
    \item An object $\cE$ in $\Mod^{\Lambda_0/T^c\Lambda_0}(\Sh^{\Lambda_0}(M))$ with $c>0$ is {\em Yoneda} if there exists an object $\cE'\in \Sh^{\Lambda_0}(M)$ such that $\cE\cong \Hom(-, \cE')\otimes_{\Lambda_0}\Lambda_0/T^c\Lambda_0$.
    \item An object $\cE$ in $\Mod^{\Lambda_0/T^c\Lambda_0}(\Sh^{\Lambda_0}(M))$ with $c>0$ is {\em locally Yoneda} if the following holds: For any point $x\in M$, there exists a local neighborhood $V$ of $x$ such that the restriction of $\cE$ to $V$ is a Yoneda module.
    \item An object $\cE$ in $\Mod(\Sh^{\Lambda_0}(M))$ is {\em locally Yoneda} if the following holds: there exists an inverse system of objects $\{\cE_c\}_{c\in \bR_{>0}}$ in $\Mod(\Sh^{\Lambda_0}(M))$ such that
    \begin{enumerate}
        \item $\cE_c$ is in $\Mod^{\Lambda_0/T^c\Lambda_0}(\Sh^{\Lambda_0}(M))\hookrightarrow \Mod(\Sh^{\Lambda_0}(M))$,
        \item locally Yoneda, and
        \item the limit is $\lim_{c\rightarrow \infty}\cE_c\simeq \cE$.
    \end{enumerate}
\end{enumerate}
We denote the subcategory of locally Yoneda modules by $\Sh^Y(M)$.
\end{definition}

Take an exact symplectic structure $\omega$ on $T^*M$ with its primitive $\lambda$. Then $(T^*M\times \bR_t, dt+\lambda)$ is a contact manifold.
We denote the standard symplectic structure of $T^*M$ by $\omega_{st}$ with its standard primitive $\lambda_{st}$. Then $\omega_{st}+d\tau\wedge dt$ is a symplectic structure of $T^*M\times T^*_{\tau>0}\bR$.
Suppose there exists an $\bR_{>0}$-action on $T^*M\times T^*_{>0}\bR$ which makes it a homogeneous symplectic manifold and its quotient is the contact manifold $(T^*M\times \bR_t, dt+\lambda)$. Then we have projections $T^*M\times T^*_{>0}\bR_t\rightarrow T^*M\times \bR_t\rightarrow T^*M$ where the second projection is the stupid projection. We denote the composition by $\rho$. By the uniqueness lemma of Viterbo~\cite{Vit}, such $\rho$ is unique for a given $\lambda$ if it exists.

The following is the fundamental example.
\begin{example}\label{rho}
For the standard Liouville structure, $\rho$ can be explicitly written as $\rho\colon T^*M\times T^*_{\tau>0}\bR_t\rightarrow T^*M$ by $(x, \xi, t, \tau)\mapsto (x, \xi/\tau)$ where $x\in M, \xi\in T^*_{x}M$.
\end{example}

Since an object of $\Sh^Y(M)$ is a module over sheaves, we can extend the definition of $\SS$ in \cite{KS} to $\Sh^Y(M)$ obviously. In the following, we fix $\rho$.
\begin{definition}
For an object $\cE\in \Sh^Y(M)$, we set
\begin{equation}
    \musupp(\cE):=\rho(\SS(\cE)\cap(T^*(M\times \bR_t)\bs T^*_{M\times \bR_t}M\times \bR_t)).
\end{equation}
\end{definition}

Suppose $\musupp(\cE)$ is a Lagrangian submanifold. For $(x_0, \xi_0)\in \musupp(\cE)$, there exists a lift $(x_0, \xi_0\tau_0, t_0, \tau_0)\in \SS(\cE)$. We take a differentiable function $f$ such that $f(x_0, t_0)=0$ and $df$ transversely intersects with $\SS(\cE)$ at $(x_0, \xi_0\tau_0, t_0, \tau_0)$. For an open neighborhood $U$ of $(x_0,t_0)$, we set $U_f:=\{(x,t)\in U, f(x,t)\geq 0\}$. Then we set
\begin{equation}\label{microstalk}
    \mu_{(x_0,\xi_0)}(\cE):=\lim_{U\ni (x_0,t_0)}\Hom\lb\bigoplus_c T_c\bK_{U_f}, \cE\rb\otimes_{\bK}\Lambda[(\tau_f+\dim X)/2]
\end{equation}
where $\tau_f$ is the Maslov index~\cite{KS}. We call $\mu_{(x,\xi)}(\cE)$ {\em the microstalk of $\cE$ at $(x, \xi)$}.

\begin{definition}[Pure and simple objects]
An object $\cE$ of $\Sh^Y(M)$ is pure (resp. simple) if $\musupp(\cE)$ has a dense open subset which is a Lagrangian submanifold and the microstalks on the open subset are concentrated in one degree (resp. pure and the microstalks are 1-dimensional).
\end{definition}

\begin{definition}
The microlocal category $\mu(T^*M, \lambda,\Lambda_0)$ is the subcategory of the category of $\Mod({\Sh}^{\Lambda_0}(M))$ which is the triangulated hull of simple objects.

We set $\mu(T^*M, \lambda, \Lambda):=\mu(T^*M, \lambda, \Lambda_0)\otimes_{\Lambda_0}\Lambda$ where $\Lambda$ is the Novikov field.
\end{definition}
We call an object of $\mu(T^*M, \lambda, \Lambda_0)$, a {\em sheaf quantization}.

\section{Sheaf quantization}
\subsection{Sheaf quantization associated to brane data}
Sheaf quantization was invented by Tamarkin~\cite{Tam} and developed in \cite{GKS}, \cite{Guillermou}, \cite{JT}.
We would like to recall the notion of sheaf quantization in a slightly general fashion. In one sentence, sheaf quantization is a procedure to assign a sheaf to a Lagrangian brane (an object of Fukaya category). Usually, for a Lagrangian brane in $T^*M$, its sheaf quantization lives in $\Sh(M)$ or $\Sh_{>0}(M\times \bR_t)$ (there are also variants, for example, Asano--Ike's construction~\cite{AI}). Our version of sheaf quantization lives in $\mu(T^*M, \lambda, \Lambda_0)$, which can be considered as a generalization of the version for $\Sh(M)$ and $\Sh_{>0}(M\times \bR_t)$.

For the moment, we restrict ourselves to the case when the underlying Lagrangians are smooth. For the singular case, we will remark in the end of this subsection.

Fix a relative Pin structure $P$ with the background class $w_2(TM)$, a Maslov grading $\tilde{\alpha}$. For a simple object $\cE\in \Sh^Y(M)$ with $\musupp(\cE)= L$ is a smooth Lagrangian submanifold, we will construct the Novikovized microlocalization (a generalization of \cite{Guillermou, JT}).

Fix an open covering $U_i$ of $L$ such that any intersection is contractible. On $U_i$, we fix a primitive $f_i$ of $\lambda_{st}|_L$. We set
\begin{equation}
    L_{f_i}:=\lc (x, \xi, t, \tau)\in U_i\times T^*\bR_t\relmid \tau>0, (x, \xi/\tau)\in L, -t=f_i(x, \xi/\tau)\rc.
\end{equation}
Then $L_{f_i}$ is a conic Lagrangian submanifold with respect to the symplectic structure $\omega_{st}+d\tau\wedge dt$. On each $U_i\cap U_j$, there exists $c_{ij}\in\bR$ such that $T_{c_{ij}}L_{f_i}=L_{f_j}$. The choice of the brane data gives a family of simple objects $\cF_i\in \mu_{L_{f_i}}(L_{f_i})$ and identifications $\rho_{ij}\colon T_{c_{ij}}\cF_i|_{U_i\cap U_j}\cong \cF_j|_{U_i\cap U_j}\in \mu_{L_{f_j}}(U_i\cap U_j)$ where $\mu_{L_{f_i}}$ is the microlocal category over the Lagrangian~\cite{Guillermou, JT, Jin}.

By the definition of $\mu_{L_{f_i}}$, there exists an inverse system of objects $\cF_i^k\in \Sh(M\times \bR_t)$ satisfying the following: Let $\cE$ be an object of $\Sh_{>0}(M\times \bR_t)$ such that $\musupp(\cE)$ is in $\bigcup T_cL_{f_i}$ in a neighborhood of $L_{f_i}$. Then
\begin{equation}
\lim_{j\rightarrow \infty}\Hom_{\Sh_{\tau>0}(M\times \bR_t)}(\cF_i^j, \cE)=\Hom_{\mu_{L_{f_i}}(L_{f_i})}(\cF_i, \mu(\cE))
\end{equation}
where $\mu(\cE)$ is the microlocalization of $\cE$ along $L_i$~\cite{JT}.

For an object $\cE\in \Sh^Y(M)$, we set
\begin{equation}
    \mu(\cE)|_{U_i}:=\lim_{j\rightarrow\infty}\Hom_{\Sh^Y(M)}(\bigoplus_cT_c\cF_i^j, \cE)\otimes_\bK\Lambda_{U_i}
\end{equation}
where $\Lambda_{U_i}$ is the rank 1 constant sheaf valued in the Novikov field $\Lambda$ on $U_i$.
We glue up this family of local systems by $T^{c_{ij}}\rho_{ij}$'s. The resulting local system is called the microlocalization of $\cE$ along $(P, \tilde{\alpha})$ and denoted by $\mu(\cE)$.

\begin{definition}
Let $(L, P, \tilde{\alpha})$ be as above and $\cL$ be a $\Lambda$-local system on $L$.

An object $\cE$ of $\Sh^Y(M)$ is a {\em sheaf quantization} of $(L, P, \tilde{\alpha}, \cL)$ if the microlocalization $\mu(\cE)$ is equal to $\cL$.
\end{definition}
The following proposition is conceptually important, but not essential in this paper. We only give a sketch of the proof.
\begin{proposition}
A sheaf quantization of a Lagrangian brane is unique if it exists.
\end{proposition}
\begin{proof}
Let $\cE$ and $\cE'$ be sheaves whose $\musupp$ are the same and $\mu(\cE)\cong \mu(\cE')$. We set $L:=\musupp(\cE)$ Take an open covering of $L$ such that each member is contractible. Consider the microlocal cutoff of $\cE$ and $\cE'$ around each $U$, then it has a form that $\cE\cong \bigoplus_{c\in \bR} T_c\cE_0$. By taking appropriate translations, we can assume that $\supp(\cE_0)=\supp(\cE_0')$ and the isomorphism $\mu(\cE)\cong \mu(\cE')$ is represented by a morphism $\cE_0\rightarrow \cE_0'$. One can argue that this gives a morphism between $\cE$ and $\cE'$ (see Guillermou~\cite{Guillermou} for such a lift). Since the microstalks of $\cE$ and $\cE'$ are simple, the cone of this morphism has zero microstalks. Hence $\cE$ and $\cE'$ are isomorphic.
\begin{comment}

For $c<c'\in \bR_{\geq 0}$, we have a canonical morphism
\begin{equation}
    \Cone(\cE\rightarrow T_c\cE)\rightarrow \Cone(\cE\rightarrow T_{c'}\cE).
\end{equation}
We consider it as an inverse system and take limit. We denote the limit by $T_{[0,+\varepsilon)}\cE$. We have an isomorphism
\begin{equation}
\Hom(T_{[0,+\varepsilon)}\cE,\cE')\cong \Hom(T_{[0,+\varepsilon)}\cE, T_{[0,+\varepsilon)}\cE').
\end{equation}
By the assumptinon $\mu(\cE)\cong \mu(\cE')$, we can find an isomorphism $f$ in $\Hom(T_{[0,+\varepsilon)}\cE, T_{[0,+\varepsilon)}\cE')$ (see \cite{Guillermou} for a similar argument).

By the construction, we also have
\begin{equation}
\Hom(T_{[0,+\varepsilon)}\cE,\cE')\cong  \lim_{c\rightarrow +0}\Hom(\cE, \cE')\otimes_{\Lambda_0}\Lambda_0/T^c\Lambda_0.
\end{equation}

Import $f$ to $\lim_{c\rightarrow +0}\Hom(\cE, \cE')\otimes_{\Lambda_0}\Lambda_0/T^c\Lambda_0$ and
take a representative of $f$ in $\Hom(\cE, \cE')$. The associated morphism between $\Lambda_0$-modules
\begin{equation}
    \Hom(\cE', -)\rightarrow \Hom(\cE, -)
\end{equation}
is an isomorphism after taking the tensor with $\Lambda_0/\la T^c\relmid c>0\ra$. Hence it is an isormophism. By the Yoneda lemma, $\cE$ and $\cE'$ are isomorphic.
\end{comment}
\end{proof}

\begin{remark}
To construct more general sheaf quantizations, we have to use $\Lambda$ as the original base field. Since we will construct some specific sheaf quantizations in this paper, we will not speak about this feature in this paper.
\end{remark}

\begin{remark}[Sheaf quantization in $\Sh(M)$ and $\Sh_{>0}(M\times\bR_t)$]
For a conic Lagrangian $L$ in $T^*M$, a sheaf quantization $\cE$ of $L$ is a constructible sheaf with $\SS(\cE)=L$.

If $L$ is an exact Lagrangian submanifold, we can take a primitive of $\lambda_{st}|_L$ globally on $M$. Then we can lift $L$ to $L_f\subset T^*M\times T^*_{>0}\bR_t$. A constructible sheaf $\cE$ on $M\times \bR_t$ is said to be a sheaf quantization of $L$ if $\SS(\cE)\bs T^*_{M\times \bR_t}M\times \bR_t\subset L_f$. Under certain assumptions on $L$, Guillermou and Jin--Treumann constructed such sheaves~\cite{Guillermou, JT}. This notion is a generalization of the above concept in the following sense: a sheaf quantization $\cE$ of a conic Lagrangian $L$ gives a sheaf on $M\times \bR_t$ by $\cE\boxtimes \bK_{t\geq 0}$, which turns out to be a sheaf quantization of $L$ in the latter sense.

Our concept of sheaf quantization includes both. For a sheaf quantization $\cE$ of a conic Lagrangian $L$, consider $\bigoplus_cT_c\cE$ with the obvious $\bR_t$-equivariancy. Then this is a sheaf quantization of $L$ in our sense.
\end{remark}
\begin{remark}
Let $L$ be a singular Lagrangian submanifold $L$. For a (local) singular exact Lagrangian, we can still define the microlocal category~\cite{Nadler}. Using this, we can define a brane structure as a compatible set of sections of local microlocal categories. Then the same uniqueness theorem of sheaf quantization can be proved in the exactly same way as above.
\end{remark}

\subsection{B-field twist}
In this subsection, we will introduce a different normalization of the microlocalization. Let $b$ be a 1-form on $T^*M$.
Let $\frakL$ be the set of Lagrangian submanifolds satisfying $db|_L=0$. On each $L$, we have a flat connection $d+b$ on the rank 1 trivial bundle. We denote the corresponding local system by $\cE_L$.

We denote the triangulated hull of the full subcategory of $\mu(T^*M, \lambda, \Lambda_0)$ spanned by sheaf quantizations of elements of $\frakL$ by $\mu(T^*M, \lambda, b, \Lambda_0)$.

For a sheaf quantization in $\mu(T^*M, \lambda, b, \Lambda_0)$, we redefine $\mu(\cE)$ with $\mu(\cE)\otimes \cE_L$.

\begin{remark}
Secretly, this convention comes from the B-field twist in Fukaya category. A more natural construction should be as follows: One can interpret a B-field $d\lambda$ as a gerbe~\cite{Hitchin}. Then one can twist the microlocal category $\mu(T^*M, \lambda, \Lambda_0)$ with respect to the gerbe. In our setting, we can always trivialize this gerbe by $b$ canonically. However, this trivialization twists our brane data by the connection $d+b$.
\end{remark}

\section{Meromorphic flat $\hbar$-connection}
\subsection{Meromorphic flat $\hbar$-connection}
In this section, we would like to set up our general setting for differential equations.

Let $C$ be a Riemann surface. Let $\cO_C$ be the structure sheaf. We consider the subring $\cD^\hbar_C$ of the sheaf of $\bC[\hbar]$-linear endomorphisms of $\cO_C[\hbar]:=\cO_C\otimes_\bC\bC[\hbar]$ generated by $\hbar\partial_z$ and $\cO_C$. We would like to consider the modules over $\cD^\hbar_C$.

Our fundamental example of such a module is given by the class meromorphic flat $\hbar$-connections:
\begin{definition}
A meromorphic flat $\hbar$-connection is given by the following data
\begin{enumerate}
    \item A meromorphic bundle $\cE$ with poles in $M\subset C$.
    \item A $\bC[\hbar]$-linear morphism $\nabla\colon \cE[\hbar]\rightarrow \cE\otimes K_C[\hbar]$ where $K_C$ is the canonical bundle and $\nabla$ satisfies the following: For any $f\in \cO_C$ and $s\in \cE$, we have $\nabla(fs)=\hbar df\otimes s+ f\otimes \nabla s$.
\end{enumerate}
\end{definition}
A meromorphic flat $\hbar$-connection $(\cE, \nabla)$ gives a $\cD^\hbar$-module as follows: The underlying $\cO[\hbar]$-module is $\cE[\hbar]$ and the action of $\cD^\hbar$ is defined as follows: For a tangent vector $\hbar v\in \cD\subset \cD^\hbar$ and $s\in \cE$, we set
\begin{equation}
    (\hbar v)\cdot s:=\nabla s(v).
\end{equation}

For $\hbarr\in \bC^\times$, we define a meromorphic connection $\cM^\hbarr:=(\cE,\nabla^\hbarr)$ by
\begin{equation}
    \nabla^\hbarr:=\frac{1}{\hbarr}\cdot \nabla\otimes_{\bC[\hbar]}\bC
\end{equation}
where the morphism $\bC[\hbar]\rightarrow\bC$ of the tensor is given by the evaluation at $\hbarr$.

\begin{remark}
The notion of $\hbar$-connection is almost the same as Deligne's lambda connections. However, we would like to avoid the terminology in this paper, because of the existence of a non-straight-forward relations between these when one starts from a point in Hitchin's base~\cite{GaiottoTBA, Mulaseetal}.
\end{remark}

\subsection{Spectral curve}
We first note that $\cD_C^\hbar/\hbar\cD_C^\hbar\cong \cO_{T^*C}$. Let $\cM$ be a $\cD^\hbar_C$-module. Then $\cM/\hbar\cM$ is a module over $\cO_{T^*C}$. We set
\begin{equation}
    \SS(\cM):=\supp(\cM/\hbar\cM)\subset T^*C.
\end{equation}
We sometimes call it the {\em spectral curve} of $\cM$.

Let us consider the case when $\cM$ is a meromorphic flat $\hbar$-connection $(\cE, \nabla)$. On a sufficiently small open subset $U$, the connection can be written as
\begin{equation}
    \hbar d+Adz
\end{equation}
where $A=A_0+A_1\hbar+\cdots +A_k\hbar^k$ is an $n\times n$ matrix valued in meromorphic functions. The following is standard.

\begin{lemma}
\begin{equation}
    \SS(\cM)\cap T^*U=\Zero(\det(\xi 1_n-A_0))
\end{equation}
where $\xi$ is the cotangent coordinate associated to the coordinate $z$.
\end{lemma}

We write the right hand side as
\begin{equation}
    \det(\xi 1_n-A_0)=\xi^n+Q_{1,0}\xi^{n-1}+\cdots +Q_{n,0}.
\end{equation}
Take a flat section $\psi$ of $\nabla$.
Then we get
\begin{equation}
    (\hbar\partial)^k\psi=A_0^k\psi+O(\hbar)\psi.
\end{equation}
By the Cayley--Hamilton theorem,
\begin{equation}
    A_0^n\psi+Q_{1,0}A_0^{n-1}\psi+\cdots +Q_{n,0}\psi=0
\end{equation}
Hence $\psi$ satisfies an equation
\begin{equation}
    (\hbar\partial)^n\psi+B_1(\hbar\partial)^{n-1}\psi+\cdots B_n\psi=0
\end{equation}
where $B_i=Q_{i,0}+O(\hbar)$.

\section{Sheaf quantizations associated to meromorphic flat $\hbar$-connections of rank 0 and 1}
In this section, as a warm-up, we would like to construct sheaf quantizations associated to rank 0 and rank 1 connections. In this section, we do not need exact WKB analysis. The material of this section can be applicable to any dimensions.

\subsection{Rank 0}
We will interpret rank 0 connection as a meromorphic flat $\hbar$-connection supported on 0-dimensional subvarieties. It means that, for a rank 0 connection $\cM$, $\SS(\cM)$ is a union of some cotangent fibers. Hence the corresponding $\cD_\hbar$-module locally looks like $\cD_\hbar/(z-z_0)^n\cD_\hbar$ for some $n$ and $z_0\in C$. As usual, it is easy to see that this is isomorphic to $(\cD_\hbar/(z-z_0)\cD_\hbar)^{\oplus n}$. The ``solution" of the equation $\cD_\hbar/(z-z_0)\cD_\hbar$ is expected to be the delta function. Then it is canonical to associate a sheaf
\begin{equation}
    \bigoplus_cT_c(\bK_{z_0}\boxtimes \bK_{t\geq 0})
\end{equation}
with a canonical equivariant structure, to $\cD_\hbar/(z-z_0)\cD_\hbar$. Here $\bK_{z_0}$ is the skyscraper sheaf on $z_0$. We also have $\musupp(\bigoplus_cT_c(\bK_{z_0}\boxtimes \bK_{t\geq 0}))=T^*_{z_0}C$.

\subsection{Rank 1}
Let $\cM$ be a rank 1 meromorphic flat $\hbar$-connection with poles in $M$. On a local coordinate $U$, we can write it as
\begin{equation}
    (\hbar\partial-Q(z,\hbar))\psi=0
\end{equation}
where $Q(z, \hbar)=\sum_{i=0}^kQ_i(z)\hbar^i$ for some finite $k$. We assume that poles of $\sum_iQ_i(z)\hbar^i$ are poles of $Q_0$. In $T^*U$, the spectral curve is defined by $\xi=Q_0$.

Take an open covering $\{U_i\}$ of $C\bs M$ such that each intersection is contractible. Fix $\hbarr\in \bC^\times$ and a base point $z_i$ in each $U_i$. We solve the equation explicitly:
\begin{equation}
    \psi(z,\hbarr)=\exp\lb \int_{z_i}^z\frac{Q(z,\hbarr)}{\hbarr}dz\rb=\exp\lb \Re\int_{z_i}^z\frac{Q_0}{\hbarr}dz\rb\exp\lb\Image\int_{z_i}^z\frac{Q_0}{\hbarr}dz\rb\exp\lb\int_{z_i}^z\frac{\sum_{i>0}Q_{i}\hbarr^i}{\hbarr}dz\rb
\end{equation}
On each $U_i$, we put $\bigoplus_cT_c\bK_{t\geq -\Re\int_{z_i}^z \frac{Q_0}{\shbarr}}$. On each overlap $U_i\cap U_j$, we consider the isomorphism between $\bigoplus_cT_c\bK_{t\geq -\Re\int_{z_i}^z \frac{Q_0}{\shbarr}}$ and $\bigoplus_cT_c\bK_{t\geq -\Re\int_{z_j}^z \frac{Q_0}{\shbarr}}$ induced by
\begin{equation}
    \exp\lb\int_{z_i}^z\frac{\sum_{i>0}Q_{i}\hbarr^i}{\hbarr}dz\rb\cdot \id\colon \bK_{t\geq -\Re\int_{z_i}^z \frac{Q_0}{\shbarr}}\rightarrow \bK_{t\geq -\Re\int_{z_i}^z \frac{Q_0}{\shbarr}}=T_{-\Re\int_{z_i}^{z_j} \frac{Q_0}{\shbarr}}\bK_{t\geq -\Re\int_{z_j}^z \frac{Q_0}{\shbarr}}.
\end{equation}
These gluing-up isomorphisms give a sheaf quantization of $\SS(\cM)$. The reason why we discarded $0$-th terms will be explained later in the context of microlocalization.

\section{Exact WKB analysis for rank 2 meromorphic flat $\hbar$-connections}
We would like to recall some basic facts of exact WKB analysis. We refer to \cite{KawaiTakei} and \cite{IwakiNakanishi} for more accounts. Although the literatures treat only the case of Schr\"odinger equation, we can apply it to $\hbar$-connection. The reason why it's possible is that Stokes regions are contractible and we can trivialize the connection on the regions.

\subsection{Rank 2 meromorphic flat $\hbar$-connections}
Let $\cM$ be a rank 2 meromorphic flat $\hbar$-connection. Locally, we can write it as
\begin{equation}
    \hbar d+\Omega
\end{equation}
where $\Omega=\Omega_0+\hbar\Omega_1+\cdots +\hbar^k\Omega_k$ is a $2\times 2$-matrix valued in meromorphic functions. We assume $\det\Omega_0\neq 0$. The spectral curve is defined by
\begin{equation}\label{speccurvelocal}
    \xi^2+\tr\Omega_0\xi+\det\Omega_0=0
\end{equation}
Note that $\tr\Omega_0$ (resp. $\det\Omega_0$) is a globally defined 1-form (resp. quadratic form), which is nothing but the Hitchin image of $\Omega_0$.

\subsection{WKB solutions}
We first would like to set up a sheaf where we put WKB solutions. Let $e^{\cO/\hbar}$ be a sheaf of $\bC$-modules isomorphic to the structure sheaf $\cO$ as a sheaf of $\bC$-modules. For $f\in \cO$, the corresponding element in $e^{\cO/\hbar}$ is denoted by $e^{f/\hbar}$. The addition of sections of $e^{\cO/\hbar}$ is denoted like a multiplication i.e., $e^{(f+g)/\hbar}:=e^{f/\hbar}e^{g/\hbar}$.

For a rank 2 meromorphic flat $\hbar$-connection $\cM:=(\cE, \nabla)$, one can linearly extend $\nabla$ to $\cE[[\hbar]]$. We can further extend it to $\cE[[\hbar]]\otimes_\bC e^{\cO/\hbar}\rightarrow \cE\otimes_\cO K[[\hbar]]\otimes_\bC e^{\cO/\hbar}$ by setting
\begin{equation}
    \nabla \colon s\otimes e^{g/\hbar}\mapsto\nabla s\otimes e^{g/\hbar}+ sdg\otimes e^{g/\hbar}.
\end{equation}

We denote the spectral curve of $\cM$ by $L_\cM$. Then $L_\cM$ is a branched covering of $C\bs M$ where $M$ is the pole divisor of $\cM$. Away from branched points, take a point $z\in C$ and a neighborhood $U$ of $C$. Then there exists a lift $\tilde{U}$ of $U$ to $L_\cM$. We denote the restriction of the Liouville one-form $\lambda$ to $\tilde{U}$ by $\xi$.

\begin{definition}[WKB solution]\label{defofWKBsol}
For a flat connection $(\cE, \nabla)$, a WKB solution $s$ associated to $\xi$ is a local section of $\cE[[\hbar]]\otimes_\bC e^{\cO/\hbar}$ satisfying the following:
\begin{enumerate}
    \item $\nabla s=0$.
    \item There exists a local holomorphic function $f$ such that $df=\xi$ and a local section $s'$ of $\cE[[\hbar]]$ such that $s=e^{f/\hbar}s':=s'\otimes e^{f/\hbar}$.
\end{enumerate}
\end{definition}

\begin{proposition}
There locally exists a WKB solution associated to $\xi$ outside the poles and turning points. A WKB solution is unique up to constant multiplication.
\end{proposition}
\begin{proof}
The following proof is an easy adaptation of the standard construction.

We first prove the existence.
Locally, one can write the equation of flats sections as
\begin{equation}
 (\hbar d+\Omega)\psi=\lb \hbar d+ \left(
    \begin{array}{cc}
      \Omega_{11}(z, \hbar) &  \Omega_{12}(z, \hbar) \\
      \Omega_{21}(z, \hbar) & \Omega_{22}(z, \hbar)
    \end{array}
  \right) \rb\left(
    \begin{array}{c}
      \psi_{1}  \\
      \psi_{2}
    \end{array}
  \right)=0.
\end{equation}
Then $\psi_1$ should satisfy
\begin{equation}\label{assocSchrodinger}
    (\hbar \partial)^2\psi_1+\hbar\partial\Omega_{11}\psi_1+\Omega_{11}\hbar\partial\psi_1+\frac{\hbar\partial\Omega_{12}}{\Omega_{12}}(-\hbar\partial\psi_1-\Omega_{11}\psi_1)+\Omega_{12}(\Omega_{21}\psi_1+\Omega_{22}(-\hbar\partial\psi_1-\Omega_{11}\psi_1))=0
\end{equation}
We can abbreviate this equation into
\begin{equation}
    (\hbar\partial)^2\psi_1+A(z, \hbar)(\hbar\partial)\psi_1(z, \hbar)+B(z, \hbar)\psi_1(z, \hbar)=0
\end{equation}
where $A(z, \hbar)=\tr\Omega_0+O(\hbar), B(z, \hbar)=\det\Omega_0+O(\hbar)$ where $\Omega=\Omega_0+O(\hbar)$.

For the given $\xi$, we consider a solution of the form
\begin{equation}\label{WKBexponential}
    \psi_1=\exp\lb \int^z \frac{\xi}{\hbar} dz\rb\exp\lb \sum_{i\geq 0} \int^z P_{i}\hbar^i\rb.
\end{equation}
Then the equation (\ref{assocSchrodinger}) inductively determines $P_i$. We also set
\begin{equation}\label{WKBexponential2}
    \psi_2:=\frac{-\hbar\partial\psi_1-\Omega_{11}\psi_1}{\Omega_{12}}=\exp\lb \int^z \frac{\xi}{\hbar} dz\rb\frac{-\lb\xi+\sum_{i\geq 0}P_i\hbar^{i+1}\rb-\Omega_{11}}{\Omega_{12}} \exp\lb \sum_{i\geq 0} \int^z P_{i}\hbar^i\rb
\end{equation}
By expanding the second exponentials of (\ref{WKBexponential}) and (\ref{WKBexponential2}), we get a WKB solution. Note that the solution of the form (\ref{WKBexponential}) is unique up to constant multiplication coming from the choice of endpoints of the integrals.

We next prove the uniqueness. Suppose locally we have a WKB solution $\psi=(\psi_1, \psi_2)$. We can write $\psi_1$ as
\begin{equation}
    \psi_1=C\exp\lb\int \frac{\xi}{\hbar} \rb\sum_{i\geq 0} p_i\hbar^i
\end{equation}
for some constant $C$. Note the following identity of formal power series:
\begin{equation}
    \sum_{i\geq 0} p_i\hbar^i=p_0\sum_{i\geq 0}\frac{p_i}{p_0}\hbar^i=p_0\exp\lb\log\lb 1+\sum_{i\geq 1}p_i\hbar^i\rb\rb.
\end{equation}
Then we can further rewrite $\psi_1$ as
\begin{equation}
\begin{split}
         \psi_1&=C\exp\lb\int \frac{\xi}{\hbar}\rb \exp\lb\log p_0+ \log\lb 1+\sum_{i\geq 1}p_i\hbar^i\rb \rb\\
         &=C\exp\lb\int \frac{\xi}{\hbar}\rb \exp\lb\log p_0+ \sum_{i\geq 1}\tilde{p}_i\hbar^i\rb\\
         &=C\exp\lb\int \frac{\xi}{\hbar}\rb \exp\lb\int^z\frac{1}{p_0}\frac{\partial p_0}{\partial z}dz+ \sum_{i\geq 1}\int^z\frac{\partial\tilde{p}_i}{\partial z}dz \hbar^i\rb\\.
\end{split}
\end{equation}
where the second equality is the definition of $\tilde{p}$. This is exactly the form of (\ref{WKBexponential}) up to multiplication by some element in $\bC[[\hbar]]$.
\end{proof}

\begin{definition}
Let $e^{f/\hbar}s$ be a WKB solution of $(\cE,\nabla)$ defined on an open subset $U$ of $C$. A resummation of $e^{f/\hbar}s$ on $U$ at $\hbarr\in \bC^\times$ is a holomorphic flat section $\fraks(z, \hbar)$ of $\cE|_U$ parametrized by $\hbar$ in a sector with the angle $|\arg\hbar-\arg\hbarr|<\pi/2$ such that
\begin{equation}
    ||e^{-f/\hbar}\fraks(z,\hbar)-\sum_{i\leq N}s_i(z)\hbar^i||\sim O(\hbar^{N+1})
\end{equation}
as $\hbar\rightarrow 0$ for any $N$, any $z\in U$, and any metric $||\cdot||$ on $\cE$. We call $\fraks(z,\hbar)$ a resummation of $e^{f/\hbar}s$.
\end{definition}
In general, there are many resummations of one formal power series. In the rest of this section, we will review the resummability.

\subsection{Gauge transformation}
First, we remark about gauge transformations. Locally, we can always reduce the problem to the equation of the form
\begin{equation}\label{schrodinger}
    (\hbar^2\partial_z^2+Q_1(z,\hbar)\hbar\partial_z+Q_2(z,\hbar))\psi(z)=0.
\end{equation}
For the equation (\ref{schrodinger}), we set
\begin{equation}
    \widetilde{\psi}:=\exp\lb\int^z\frac{Q_1(z, \hbar)}{2\hbar}\rb\psi.
\end{equation}
This is only a local transform.

Then $\widetilde{\psi}$ satisfies
\begin{equation}\label{gaugetransformofSchrodinger}
    \lb (\hbar\partial)^2-\lb\frac{Q_1^2}{4}-Q_2+\hbar\frac{ Q_1'}{2}\rb\rb\widetilde{\psi}=0.
\end{equation}

\begin{lemma}
    A WKB solution $\tilde{\psi}$ for $(\ref{gaugetransformofSchrodinger})$ can be written by using a WKB solution $\psi$ for $(\ref{schrodinger})$ by
\begin{equation} \widetilde{\psi}=\exp\lb\int^z\frac{Q_1(z, \hbar)}{2\hbar}\rb\psi.
\end{equation}
\end{lemma}
\begin{proof}
This is obvious.
\end{proof}
As we will see in the next section, gauge transformations do not change the Stokes geometry.

\subsection{Stokes geometry}
We introduce the languages of Stokes geometry in this setting. We fix a rank 2 meromorphic flat $\hbar$-connection $\cM=(\cE, \nabla)$ and set $L:=\SS(\cM)$.
\begin{definition}
We fix $\hbarr\in \bC^\times$.
\begin{enumerate}
    \item A turning point of $\cM$ is a branched point of $L$. We denote the set of turning points of $\cM$ by $\Zero(\cM)$.
    \item For a turning point $v$ of $\cM$, the $\hbarr$-Stokes curve emanating from $v$ is defined by
    \begin{equation}
        \lc z\relmid \Im\lb \int_v^z\lb \frac{\xi_+(z')-\xi_-(z')}{\hbarr}\rb dz'\rb=0\rc
    \end{equation}
    where $\xi_+$ and $\xi_-$ are the restriction of the Liouville form on the sheets of $L$.
   \item The $\hbarr$-Stokes graph $G_\cM$ of $\cM$ is the union of the Stokes curves of $\cM$.
   \item An $\hbarr$-Stokes region is a connected component of the complement of the $\hbarr$-Stokes graph in $C$.
   \item A subset of the $\hbarr$-Stokes graph is called an $\hbarr$-Stokes segment if it is diffeomorphic to a closed interval $[a,b]\subset \bR$ ($a<b$) and it connects two turning points.
   \end{enumerate}
\end{definition}

We consider the following {\em gauge transformation} of spectral curve. Using the notation of (\ref{speccurvelocal}), we consider the quadratic differential $Q_{\cM}:=\frac{(\tr\Omega_0)^2}{4}-\det\Omega_0$. Since the notion of Stokes graphs only depend on the spectral curve, we can speak about the Stokes geometry of $Q_\cM$ by considering the curve $\xi^2+Q_\cM=0$. It is easy to see the following.
\begin{lemma}
The Stokes graph of the quadratic differential $Q_\cM$ is the same as the Stokes graph of $L$.
\end{lemma}

We would like to introduce an easy class of connections.
\begin{definition}\label{weaklyGMN}
We say a quadratic differential $Q$ is weakly GMN if
\begin{enumerate}
 \item the order of any pole of $Q$ is more than or equal to 2
    \item $Q$ has at least one pole with order $\geq 2$, and
    \item $Q$ has at least one branch point.
\end{enumerate}
We say that a meromorphic flat $\hbar$-connection is weakly GMN if $Q_\cM$ is weakly GMN.
\end{definition}

For weakly GMN connections, their Stokes regions are easy:
\begin{theorem}[{Strebel~\cite{Strebel}, Bridgeland--Smith~\cite{BridgelandSmith}}]
Suppose $\cM$ is weakly GMN and the $\hbarr$-Stokes graph of $Q_\cM$ does not have any $\hbarr$-Stokes segments. Then an $\hbarr$-Stokes region of $\cM$ has one of the following forms:
\begin{enumerate}
    \item(horizontal strip). a square surrounded by four edges of the $\hbarr$-Stokes graph. Two of four vertices are poles, and the others are zeros.
    \item(half plane). a region surrounded by two edges of the $\hbarr$-Stokes graph. One of the vertices is a zero, and the others are poles.
\end{enumerate}
\end{theorem}
\begin{proof}
Since $Q_\cM$ is weakly GMN, $L_\cM$ has a branch point and a pole. Then the proof of \cite[Lemma 3.1]{BridgelandSmith} works.
\end{proof}

\subsection{WKB regularity}
To obtain the resummability, we have to assume the higher degree terms with respect to $\hbar$ in the differential operator should be tame in certain sense.

Locally around a pole $p$ of $\cM$, the equation can be written as (\ref{schrodinger}). Consider the gauge transformation to (\ref{gaugetransformofSchrodinger}). We write it as
\begin{equation}\label{usualschrodinger}
    ((\hbar\partial)^2-Q(z, \hbar))\psi=0.
\end{equation}

\begin{assumption}[WKB-regularity]\label{WKBregular}
We say $\cM$ is WKB-regular if, around any pole $p$, $Q(z,\hbar)=Q_0(z)+\hbar Q_1(z)+\hbar^2Q_2(z)+\cdots$ satisfies the following:
\begin{enumerate}
    \item $p$ is a pole of $Q_0$.
    \item If $Q_0$ has a pole $p$ of order $m\geq 3$, then the pole order of $Q_i (i\geq 1)$ at $p$ is less than $1+m/2$.
    \item If $Q_0$ has a pole of order $m=2$, the following conditions hold: $Q_i$ has at most one simple pole at $p$ for all $i\geq 1$ except for $n=2$, and $Q_2(z)$ has a double pole at $p$ and satisfies $Q_2(z)=-\frac{1}{4(z-p)^2}(1+O(z-p))$ as $z\rightarrow p$.
\end{enumerate}
\end{assumption}
This is defined in \cite{IwakiNakanishi}, which attributes it to Koike--Sch\"afke.

The following material in this subsection is not strictly necessary needed in the main result of paper.

We would like to have a more intrinsic definition of WKB-regularity. For that, we prepare some notations. For a singularity $z$ of a flat connection $\cN$ (without $\hbar$), the formal completion of $\cN$ at $z$ is isomorphic to the form
\begin{equation}
    \bigoplus_i \cE^{f_i}\otimes \cL_i
\end{equation}
where $i$ runs through finite indices, each $\cL_i$ is regular singular, $f_i$ is a finite Laurent Puiseux polynomial, and $\cE^{f_i}$ is rank 1 connection defined by $d+ df_idz$. This is known as Hukuhara--Levelt--Turrittin theorem. The set is $\{f_i\}_{i\in I}$ is uniquely determined from $\cN$ up to the following operations: (i) for some $f_i$, replace it with $f_i+g$ where $g$ is a regular function (ii) Galois transformation. We say the class $\{f_i\}_{i\in I}$ modulo the above operations {\em the irregularity type of $\cN$}.

For a fixed $\hbarr$, associated to (\ref{usualschrodinger}), we consider the equation
\begin{equation}\label{hbarrschrodinger}
    \lb\partial^2-\frac{Q}{\hbarr^2}\rb\psi=0.
\end{equation}
\begin{proposition}\label{constantformal}
The irregularity type of (\ref{hbarrschrodinger}) at a singularity is given by $\xi_\pm/\hbarr$ for any $\hbarr$.
\end{proposition}
\begin{proof}
In this proof, we will use the notations in Section 9.
In a local coordinate around a pole, $Q_0$ can be written as $\xi=(\frac{c_1}{z^{n/2}}+\frac{c_2}{z})dz$ with $n\geq 2$ for some $c_1, c_2\in \bC$~\cite{Strebel}. Recall (or see Section 9) that the WKB solution can be written as
\begin{equation}
    \frac{1}{\sqrt{P_{odd}}}\exp\lb \int P_{odd}\rb.
\end{equation}
By \cite[Proposition 2.8]{IwakiNakanishi}, $\int P_{odd, i}$ for $i\geq 0$ does not have a pole. Hence $\exp\lb \int P_{odd, \geq 0}\rb$ is a (non-Laurent) formal power series.

Also, we have
\begin{equation}
\begin{split}
        \frac{1}{\sqrt{P_{odd}}}&= \frac{1}{\sqrt{c_1z^{-n/2}+c_2z^{-1}+P_{\geq 1, odd}}}\\
        &=\frac{1}{z^{-n/4}}\frac{1}{\sqrt{c_1+c_2z^{n/2-1}+z^{n/2}P_{\geq 1, odd}}}\\
\end{split}
\end{equation}
Again, by \cite[Proposition 2.8]{IwakiNakanishi}, $z^{n/2}P_{\geq 1, odd}$ is a formal power series. Hence the WKB solution is
\begin{equation}
    \exp\lb \int (c_1z^{-n/2}+c_2z^{-1})/\hbarr\rb F
\end{equation}
where $F$ is a (non-Laurent) Puiseux series. This confirms the claim.
\end{proof}

\begin{lemma}[Characterization of WKB-regularity]
For a pole $p$, let $\{f_i\}$ be the irregularity type of the equation at $p$. Then it is WKB-regular at $p$ if and only if $\{\hbarr f_i\}$ does not depend on $\hbarr$.
\end{lemma}
\begin{proof}
By gauge transformation, we can work with quadratic differentials.

The necessity is already proved, hence we will prove the sufficiency.
By the proof of \cite{IwakiNakanishi}, if one of the conditions in Assumption~\ref{WKBregular} is violated, the part $\sum \int P_{odd, \geq 0}$ will have a singularity. This changes the irregularity type of the connection from $\xi_\pm/\hbarr$.
\end{proof}

\begin{remark}
The regularity here is not related to the regularity of holonomic $\cD$-modules. In particular, it is possible that a WKB regular module has an irregular singularity after specializing $\hbar$.
\end{remark}

\subsection{Resummation}
We first show that there exists at least one resummation of a WKB solution. For a weakly GMN connection $(\cE, \nabla)$, pick up a Stokes region $D$. Since $D$ is contractible, by a trivialization and a gauge transform, we can reduce the problem to a Schr\"odinger equation. Then we will use the notation of Section 9.

To implement the Borel resummation, we first expand (\ref{WKB}) as
\begin{equation}
    \psi_\pm(z, \hbar)=\exp\lb\int_{z_0}^z\frac{\xi_\pm}{\hbar}dz\rb \sum^\infty_{m=0}\hbar^{m+\frac{1}{2}}\psi_m^\pm(z).
\end{equation}
The formal Borel transform with respect to $\hbar^{-1}$ is
\begin{equation}
    \psi_{\pm, B}(z, y)=\sum_{m=0}^\infty \frac{\psi^\pm_m(z)}{\Gamma(m+\frac{1}{2})}(y\pm a(z))^{m+\frac{1}{2}}
\end{equation}where $a(z):=\int^z_{z_0}\sqrt{Q}dz$. This series is convergent. We set
\begin{equation}
    \cS[\psi_\pm](z,\hbar):=\int_{\mp a(z)}^\infty e^{-y/\hbar}\psi_{\pm, B}(z, y)dy.
\end{equation}

\begin{theorem}[{Koike--Sh\"afke, \cite[Theorem 2.18]{IwakiNakanishi}}]\label{Koike}
Suppose $\cM$ is weakly GMN and WKB-regular. Suppose the $\hbarr$-Stokes graph of $\cM$ does not have any $\hbarr$-Stokes segments.
Fix $z\in C\bs G_\cM$. Then $\psi_{\pm, B}(z, \hbar)$ is convergent in a Stokes region containing $z$ and $S[\psi_\pm]$ is analytic for $|\arg(\hbar)-\arg(\hbarr)|<\pi/2, |\hbar|\ll 1$. The functions $\cS[\psi_\pm]$ give local analytic solutions of (\ref{schrodinger}). The asymptotic expansion of $\cS[\psi_\pm]$ along $\bR_{>0}\hbarr\ni \hbar\rightarrow 0$ recovers $\psi_\pm$.
\end{theorem}

\begin{remark}
The sketch of the proof of the above theorem is available in Takei~\cite{Takei}, which is a generalization of the argument in \cite{DLS}.
\end{remark}

The following is almost well-known.
\begin{proposition}
Let $\Psi_+(z, \hbar)$ be a resummation of a WKB solution associated to the sheet $\xi_+$ on a Stokes region $D$ at $\hbarr$. Then there exists a complex number $c$ such that $\Psi_+(z, \hbarr)=c\cS[\psi_+](z, \hbarr)$
\end{proposition}
\begin{proof}
On a Stokes region $D$, let $\Psi_+$ be a resummation of $\psi_+$. Since $\Psi_+$ is a flat section, we can write it as
\begin{equation}
    \Psi_+=a\cS[\psi_+]+b\cS[\psi_-]
\end{equation}
for some $a,b\in \bC$. Since $\Psi_+$ is a resummation of $\psi_+$, if $b\neq 0$, $\Re(\xi_+/\hbar) >\Re(\xi_-/\hbar)$ for any possible $\hbar$.

We shall confirm this does not happen. Since we are not on the Stokes curves, we have $\Im((\xi_+-\xi_-)/\hbarr)\neq 0$. Hence there exists $\theta\in (-\pi/2, \pi/2)$ such that $\arg((\xi_+-\xi_-)/\hbarr e^{\sqrt{-1}\theta})\in (\pi/2, 3\pi/2)$. Therefore there exists $\hbar$ such that $|\arg \hbar-\arg \hbarr|<\pi/2$ (Theorem~\ref{Koike}) and $\Re(\xi_+/\hbar) -\Re(\xi_-/\hbar)<0$. This contradicts to the previous assertion. Hence $b=0$ and this completes the proof.
\end{proof}

\subsection{Connection formulas}
In the rest of this section, we fix $\hbarr\in \bC^\times$.

Another important result of exact WKB analysis is Voros' connection formula (see Section 9). Here we only need a weaker general version of the statement, which is applicable to higher order turning points.

Let $D_1, D_2$ be adjacent Stokes regions. Let $\Psi_i=(\psi_{i+}, \psi_{i-})$ ($i=1,2$) be a set of solutions obtained by resummation at $\hbarr$. Consider the analytic continuation of the solutions across the separating Stokes edge $l$ and compare them. Then we get the following. We denote the turning point on $l$ by $v$.

\begin{proposition}\label{Vorosnaive}
The analytic continuation of $\Psi_1$ crossing the Stokes curve is related to $\Psi_2$ by
\begin{equation}
    \Psi_1= \Psi_2\cdot S
\end{equation}
where
\begin{equation}
    S=\begin{cases}
    \left(
    \begin{array}{cc}
     d_1 & 0  \\
     a & d_2
    \end{array}
  \right) \text{ if $\int_v^z\lb\frac{\xi_+-\xi_-}{\hbarr} \rb dz>0$ on the Stokes curve, }
  \\
  \\
  \left(
    \begin{array}{cc}
      d_1 & a  \\
      0 & d_2
    \end{array}
  \right)\text{ otherwise.}
    \end{cases}
\end{equation}
where $d_1, d_2\neq 0$.
\end{proposition}

\begin{proof}
This is a direct corollary of Proposition~\ref{constantformal}.
\end{proof}

\subsection{Summary}
We would like to summarize this section as follows:
\begin{theorem}
Let $C$ be a closed Riemann surface. Let $\cM$ be a weakly GMN rank 2 meromorphic flat $\hbar$-connection without $\hbar$-Stokes segments.
Fix sufficiently small $\hbarr\in \bC^\times$. Then, on each Stokes region, there exists a canonical splitting $V=V_+\oplus V_-$ of the space of flat sections $V$ of $\cM_\hbarr$ on the Stokes region. Here the subscripts $\pm$ is identified with the sheets of the spectral curve. The splittings for adjacent Stokes regions are related by Proposition~\ref{Vorosnaive}.
\end{theorem}

\begin{remark}[Non weakly GMN]
It may be easy to treat simple poles with the results in \cite{IwakiNakanishi2}. To remove the other conditions, it will require more delicate analysis like in higher order cases.
\end{remark}

\section{Planck constant in complex and real symplectic geometry}
\subsection{Complex symplectic geometry and deformation quantization}
Let $X$ be a complex manifold. Then the cotangent bundle $T^*X$ is a complex exact symplectic manifold. The canonical symplectic form can be locally written as $\sum d\zeta_i\wedge dz_i$. Here $z_i$ is a local coordinate of $X$ and $\zeta_i$ is the corresponding cotangent coordinate. The form $\omega$ is exact and has a canonical primitive $\lambda$, locally written as $\sum \zeta_i dz_i$.

We will consider the canonical quantization of $T^*X$ from local pieces. The canonical deformation quantization of $T^*X$ is given by the ring $\cD_\hbar:=\bC[\hbar][z_i, \hbar \partial_{z_i}]$. In the classical limit, $\hbar\partial_z$ corresponds to $\zeta$.

This gives a hint to make a Lagrangian into a conic Lagrangian: The original coordinate $\zeta$ splits into the product $\hbar\partial_z$ in the setting of quantization. Hence we will reconsider $\hbar$ and $\partial_z$ as independent variables. Then a Lagrangian submanifold $L\subset T^*M$ lifts to
\begin{equation}
    \widetilde{L}:=\lc (z, p,\hbar)\in T^*X\times \bC^\times_\hbar \relmid (z, \hbar p)\in L\rc.
\end{equation}
For the later use, it is convenient to work with $\eta:=\hbar^{-1}$. To get a Lagrangian submanifold, we moreover consider the Fourier dual coordinate $y$ of $\eta$. We set
\begin{equation}
    {L}_f:=\lc (z, p, \hbar, y)\in T^*X\times \bC^\times_\hbar\times \bC_y \relmid (z, \hbar p)\in L, y= -f(z, \hbar p)\rc
\end{equation}
where $f$ is a primitive of $L$ i.e., $df=\lambda|_L$.
We consider $\bC_y\times \bC^\times_\hbar\cong T^*_{\eta\neq 0}\bC_y$. Then ${L}_f$ is a Lagrangian manifold in $T^*X\times T^*_{\eta\neq 0}\bC_y$ with respect to the standard symplectic structure and is conic i.e., invariant under the scaling action of $\bC^\times_\hbar$ on the fibers of $T^*X\times T^*_{\eta\neq 0}\bC_y$.

There exists a projection
\begin{equation}
    \rho^\bC\colon T^*X\times T^*_{\eta\neq 0}\bC_y\rightarrow T^*X; (z, \zeta, y, \eta)\mapsto (z, \zeta/\eta)=(z, \hbar\zeta).
\end{equation}
Then the image of $L_f$ is precisely $L$. Conversely, $L_f$ is a Lagrangian leaf of $\rho^{-1}(L)$. This procedure is a classical trick in the theory of deformation quantization (for example, \cite{PS}).

\subsection{From complex symplectic geometry to a family of real symplectic geometries}
Considering $X$ as a real manifold, we have $T^*X_\bR$ the real cotangent bundle equipped with the standard real symplectic structure $\omega_\bR$.

In a holomorphic local coordinate $\{z_i\}$ of $X$, we have the associated real coordinate $(x_i, y_i)$ with $z_i=x_i+\sqrt{-1}y_i$. We can further identify the real and complex cotangent bundles via
\begin{equation}
dx_i\mapsto dz_i, dy_i\mapsto -\sqrt{-1}dz_i.
\end{equation}
Hence in the cotangent coordinate, we have $(\xi_i, \eta_i)\mapsto \xi_i-\sqrt{-1}\eta_i=\zeta_i$. This implies that $\Re(\omega)=\omega_\bR$. In the following, we will not distinguish $T^*X$ with $T^*X_\bR$.

Hence a holomorphic Lagrangian submanifold with respect to $\omega$ gives a real Lagrangian submanifold with respect to $\omega_\bR$.

Now we are going to vary the holomorphic symplectic form. For $\hbarr\in\bC^\times_\hbar$, we have another holomorphic symplectic form $\omega/\hbarr$. This induces a new real symplectic form $\Re(\omega/\hbarr)$ on $T^*X$. A holomorphic Lagrangian submanifold with respect to $\omega$ is also a holomorphic Lagrangian with respect to $\omega/\hbarr$. Hence it is also a real Lagrangian submanifold with respect to $\Re(\omega/\hbarr)$.

If $L$ is a holomorphic exact Lagrangian with respect to $\lambda$, then $L$ is also a real exact Lagrangian with respect to $\Re(\lambda/\hbarr)$. In this case, a real primitive function $f_\hbarr\colon L\rightarrow \bR$ can be also considered as a real primitive function of $L/\hbarr$ with respect to $\Re(\lambda)=\lambda_\bR$.

\begin{remark}
Since microlocal sheaf theory knows only about the standard symplectic form, to speak about $\Re(\omega/\hbarr)$, we will consider the following identification.
\begin{equation}
    (T^*X, \Re(\omega/\hbarr))\cong (T^*X, \Re(\omega/\hbarr))\xrightarrow{\cdot (1/\hbarr)} (T^*X, \Re\omega)\cong (T^*X, \omega_\bR)
\end{equation}
Hence a sheaf quantization of a real Lagrangian submanifold $L$ on the leftmost side is interpreted as a sheaf quantization of $L/\hbar$ in the rightmost side.
\end{remark}

For the Liouville form $\Re(\lambda/\hbarr)$, the associated $\rho$ can be written as follows:
\begin{equation}
\rho_\hbarr\colon T^*X\times T^*_{>0}\bR\rightarrow  T^*X; (z, \zeta, t, \tau)\mapsto (z, \hbar\zeta/\tau).
\end{equation}
If the context is clear, we will omit the subscript $\hbarr$.

We can compare this picture with the story of deformation quantization. We will consider $\bR_{\tau>0}$ is a ray in $\bC^\times_\eta$ defined by $\hbarr^{-1}\bR_{>0}$. Then we set the following morphisms
\begin{equation}
\begin{split}
    \iota\colon T^*X\times T^*_{>0}\bR\rightarrow T^*X\times T^*_{\upsilon \neq 0}\bC_\eta; (z, \zeta, t, \tau)\mapsto (z, \zeta, \hbarr t, \hbarr^{-1}\tau)\\
\end{split}
\end{equation}
Then $\rho\circ \iota=\rho_\hbarr$. This means $\rho_\hbarr$ precisely corresponds to a rotation of $\rho$ in Example~\ref{rho} by $\hbarr$.

\section{Sheaf quantization from exact WKB analysis}
We would like to construct a sheaf quantization of the spectral curve of a given meromorphic flat $\hbar$-connection using exact WKB analysis. Our sheaf quantization will be an object of $\mu(T^*C, \Re(\lambda/\hbarr), \sqrt{-1}\Im(\lambda/\hbarr), \Lambda_0)$.
\subsection{Construction}
We take and fix a rank 2 meromorphic flat $\hbar$-connection $\cM:=(\cE, \nabla)$. We set $L:=L_\cM$ to be the spectral curve of $\cM$.

Let us recall the WKB solutions:
\begin{equation}
\psi_\pm(z,\hbar)=\exp\lb\Re\int_{z_0}^z\frac{\xi_{\pm}}{\hbar}dz\rb\exp\lb\sqrt{-1}\Im\int_{z_0}^z\frac{\xi_{\pm}}{\hbar}dz \rb\sum_{m=0}^{\infty}\hbar^{m+\frac{1}{2}}\psi_m^\pm(z).
\end{equation}
We will treat the information of exponential factors separately. Fix $\hbarr\in \bC^\times_\hbar$ with sufficiently small $|\hbarr|$ as in Theorem~\ref{Koike}. The exponent of the first factor
\begin{equation}
    \exp\lb \Re\int_{z_0}^z\frac{\xi_\pm}{\hbarr}dz\rb
\end{equation}
is exactly a local primitive of the spectral curve with respect to the Liouville form $\Re(\lambda/\hbarr)$. The exponent of the second factor
\begin{equation}
    \exp\lb\sqrt{-1}\Im\int_{z_0}^z\frac{\xi_\pm}{\hbarr}dz \rb
\end{equation}
will be considered as an effect of the $B$-field $\Im(\omega/\hbarr)$.

We will first construct on a Stoke region $D$. We set
\begin{equation}
    s_{L}|_{D}:=\bigoplus_\pm \bK_{D_\pm}
\end{equation}
where
\begin{equation}
    D_\pm:=\lc (z, t)\in D'\times \bR_t \relmid t\geq -  \Re\int_{z_D}^z(\xi_\pm/\hbarr) dz \rc,
\end{equation}
$z_0$ is some point in the closure of $D$, and $D'$ is an open subset which contains $D$ and the Stokes edges of $D$ but not vertices (turning points). Let $T_c$ ($c\in \bR$) be the translation by $c$. We set
\begin{equation}
    S_{L}|_{D}:=\bigoplus_{c\in \bR} T_cs_{L}|_{D}.
\end{equation}
This defines an object of $\Sh^\bR_{\tau>0}(D'\times \bR_t)$ with a trivial equivariant structure and does not depend on the choice o f $z$. We view it as an object of $\mu(T^*D', \Re(\lambda/\hbarr), \sqrt{-1}\Im(\lambda/\hbarr), \Lambda_0)$.

For each Stokes region $D$, we fix resummed WKB solutions $\psi^D_\pm$. Take $z_D\in D$ and set
\begin{equation}
    \psi_{\pm, \geq 0}^{D}:=\exp\lb-\int_{z_D}^z \xi/\hbarr \rb\psi_{\pm}^D.
\end{equation}
This is again unique up to constant multiplication. When we go across a Stokes edge, we glue up the sheaves by the following morphisms: For adjacent Stokes regions $D_1$ and $D_2$, the Stokes edge separating two regions satisfies $\Re\int^x_v(\xi_+-\xi_-)/\hbarr dz <0$ or $\Re\int^x_v(\xi_+-\xi_-)/\hbarr dz >0$. Then we have
\begin{equation}
\End\lb \bigoplus_\pm \bK_{(D'_1\cap D'_2)_\pm^v}\rb\\
        =\bK\cdot\id_{+}\oplus \bK\cdot \id_{-}\oplus \bK\cdot e
\end{equation}
where $\id_\pm\in \End\lb\bK_{(D'_1\cap D'_2)_\pm^v}\rb$ are the identities of the components and
\begin{equation}
    e\in \begin{cases}
        &\Hom\lb \bK_{(D'_1\cap D'_2)_+^v}, \bK_{(D'_1\cap D'_2)_-^v}\rb \text{ if $\Re\int^x_v(\xi_+-\xi_-)/\hbarr dz >0$} \\
        & \Hom\lb \bK_{(D'_1\cap D'_2)_-^v}, \bK_{(D'_1\cap D'_2)_+^v}\rb \text{ if $\Re\int^x_v(\xi_+-\xi_-)/\hbarr dz <0$}
    \end{cases}
\end{equation}
is a canonical nontrivial basis coming from the inclusion of the closed sets. We can reflect the connection formula of Proposition~\ref{Vorosnaive} using this:
\begin{equation}
    \varphi_{D_1, D_2}:=d_1\id_++d_2\id_--ae\in \End\lb \bigoplus_\pm \bK_{(D'_1\cap D'_2)_\pm^v}\rb.
\end{equation}
We consider this morphism as an isomorphism $\varphi\colon (S_{L}|_{D_1})|_{D_1\cap D_2}\rightarrow (S_{L}|_{D_2})|_{D_1\cap D_2}$.

\begin{lemma}
The sheaves $S_{L}|_{D}$'s can be glued up by $\varphi$'s and give an object in $\Sh_{\tau>0}^\bR(( C\bs (M\cup \Zero(\cM)))\times \bR_t)$. We denote this object by $S^{\shbarr}_{\cM}|_{C\bs (M\cup \Zero(\cM))}$. The resulting object does not depend on the choice of $z_D$ and $\psi_\pm^D$.
\end{lemma}
\begin{proof}
The first assertion is clear from the consistency of the connection formulas. Since the resummation of WKB solutions are unique up to constants, it does not affect the construction.
\end{proof}

We would like to extend the object to $C\bs M$. Around a turning point $v$, we have several Stokes regions $D_1, D_2, ..., D_n$. Hence $s_{L}|_{D_j}$ for $j=1,2,..., n$ will glue up to a sheaf $s_{L}$ and $S_{L}|_{\bigcup _{i=1}^nD_i}=\bigoplus_c T_cs_L$. Note that there are no monodromies of the solutions, hence we have
\begin{equation}
    \Ext^1\lb \bK_v\boxtimes \bK_{t\geq 0}^{\oplus 2}, s_{L}\rb=\bK^{\oplus 2}.
\end{equation}
where $\bK_v$ is the skyscraper sheaf. By using $1\oplus 1$ in this extension space for each turning point, we get an object on $(C\bs M)\times \bR_t$.

\begin{remark}
Under the microsupport condition $\musupp=L$, this is a unique extension. One can also obtain the same object by the push-forward from the complement of $\{v\}\times\bR_t$.
\end{remark}

By the zero extension, we get an object $S^{\shbarr}_{\cM}\in \Sh^\bR_{\tau>0}(C\times \bR_t)$. As a summary, we have the following:

\begin{theorem}
We have a simple object $S^\shbarr_{\cM}\in \mu(T^*C, \Re(\lambda/\hbarr), \sqrt{-1}\Im(\lambda/\hbarr), \Lambda_0)$ with $\musupp(S_{\cM}^\shbarr)=L_\cM$.
\end{theorem}

\section{Schr\"odinger equations}
In this section, we would like to implement the construction in the last section more concretely for Schr\"odinger equations. The explicit construction will enable us to see Voros symbols from sheaf quantizations. There are some overlaps with the last section, but we would like to repeat it to keep the concreteness.

\subsection{Notations and Voros formula}
We consider a second order differential operator $\cQ$ on $C$ with a parameter $\hbar$ locally written as
\begin{equation}\label{globalschrodinger}
    ((\hbar\partial)^2-Q)\psi=0.
\end{equation}
We call this equation {\em Schr\"odinger equation}. Here $Q(z,\hbar)=\sum_{i=0}^kQ_i(z)\hbar^i$ for some $k$. Then $Q_0$ defines a global quadratic differential on $C$.
\begin{assumption}
In this section, we assume that the quadratic differential $Q_0$ is weakly GMN and any zeros are simple.
\end{assumption}

Consider a formal solution of the form $\psi(z, \hbar)=\exp{\lb\int^z \sum_{m=-1}^\infty\hbar^mP_m dz\rb}$, then one can derive a system of equations governing $P_m(z)$'s:
\begin{equation}\label{EquationP}
    \begin{split}
        &P_{-1}^2=Q_0\\
        &2P_{-1}P_m+\sum_{\substack{m_1+m_2=m-1\\ m_1, m_2\geq 0}}P_{m_1}P_{m_2}+\frac{dP_{m-1}}{dz}=Q_{m-1}(z).
    \end{split}
\end{equation}
The first equation has two solutions: $P_{-1}^{\pm}=\pm \sqrt{Q_0}$. The second equation has unique solutions for a choice of $\pm$. We write these two solutions by
\begin{equation}
\begin{split}
        P^{\pm}(z, \hbar)&:=P^{\pm}_{od}(z, \hbar)+P_{ev}(z, \hbar)\\
        P^{\pm}_{od}&:=\pm \frac{1}{2}(P^+-P^-)\\
        P_{ev}&:=\frac{1}{2}(P^++P^-)\\
\end{split}
\end{equation}
Note that $P_{ev}$ does not depend on the choice of $\pm$, and $P_{od}^+$ differs from $P_{od}^-$ only by the overall $\pm$. We also have
\begin{equation}
    P_{ev}=-\frac{1}{2}\frac{1}{P_{od}}\frac{dP_{od}}{dz}.
\end{equation}
This means $P_{ev}=\frac{\partial}{\partial z}\log(1/ \sqrt{P_{od}})$ formally.
Hence we can formally rewrite $\psi(z, \hbar)$ as
\begin{equation}\label{WKB}
    \begin{split}
        \psi_{\pm}(z, \hbar)&=\exp\lb\pm \int^zP_{od}dz\rb\exp \lb\int^zP_{ev}\rb\\
        &=\frac{1}{\sqrt{P_{od}}}\exp\lb\pm \int^zP_{od}\rb
    \end{split}
\end{equation}
These two solutions are called WKB solutions of (\ref{globalschrodinger}).

To treat this kind of WKB solutions, it is conceptually necessary to fix a spin structure $\sqrt{K}$ of $C$. Then we can upgrade the Sch\"odinger equation to a meromorphic flat $\hbar$-connection:
\begin{equation}
    \hbar d+\lb\begin{array}{cc}
      0 & Q \\
      1 & 0
    \end{array}\rb
\end{equation}
on the bundle $K^{1/2}\oplus K^{-1/2}$. We denote the spectral curve of the connection by $L$. Associated to the spin structure, we also have a (topological) branched covering $\widetilde{\pi}\colon \widetilde{L}\rightarrow L$ branched at the branch points of $\pi\colon L\rightarrow C$.

Since (\ref{WKB}) involves $Q_0^{1/4}$, to fix a branch of (\ref{WKB}), it is necessary to designate a sheet of $\widetilde{L}$. Then after multiplying $\hbar^{-1/2}$, (\ref{WKB}) gives a WKB solution in the sense of Definition~\ref{defofWKBsol}. Since it is almost harmless to work without multiplying $\hbar^{-1/2}$, we will sometimes omit this.

Now we would like to introduce Voros' connection formula. Let $D_1, D_2$ be adjacent Stokes regions. Let $z_0$ be a turning point in the Stokes curve separating $D_1$ and $D_2$. We will use the following normalization in the expression (\ref{WKB}). Fix $z\in D_1$ and $(z, \zeta)\in \pi^{-1}(z)\subset L$. Let us denote the corresponding branch of $P_{od}$ by $P_{od}^{(z, \zeta)}$. Let $\gamma_z$ be a minimal counterclockwise loop starts from $(z,\zeta)$ surrounding $z_0$ on $L$. Then we set
\begin{equation}\label{normalization}
    \int_{z_0}^{(z,\zeta)}P_{od} dz:=\frac{1}{2}\int_{{\gamma_{z}}} P_{od}^{(z,\zeta)} dz.
\end{equation}
We also fix a branch of $\sqrt{P_{od}}$, which is given by a choice of $\tilde{z}\in \tilde{\pi}^{-1}(z,\zeta)$.
We will write the corresponding resummed solutions on $D_1$ by $\Psi_{z_0, \tilde{z},1}^{D_1}$ i.e.,
\begin{equation}
    \Psi_{z_0, \tilde{z},1}^{D_1}:=\frac{1}{(\sqrt{P_{od}})_{\tilde{z}}}\exp\lb\int_{z_0}^{(z,\zeta)}P_{od}dz \rb.
\end{equation}

We also take a lift of $\gamma_z$ to $\widetilde{L}$ starting from $\tilde{z}$ and denote the endpoint by $\widetilde{z}'$. We set
\begin{equation}
    \Psi_{z_0, \tilde{z},2}^{D_1}:=\frac{1}{(\sqrt{P_{od}})_{\tilde{z}'}}\exp\lb\int_{z_0}^{\tilde{\pi}(\tilde{z}')}P_{od}dz \rb.
\end{equation}
This is also understood to be resummed. We set $\Psi_{z_0. \tilde{z}}^{D_1}:=(\Psi_{z_0, \tilde{z},1}^{D_1},\Psi_{z_0, \tilde{z},2}^{D_1})$.

Let $\gamma_{12}$ be an interval which starts from $z$, ends at $z_2\in D_2$, and intersects with the Stokes graph exactly once at a point in the Stokes curve separating $D_1$ and $D_2$. Take the lift $\widetilde{\gamma_{12}}$ of $\gamma_{12}$ to $\widetilde{L}$ starting from $\widetilde{z}$. We denote the end point of $\widetilde{\gamma_{12}}$ by $\widetilde{z_2}$. Then we define $\Psi^{D_2}_{z_0, \widetilde{z_2}}$ in the same way as $\Psi^{D_1}_{z_0, \widetilde{z_2}}$ replacing $\widetilde{z}$ with $\widetilde{z_2}$.

\begin{theorem}[{Voros~\cite{Voros}, Aoki--Kawai--Takei~\cite{AKT}}]\label{Voros}
The analytic continuation of $ \Psi_{z_0, \tilde{z}}^{D_1}$ crossing the Stokes curve is related to $\Psi_{z_0, \tilde{z}_2}^{D_2}$ by
\begin{equation}
    \Psi_{z_0, \tilde{z}}^{D_1}= \Psi_{z_0, \tilde{z}_2}^{D_2}\cdot S
\end{equation}
where
\begin{equation}
    S=\begin{cases}
    \left(
    \begin{array}{cc}
      1 & 0  \\
     -1 & 1
    \end{array}
  \right) \text{ if $\int_v^z\frac{\sqrt{Q_0}}{\hbarr}dz>0$ on the Stokes curve, }
  \\
  \\
  \left(
    \begin{array}{cc}
      1 & -1  \\
      0 & 1
    \end{array}
  \right)\text{ otherwise.}
    \end{cases}
\end{equation}
\end{theorem}

On a horizontal strip $D$, we have two turning points in its closure. We denote them by $v_1, v_2$. Take a path $\gamma_{v_1v_2}$ in $D$ connects $v_1$ and $v_2$. Take a lift $\widetilde{\gamma_{v_1v_2}}$ of $\gamma_{v_1v_2}$ on $\widetilde{L}$. For $\tilde{z}\in \widetilde{\gamma}$, we have two solutions normalized at $v_1$ and $v_2$ and denote them by $\Psi^D_{v_1, \tilde{z}}$ and $\Psi^D_{v_2, \tilde{z}}$. Then we have $\Psi^D_{v_1, \tilde{z}}=\exp(\int_{\tilde{\pi}(\widetilde{\gamma_{v_1v_2}})} P_{od}dz) \Psi^D_{v_2, \tilde{z}}$.

\subsection{Construction}
Note that, in the present case, $\xi_\pm=\pm\sqrt{Q_0}$.
Let us recall the WKB solutions:
\begin{equation}
    \psi(z,\hbar)=\exp\lb \Re\int_{z_0}^z\frac{\sqrt{Q_0}}{\hbar}dz\rb\exp\lb\sqrt{-1}\Im\int_{z_0}^z\frac{\sqrt{Q_0}}{\hbar}dz \rb\sum_{m=0}^{\infty}\hbar^{m+\frac{1}{2}}\psi_m^\pm(z).
\end{equation}
We will treat the information of exponential factors separately. Fix $\hbarr\in \bC^\times_\hbar$ with sufficiently small $|\hbarr|$ as in Theorem~\ref{Koike}. Then the exponent of the first factor
\begin{equation}
    \exp\lb \Re\int_{z_0}^z\frac{\sqrt{Q_0}}{\hbarr}dz\rb
\end{equation}
is exactly a local primitive of the spectral curve with respect to the Liouville form $\Re(\lambda/\hbarr)$. The exponent of the second factor
\begin{equation}
    \exp\lb\sqrt{-1}\Im\int_{z_0}^z\frac{\sqrt{Q_0}}{\hbarr}dz \rb
\end{equation}
will be considered as an effect of the $B$-field $\Im(\omega/\hbarr)$.

We will first construct on a Stoke region $D$. Then $\widetilde{L}|_D\rightarrow D$ is a trivial covering of degree 4. Fix a sheet $i$. We set
\begin{equation}
    s^v_{L}|_{D,i}:=\bigoplus_\pm \bK_{D^v_\pm}
\end{equation}
where
\begin{equation}
    D^v_\pm:=\lc (z, t)\in D'\times \bR_t \relmid t\geq \mp \Re\int_v^z(\sqrt{Q_0}/\hbarr) dz \rc,
\end{equation}
$v$ is a turning point in the closure of $D$, and $D'$ is an open subset which contains $D$ and the Stokes edges of $D$ but not vertices (turning points). This object does not depend on $i$, but we will keep the indices. Let $T_c$ ($c\in \bR$) be the translation by $c$. We set
\begin{equation}
    S^v_{L}|_{D,i}:=\bigoplus_{c\in \bR} T_cs^v_{L}|_{D,i}.
\end{equation}
This defines an object of $\Sh^\bR_{\tau>0}(D'\times \bR_t)$ with a trivial equivariant structure. We view it as an object of $\mu(T^*D', \Re(\lambda/\hbarr), \sqrt{-1}\Im(\lambda/\hbarr), \Lambda_0)$.

We would like to list up the gluing-up isomorphisms:
\begin{enumerate}
\item (Change of sheets) For a sheet $i$, we denote the sheet where we will arrive by one counter clockwise loop around $v$ by $i'$, Then we define
\begin{equation}
    \varphi_{ii'}\colon  S^v_{L}|_{D,i}\rightarrow  S^v_{L}|_{D,i'}
\end{equation}
by $T$: namely, the multiplication by $\sqrt{-1}$.
\item (Change of normalizations) For $v_1, v_2\in D$, the isomorphism $\varphi_{v_1v_2}\colon S^{v_1}_{L}|_{D,i}\rightarrow S^{v_2}_{L}|_{D,i}$ is induced by the scalar multiplication
\begin{equation}
        \exp\lb \int_{\gamma_{v_1,v_2}}{P_{od}^{\geq 0}}\rb \cdot (-) \colon
        T_c \bK_{D^{v_1}_\pm}\rightarrow T_{c\mp\Re\int_{\gamma_{v_1v_2}}(\sqrt{Q_0}/\shbarr)dz} \bK_{D_{\pm}^{v_2}}
\end{equation}
where $P_{od}^{\geq 0}:=\sum_{m=0}^{\infty}\hbar^{2m+1}P_{2m+1}$.
\item (Change of regions) For adjacent Stokes regions $D_1$ and $D_2$, the Stokes edge separating two regions satisfies $\Re\int^x_v\sqrt{Q_0}/\hbarr dz <0$ or $\Re\int^x_v\sqrt{Q_0}/\hbarr dz >0$. Then we have
\begin{equation}
\End\lb \bigoplus_\pm \bK_{(D'_1\cap D'_2)_\pm^v}\rb\\
        =\bK\cdot\id_{+}\oplus \bK\cdot \id_{-}\oplus \bK\cdot e
\end{equation}
where $\id_\pm\in \End\lb\bK_{(D'_1\cap D'_2)_\pm^v}\rb$ are the identities of the components and
\begin{equation}
    e\in \begin{cases}
        &\Hom\lb \bK_{(D'_1\cap D'_2)_+^v}, \bK_{(D'_1\cap D'_2)_-^v}\rb \text{ if $\Re\int^x_v\sqrt{Q}/\hbarr dz >0$} \\
        & \Hom\lb \bK_{(D'_1\cap D'_2)_-^v}, \bK_{(D'_1\cap D'_2)_+^v}\rb \text{ if $\Re\int^x_v\sqrt{Q}/\hbarr dz <0$}
    \end{cases}
\end{equation}
is a canonical nontrivial basis coming from the inclusion of the closed sets. We can reflect the Voros connection formula using this:
\begin{equation}
    \varphi_{D_1, D_2}:=\id_++\id_--e\in \End\lb \bigoplus_\pm \bK_{(D'_1\cap D'_2)_\pm^v}\rb.
\end{equation}
We consider this morphism as an isomorphism $\varphi\colon S^v_{L}|_{D_1, i}|_{D_1\cap D_2}\rightarrow S^v_{L}|_{D_2, i}|_{D_1\cap D_2}$.
\end{enumerate}

\begin{lemma}
The sheaves $S^v_{L}|_{D,i}$'s can be glued up by $\varphi$'s and give an object in $\Sh_{\tau>0}^\bR(( C\bs (M\cup \Zero(\cM)))\times \bR_t)$. We denote this object by $S^{\shbarr}_{\cQ,\sqrt{K}}|_{C\bs (M\cup \Zero(\cM))}$.
\end{lemma}
\begin{proof}
This is clear from the consistency of the connection formulas.
\end{proof}

We would like to extend the object to $C\bs M$. Around a turning point, we have three Stokes regions $D_1, D_2, D_3$. Hence $s^v_{L}|_{D_j,i}$ for $j=1,2,3$ will glue up to a sheaf $s^v_{L}$ and $S^v_{L}|_{D_1\cup D_2\cup D_3}=\bigoplus_c T_cs^v_L$. Note that there are no monodromies of the solutions, hence we have
\begin{equation}
    \Ext^1\lb \bK_v\boxtimes \bK_{\bR_{t\geq 0}}^{\oplus 2}, s^v_{L}\rb=\bK^{\oplus 2}.
\end{equation}
where $\bK_v$ is the skyscraper sheaf. By using $1\oplus 1$ in this extension space for each turning point, we get an object on $(C\bs M)\times \bR_t$. By the zero extension, we have an object $S^{\shbarr}_{\cQ,\sqrt{K}}\in \Sh^\bR_{\tau>0}(C\times \bR_t)$. As a summary, we have the following:

\begin{theorem}
We have a simple object $S^\shbarr_{\cQ,\sqrt{K}}\in \mu(T^*C, \Re(\lambda/\hbarr), \sqrt{-1}\Im(\lambda/\hbarr), \Lambda_0)$ with $\musupp(S_{\cQ,\sqrt{K}}^\shbarr)=L$.
\end{theorem}

\subsection{Voros symbol}
We would like to recall the notion of Voros symbol. We refer to \cite{DDP} and \cite{IwakiNakanishi} for more accounts.

The ``dual" of a segments-free $\hbarr$-Stokes graph gives an ideal triangulation of $C$ with vertices in $M$. In this situation, $H_1(L, \bZ)$ is spanned by the following~\cite{BridgelandSmith}:
\begin{enumerate}
    \item The pull back of cycles in $C\bs M$.
    \item In an $\hbarr$-Stokes region of horizontal strip type, take a path connecting two turning points. This lifts to a cycle in the spectral curve. We call this cycle, a {\em Voros edge cycle}.
\end{enumerate}
\begin{definition}
For a cycle $\gamma\in H_1(L, \bZ)$, the Voros symbol is defined by
\begin{equation}
    V_{\gamma}(\hbarr):=\oint_\gamma P_{od}(\gamma, \hbarr).
\end{equation}
The local system on $L$ whose monodromy along $\gamma$ given by $e^{V_\gamma(\shbarr)}$ is called the $\hbarr$-Voros local system.
\end{definition}
It was proved that the collection $\{e^{V_\gamma(\shbarr)}\}$ is a set of cluster variables. Regarding this aspect, we sometimes call the set $\{e^{V_\gamma(\shbarr)}\}$ (or the Voros local system) {\em Voros--Iwaki--Nakanishi coordinate}.

Note that there is another class of Voros symbols associated to {\em paths}. We do not recall it here, since we do not use it in this paper.

\subsection{Microlocalization}
We would like to equip $L$ with a Maslov grading and a relative Pin structure.

We will use the standard Maslov map $\alpha$ on $T^*C$~\cite{NZ}. Then the zero section canonically equipped with a Maslov grading $\tilde{\alpha}=0$. For any Riemann surface, the second Stiefel--Whitney class always vanishes. Hence the notion of relative Pin structures and Spin structures coincide.

Let $B$ be the set of branched points in $L$.
Since we have a trivial grading on the zero section, we can induce a grading on $L\bs B$. In other words, we have a lift of $L\bs \pi^{-1}(M)\rightarrow U(1)$ to $L\bs B\rightarrow \bR$. The obstruction class $c\in H^1(L, L \bs B ,\bZ)$ to extend this lift to $L$ is zero, since the grading can be extended to $B=\Zero(\cM)$ on the base space. Hence we have a unique extension.

For a spin structure, we use $\sqrt{K}|_{C\bs M}$ to have a spin structure on $L\bs B$. We have a diagram
\[
\xymatrix{
L\bs B \ar[d] \ar[r] & B\mathrm{Spin}(2) \ar[d]  \\
L \ar[r] & B\mathrm{SO}(2)
}
\]
The obstruction to extend the spin structure lives in $H^2(L, L\bs \pi^{-1}(v), \bZ/2\bZ)$, which trivially vanishes. The classes of the extensions is a torsor over $H^1(L, L\bs B, \bZ/2\bZ)$. As a spin structure on $L$, these ambiguities are the images of $H^1(L, L\bs \pi^{-1}(v),\bZ/2\bZ)\rightarrow H^1(L, \bZ/2\bZ)$, which is zero. Hence we have a unique spin structure on $L$ extending $\sqrt{K}$.

The space of spin structures is a torsor over $H^1(L, \bZ/2\bZ)$. We consider the element of $H^1(L, \bZ/2\bZ)$ corresponding to the $\bZ/2$-valued rank 1 local system whose monodromy along the Voros edge cycles are $-1$ and those along the pull-backed cycles are $1$. We act this element on the above spin structure to get another spin structure. We will use the resulting spin structure as a part of the brane data.

\begin{corollary}
The $\Lambda$-local system $\mu(S_{L, \sqrt{K}}^\shbarr)$ along the above brane structure is convergent, i.e., there exists a $\Lambda^{conv}$-local system $\cE$ such that $\cE\otimes_{\Lambda^{conv}}\Lambda\cong \mu(S_{L,\sqrt{K}}^\shbarr)$. Moreover, $\cE\otimes_{\Lambda^{conv}}\bC$ where the action of $\Lambda^{conv}$ on $\bC$ is given by inserting $T^c=e^{-c}$ is the Voros local system.
\end{corollary}
\begin{proof}
On each Stokes region, we have two sheets of $L$. Further, we name two sheets of $\widetilde{L}\rightarrow L$ by $\pm$. Then comparing $\pm$ on each Stokes edge, we get a $\mathrm{\check{C}ech}$ representation of the spin structure, namely $\pm$ on each Stokes edge. This gives gluing data of local microlocal sheaves, in our situation, which is the microlocalization of $\bK_{t\geq \mp \Re\int^z_{v_0}\sqrt{Q_0}/\hbarr dz}$.

If a Stokes edge has $+$, it means the name of sheets are the same on the both sides of the Stokes edge. In this case, the corresponding sheaves ($S^v_{L}|_{D,i}$'s) were glued by $+$.

If a Stokes edge has $-$, it means the name of sheets are different on the both sides of the Stokes edge.  In this case, the corresponding sheaves ($S^v_{L}|_{D,i}$'s) were glued by $-$. However this sign will be canceled out with the gluing of the reference microlocal sheaves.

Going around a turning point once changes the sheet of $\widetilde{L}$ nontrivially. Hence, along a Voros edge cycle, we see the multiplication by $-1$. The spin structure we chose cancels this $-1$.

Hence, by comparing the connection formulas of the solutions of the Schr\"odinger equations with $\varphi$'s used in the gluing, we can see that a monodromy of $\mu(S^\shbarr_{L, \sqrt{K}})$ along $\gamma$ is $e^{\int_\gamma P_{od}^{\geq 0}dz}\cdot T^{-\int_\gamma\Re(\sqrt{Q_0}/\shbarr)dz}\cdot e^{\sqrt{-1}\int_\gamma(\Im(\lambda/\shbarr)dz)}$. After inserting $T^c=e^{-c}$, we get the Voros monodromy.
\end{proof}

\section{Irregular Riemann--Hilbert correspondence}
In this section, we would like to relate our sheaf quantization to some known formalisms of irregular Riemann--Hilbert correspondence. The Riemann--Hilbert point of view will be again treated in the final section. In the following, we fix a sufficiently small $\hbarr\in \bC^\times$. In this section, it is convenient to think $S^\shbarr_{\cM}$ as a sheaf quantization of $L/\hbarr$ with respect to the standard symplectic structure.

\subsection{D'Agnolo--Kashiwara's formalism}
We would like to relate our sheaf quantization to D'Agnolo--Kashiwara's holonomic Riemann--Hilbert correspondence~\cite{DK}.

We briefly recall the statement of \cite{DK}. For a complex manifold $X$, let us denote the derived category of holonomic $\cD$-modules by $D^b_{hol}(\cD_X)$.

For the topological side, we have to introduce the notion of enhanced ind-sheaves. Let $\overline{\bR}$ be the two point compactification of $\bR$ i.e. $\bR\cong (0,1)\hookrightarrow [0,1]=\overline{\bR}$. The category of enhanced ind-sheaves is defined in two steps: First, we set
\begin{equation}
D^b(\mathrm{I}\bC_{X\times(\overline{\bR},\bR)}):=D^b(\mathrm{I}\bC_{X\times\overline{\bR}})/D^b(\mathrm{I}\bC_{X\times{\overline{\bR}\bs \bR}})
\end{equation}
where $D^b(\mathrm{I}\bC_X)$ is the bounded derived category of ind-sheaves over $X$.
We set $\bC_{t\lesseqgtr 0}:=\bC_{\lc (x,t)\in X\times \overline{\bR}\relmid t\in \bR, t\lesseqgtr 0 \rc}$. We denote the convolution product along $\bR_t$ by $\potimes$. We set
\begin{equation}
\mathrm{IC}_{t^*=0}:=\lc K\relmid K\potimes \bC_{\leq 0}\simeq 0, K\potimes \bC_{\geq 0}\simeq 0\rc.
\end{equation}
The category of enhanced ind-sheaves over $X$ is defined by
\begin{equation}
E^b({\mathrm{I}\bC_X}):=D^b(\mathrm{I}\bC_{X\times (\overline{\bR}, \bR)})/\mathrm{IC}_{t^*=0}.
\end{equation}
We set
\begin{equation}
\bC^E_X:=\underset{a \rightarrow\infty}{\forlim}\bC_{t\geq a}
\end{equation}
as an object of $E^b({\mathrm{I}\bC_X})$. As usual, $\forlim$ means Ind-colimit. An object of $E^b({\mathrm{I}\bC_X})$ is said to be $\bR$-constructible if there locally exists an $\bR$-constructible sheaf $\cE$ and the object is isomorphic to $\cE\potimes \bC^E_X$.
\begin{theorem}[\cite{DK}]
There exists a contravariant fully faithful functor $\mathrm{Sol}^E\colon D^b_{hol}(\cD_X)\hookrightarrow E^b(\mathrm{I}\bC_X)$.
\end{theorem}

We would like to explain how to construct an object in $E^b(\mathrm{I}\bC_X)$ from a simple object of ${\Sh}^{\bR}(X)$. For an object $\cE\in {\Sh}^{\bR}(X)$, we have local trivializations $\cE|_{U_i}\cong \bigoplus_c T_c\cE_i$ on some open covering $\{U_i\}$. By appropriate translations, we can assume that $\cE_i\cong T_{c_{ij}}\cE_j$ on each overlap.
On some overlap $U_i\cap U_j$, we have a gluing isomorphism $f$ in $\Hom(\cE|_{U_i}, \cE|_{U_j})\cong \Hom(\cE_i, \bigoplus_cT_c\cE_j)$. The morphism $f$ can be written as $f=\sum f_c=T^{c_{ij}}f_0+\sum_{c>c_{ij}} f_{c}$ where $f_c\colon \cE_i\rightarrow T_c\cE_j$ and $f_0$ is an automorphism of $\cE_i$.

\begin{definition}
We say $\cE$ is finite if there exists an open covering such that each gluing isomorphism $\sum f_c$ has maximal $c'$ among $c$ with $f_c\neq 0$.
\end{definition}

\begin{remark}
This notion is not enough to apply to higher order cases. For such cases, we should define the weaker property ``convergent" saying the gluing isomorphisms are convergent under the substitution $T=e^{-1}$. We will not explore further here, since we do not treat the convergence for higher order cases in this paper.
\end{remark}

In the following, we assume $\cE$ is finite. Let $c'$ be the maximum among $c$ with $f_c\neq 0$. Then we set $\tilde{f}:=\sum_c T^{c'-c}e^{c_{ij}-c}f_c\in \Hom(\cE_i, T_{c'}\cE_j)$. Then $\tilde{f}$ induces a morphism $\cE_i\potimes \bC^E_M\rightarrow \cE_j\potimes \bC^E_M$.
\begin{lemma}\label{En}
The induced morphisms define a descent data for $E^b(\mathrm{I}\bC_M)$. We denote the resulting object by $E(\cE)$.
\end{lemma}
\begin{proof}
Each descent condition is expressed as ``some composition is equal to $\id$". Since the original data satisfies the descent condition, the modified morphisms ($\tilde{f}$'s) satisfy the condition ``some compositions is equal to $T^c\id$". After taking $\potimes \bC^E_M$, this becomes $\id$.
\end{proof}

We will now set up our comparison. For a meromorphic flat $\hbar$-connection $\cM$, we consider $\cM^\hbarr$ as in Section 4.1.

\begin{proposition}
We have $\mathrm{Sol}^E(\cM^\hbarr)\cong E(S^\hbarr_{\cM})$.
\end{proposition}
\begin{proof}
By the construction, $S_\cM^\hbarr$ is finite. Since the gluing comes from the Stokes data, this follows from \cite{DK}.
\end{proof}

\subsection{Irregularity knots and non-Novikovized quantization}
We next compare our sheaf quantization with the Legendrian knot description of Stokes data~\cite{STWZ}.

We first define the notion of irregularity knot. For an irregular singularity $p$ of a differential equation over $C$, we denote the irregularity type of $p$ by $\{f_1,..., f_k\}$ where $f_i$ is a finite Puiseux series. Take a small disk $D$ around the singularity and fix a coordinate $z$ on the disk. For each $f_i$, we give an immersion of $\sqcup_k S^1$ in $D$ by
\begin{equation}
    \bigcup_{i=1}^k\{(\theta, \epsilon\cdot \Re(f_i(e^{\sqrt{-1}\theta}/r))\}
\end{equation}
in the polar coordinate. Here $\epsilon$ is taken sufficiently small so that the immersion is into $D$ and $r$ is some large positive number. Putting the outward conormals on the immersion, we have a Legendrian link on $S^*D$. For a sufficiently large $r$, the Legendrian isotopy class of the link does not depend on $r$. We call this link, the irregularity knot and denote it by $L_{\{f_i\}}$. We identify the Legendrian link with the associated conical Lagrangian in $T^*D\subset T^*C$.

\begin{theorem}[\cite{STWZ}]\label{STWZRH}
There exists an equivalence between the category of meromorphic connections with the irregularity type $\{f_i\}$ and the subcategory of constructible sheaves spanned by the objects whose microsupports are in $L_{\{f_i\}}\cup T^*_CC$ and the microstalks at $L_{\{f_i\}}$ are pure and concentrated in degree $0$. We denote the equivalence by $M$.
\end{theorem}

We will consider the following modification of $S^{\hbarr}_{\cM}$ around the poles.

Let $p$ be a pole of the differential equation and take a local coordinate $z$ around $p$. Then we consider the following curved cylinder
\begin{equation}
    \cC_p:=\lc(z,t)\relmid |z-p|\leq \eta \rc
    \cup \lc (z, t)\relmid \substack{|z-p|>\eta, 0< \epsilon(|z-p|-\eta)< \pi,\\t\geq \tan(-\pi/2+\epsilon(|z-p|-\eta))}\rc \subset C\times \bR_t
\end{equation}
where $\epsilon$ (resp. $\eta$)is sufficiently large (resp. small) positive number. On each Stokes region $D$, we take
\begin{equation}
    D^\pm:=\pi^{-1}\pi(\lc(x, t)\in D \times \bR_t\relmid t\geq -\Re \int \frac{\xi_\pm}{\hbarr}dz\rc \bs \bigcup_{p\in \overline{D}}\cC_{p})\cap \lc(x, t)\in D\times \bR_t\relmid t\geq -\Re\int \frac{\xi_\pm}{\hbarr}dz\rc.
\end{equation}
where $\pi\colon D\times \bR\rightarrow D$ is the projection. Figure~\ref{modification} is an example of the situation where the gray shaded region is an example of $D^\pm$.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=10.0cm]{pic11.png}
\end{center}
\caption{An example of $D^\pm$}
\label{modification}
\end{figure}
We set
\begin{equation}
    S_{L_c}^D:=\bigoplus_c T_c\lb\bigoplus_\pm \bK_{D^\pm}\rb.
\end{equation}

We can glue up this by the same rule as in the construction of $S^{\shbarr}_{\cM}$, except for $\varphi_{v_1v_2}$. In the definition of $\varphi_{v_1v_2}$, we replace $\exp(\int_{\gamma_{v_1v_2}}P_{od}^{\geq 0})$ with $\exp(\int_{\gamma_{v_1v_2}}\frac{\xi_\pm}{\hbarr})\exp(\int_{\gamma_{v_1v_2}} P^{\geq 0}_{od})$. We denote the resulting sheaf $S^{\shbarr}_{\cM, c}$ and we set $L_c:=\musupp(S^{\shbarr}_{\cM,c})$. Note that $L_c$ is conical and equal to $L_{\{f_i\}}$ outside some neighborhood of the zero section.

\begin{proposition}\label{conicVoros}
\begin{enumerate}
    \item There exists a Lagrangian homotopy between $L_\cM/\hbarr$ and $L_c$.
    \item The microlocalization of $S^{\shbarr}_{\cM, c}$ is the same as the microlocalization of $S^{\shbarr}_{\cM}$ along the Lagrangian homotopy.
\end{enumerate}
\end{proposition}
\begin{proof}
\begin{enumerate}
    \item Since, the both $L_\cM$ and $L_c$ are Lagrangian homotopic to $L_\cM\bs T^*\pi(\cC)$ where $\cC=\bigcup_p \cC_p$.
    \item The microlocalization is determined by the microlocalization at $L_\cM\bs T^*\pi(\cC)$.
\end{enumerate}
\end{proof}

Let $p\colon C\times\bR\rightarrow C$ be the usual projection.
By $p^\bR_*$, we mean the following: for an $\bR$-equivariant sheaf $\cE$ on $M\times \bR_t$, we set
\begin{equation}
    p_*\cHom(\bigoplus_cT_c\bK_{t\geq 0}, \cE).
\end{equation}
where $p_*$ is the $\bR$-equivariant push-forward. Here one can see a similarity to the construction in \cite{Kuw}.

For the reader's convenience, we would like to see an example of $p^\bR_*$. Let us consider $\bigoplus_c T_c\bK_{t\geq 0}$ as a trivially $\bR_t$-equivariant sheaf on $\bR_t$. Consider a point $*$ equipped with the trivial $\bR_t$-action. Applying $p_*^\bR$ to $\bigoplus_c T_c\bK_{t\geq 0}$ gives $\Lambda_0$.

\begin{proposition}\label{projection}
The $\bR$-equivariant push-forward of $p_*^\bR S^\shbarr_{\cM, c}$ is $M(\cM)\otimes_\bK\Lambda_0$.
\end{proposition}
\begin{proof}
Note that $\pi(\{t=\Re\int\frac{\xi_\pm}{\shbarr}\}\cap \partial \cC)$ coincides with the projection of the irregularity knot. The gluing maps are responsible for Stokes phenomena.
\end{proof}

\section{Application to wild character varieties}
In this section, we will restrict ourselves to the case of Schr\"odinger equations associated to $\cQ$. We always assume that $\cQ=Q_0$ (i.e., without higher order terms with respect to $\hbar$) and $Q_0$ is weakly GMN and only has simple zeros.
We will use the above constructions to give a proof of the identification of Voros--Iwaki--Nakanishi cluster variables~\cite{IwakiNakanishi} with Fock--Goncharov cluster variables~\cite{FG}, which is previously proved by Allegretti by a different method~\cite{Allegretti}.

\subsection{Shende--Treumann--Williams--Zaslow's construction}
We would like to recall Shende--Treumann--Williams--Zaslow's construction of cluster coordinates of wild character varieties.

We fix a set of punctures on $C$ and fix an irregularity type $\{f^p_1,..., f^p_k\}$ for each puncture $p$. Then we have a set of irregularity knots on $C$. Each meromorphic connection with the given irregularity type gives a pure constructible sheaf microsupported on the irregularity knot. We also have an associated marked bordered surface.

Let $r_i^p$ be the ramification degree of $f_i^p$. We assume $n:=\sum_i r_i^p$ does not depend on $p$.

\begin{figure}[htbp]
\begin{center}
\includegraphics[width=7.0cm]{pic3.png}
\end{center}
\caption{An example of irregularity knots and the associated marked border}
\label{Irregularityknot}
\end{figure}

Take an ideal triangulation of $C$ as a marked bordered surface. Then each triangle has a portion of the Legendrian (Figure~\ref{Irregularityknot}, left, in the case of $\rank=3$).
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=7.0cm]{pic1.png}
\end{center}
\caption{Legendrian isotopy}
\label{Legendrianisotopy}
\end{figure}
Then there exists a Legendrian isotopy $\psi$ which sends the LHS of Figure~\ref{Legendrianisotopy} to the RHS of Figure~\ref{Legendrianisotopy}. We denote the latter Legendrian $L'_{\{f_i\}}$.

The {\em alternating surface} $L_a$ associated with the RHS of Figure~\ref{Legendrianisotopy} is an exact Lagrangian filling of the Legendrian whose projection to the base space is the shaded region of the LHS of Figure~\ref{Alternatingsurface}. Topologically, it looks like the RHS of Figure~\ref{Alternatingsurface}, where each crossing is replaced by a ``twist" of the surface. For an exact Lagrangian realization, see \cite{STWZ}.

The argument of \cite{STWZ} goes as follows. By the result of Guillermou--Jin--Treumann, there exists a sheaf quantization of the exact alternating Lagrangian. With a choice of a brane structure, the moduli space of such sheaf quantization is $(\bC^\times)^{H_1(L_a)}$. Then each sheaf quantization gives a sheaf microsupported at the LHS of Figure~\ref{Legendrianisotopy} whose support is the shaded region of Figure~\ref{Alternatingsurface}.

On the other hand, there exists the sheaf quantization of conic Hamiltonian isotopy in Figure~\ref{Legendrianisotopy} which maps a sheaf microsupported on the RHS to the LHS of Figure~\ref{Legendrianisotopy}. More precisely,
\begin{proposition}[\cite{GKS, STWZ}]\label{SQofHam}
There exists a sheaf $S_\psi$ on $C\times C$ such that the integral transform by $S_\psi$ gives an equivalence $\Psi\colon \Sh_{L_{\{f_i\}}}^1(C)\rightarrow \Sh^1_{L'_{\{f_i\}}}(C)$, where $\Sh^1_{L_{\{f_i\}}}(C)$ is the category of constructible sheaves microsupported on $L_{\{f_i\}}\cup T^*_CC$ and are simple at $L_{\{f_i\}}$.
\end{proposition}

As a result, for a fixed ideal triangulation, each sheaf quantization of $L_a$ gives a meromorphic connection of the given irregular type. \cite{STWZ} showed that this $(\bC^\times)^{H_1(L_a)}$ is a cluster coordinate and a cluster transformation is given by a spider move of the Legendrian.

\begin{figure}[htbp]
\begin{center}
\includegraphics[width=7.0cm]{pic2.png}
\end{center}
\caption{Alternating surface}
\label{Alternatingsurface}
\end{figure}

We have the following classes of cycles on the alternating Lagrangian surface.
\begin{enumerate}
    \item For each edge of the triangulation, the alternating surface has $n$ shapes of the region surrounded by the black square in Figure~\ref{Edge}. One also has a distinguished cycle associated to the shape depicted by the black thin line.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=8.0cm]{pic4.png}
\end{center}
\caption{Alternating surface on an edge}
\label{Edge}
\end{figure}
    \item For each face of the triangulation, the alternating surface has $(n-1)(n-2)/2$ shapes of the LHS of Figure~\ref{Face}. One also has a distinguished cycle associated to the shape depicted in the RHS of Figure~\ref{Face}.
    \begin{figure}[htbp]
\begin{center}
\includegraphics[width=8.0cm]{pic5.png}
\end{center}
\caption{Alternating surface on a face}
\label{Face}
\end{figure}
\end{enumerate}

\begin{theorem}[\cite{STWZ}]
The microlocal monodromies along the above cycles of the sheaf associated to the alternating Lagrangian gives a cluster coordinate.
\end{theorem}

\subsection{Fock--Goncharov and Shende--Treumann--Williams--Zaslow}
This section is essentially due to \cite{STWZ}. For the simplicity, we only explain $n=2,3$.

For a compact Riemann surface, take a meromorphic quadratic differential (or a point of the Hitchin base for $n\geq 3$). By seeing the singularities, we have an associated marked bordered surface. We would like to associate a framed $PGL(n)$-local system.

Without framing, the base $PGL(n)$-local system is just the monodromy local system of the equation ((\ref{schrodinger}) for $n=2$). The reason why we only have a projective representation is that we now forget about spin structures.

On each marked point at a border, we have a Stokes filtration. This gives a framing.

Fix an ideal triangulation. For an edge, we have a unique quadrilateral (in the universal covering) whose diagonal is the given edge. We name the vertices of the quadrilateral in a counterclockwise manner $a_1, a_2, a_3, a_4$ where the edge connects $a_1$ and $a_3$.

We take the trivialization of the local system on the quadrilateral. For $n=2$, we have $4$ vectors $v_1, v_2, v_3, v_4$ in $\bC^2$. The Fock--Goncharov coordinate associated to this edge is
\begin{equation}\label{crossratio}
\frac{\det(v_1v_2)\det(v_3v_4)}{\det(v_2v_3)\det(v_1v_4)}.
\end{equation}
Associated to the quadrilateral, we have the configuration of the irregularity Legendrian as in Figure~\ref{quadrilateral}. We calculate the microlocal monodromy along the edge cycle for the edge $a_1a_3$, which is from the upper left vertex to the right lower vertex.

\begin{figure}[htbp]
\begin{center}
\includegraphics[width=12.0cm]{pic9.png}
\end{center}
\caption{Left: Trivialization of the local system, Right: Monodromy calculation}
\label{quadrilateral}
\end{figure}
To calculate the microlocal monodromy, we fix a trivialization of the constructible sheaf on the gray region of the LHS of ~\ref{quadrilateral}. Then we have a rank 2 constant sheaf. Then on the RHS, the rank 1 sheaf on $A$ is spanned by $v_1$ and the rank 1 sheaf on $B$ is spanned by $v_3$. Along the edge cycle, the microlocal transport gives a sequence of isomorphism $\bC v_1\cong \bC^2/\bC v_2 \cong \bC v_3\cong \bC^2/v_ 4\cong \bC v_1$. Here, for example, the second term is the microstalk at the right upper boundary of $A$. One can calculate the composition as in \cite{STWZ} and conclude that it coincides with (\ref{crossratio}) times $-1$. We will modify the sign by introducing a spin structure.

For $n=3$, we do similarly. We have two vectors $v_i^1, v_i^2$ on each vertex on a face. We trivialize the constructible sheaf on the center of LHS of Figure~\ref{Legendrianisotopy} and have a rank $3$ constant sheaf. Then apply the Hamiltonian isotopy. For example, on the region $A$
in Figure~\ref{triangle}, the rank 1 constant sheaf is spanned by $v_1^1$.
Along the face cycle, we have isomorphisms
\begin{equation}
    \bC v_1^1\rightarrow \bC^3/(\bC v_3^1+\bC v_3^2) \rightarrow \bC v_2^1\rightarrow \bC^3/(\bC v_1^1+\bC v_1^2)\rightarrow \bC v_3^1\rightarrow \bC^3/(\bC v_2^1+\bC v_2^2)\rightarrow \bC v_1^1.
\end{equation}
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=7.0cm]{pic10.png}
\end{center}
\caption{Face monodromy}
\label{triangle}
\end{figure}
The composition is nothing but the triple ratio of Fock--Goncharov~\cite{FG}.

Hence we have
\begin{theorem}\label{STWZisFG}
The STWZ coordinate is exactly the same as the Fock--Goncharov coordinate after choosing an appropriate spin structure.
\end{theorem}

\subsection{Voros--Iwaki--Nakanishi and Fock--Goncharov}
In this section, we assume that the order of each pole of $Q_0$'s is greater than or equal to $3$. We fix a sufficiently small $\hbarr\in \bC^\times$. We will fix a spin structure once and for all and get a $\cD$-module $\cM^\shbarr_{Q_0,\sqrt{K}}$. Suppose $Q_0$ has no $\hbarr$-Stokes segments. Then $Q_0$ associates an ideal triangulation of the associated marked bordered surface. Then we have two sets of cluster coordinates: Fock--Goncharov coordinate and Voros--Iwaki--Nakanishi coordinate.

\begin{theorem}\label{VorosisFG}
These two coordinates are the same.
\end{theorem}
This is previously proved by Allegretti by a direct computation~\cite{Allegretti} (see also \cite{Iwaki} for an earlier obeservation). We will present a more geometric proof in this section.

We would like to outline the proof first. Associated to a quadratic differential $Q_0$, we have two sheaves
\begin{enumerate}
    \item the modified sheaf quantization of the modified spectral curve $S_{L_c}$.
    \item the sheaf microsupported at the irregularity knot obtained by Theorem~\ref{STWZRH}
\end{enumerate}
and two Lagrangians
\begin{enumerate}
    \item the modified spectral curve $L_c$
    \item the Hamiltonian-isotopy class of the alternating Lagrangian.
\end{enumerate}
 Although two Lagrangians are never Hamiltonian isotopic, but the projections to the base space coincide. Hence we reduce the computations of the microlocal monodromies to some computations on the base space. Since two sheaves are the same on the base space (Proposition~\ref{projection}), we will get a desired identification.

Now we would like to explain more details. We lift $S_\psi$ to $S_\psi\boxtimes \bC_{t\geq 0}\in \Sh_{\tau>0}(C\times C\times \bR_t)$. We denote the integral transform by this object of ${\Sh}^\bR(C)$ by the same notation $\Psi$.

For a trivial equivariant sheaf $\cE=\bigoplus_cT_c\cE'$ on $C\times \bR_t$, we consider $\bigoplus T_c(S_\psi\star \cE')$ which is again trivially equivariant. We can define the action of $\Psi$ for locally trivially equivariant sheaf in a similar way. Then it is clear that
\begin{lemma}\label{prep1}
$\Psi\circ p_*^\bR\cong p_*^\bR\circ \Psi$.
\end{lemma}

\begin{proposition}
The image $p^\bR_*\circ \Psi(S_{L_c})$ is a sheaf quantization of the alternating Lagrangian.
\end{proposition}
\begin{proof}
Fix a triangle. Along an edge of the triangle, the modified sheaf looks like the LHS of Figure~\ref{alongedge}. Applying $\Psi$ to the sheaf, we get the RHS of Figure~\ref{alongedge}. Hence  $p_*^\bR\circ \Psi(S_{L_c})$ has no stalks on the non-shaded region of the RHS of Figure~\ref{quadrilateral}. Combining this fact with Lemma~\ref{prep1}, we can conclude $p_*^\bR\circ \Psi(S_{L_c})$ is an alternating sheaf in the sense of \cite{STWZ}. By \cite[Proposition 4.18]{STWZ}, this is a sheaf quantization of the alternating Lagrangian.
\end{proof}

\begin{figure}[htbp]
\begin{center}
\includegraphics[width=10.0cm]{pic12.png}
\end{center}
\caption{Along edge}
\label{alongedge}
\end{figure}

\begin{lemma}\label{prep0}
We have $\musupp(\Psi(\cE))\bs T^*_CC= \psi(\musupp(\cE)\bs T^*_CC)$.
\end{lemma}
\begin{proof}
This follows from usual microsupport estimates.
\end{proof}

Hence we have $\psi(L_c\bs T^*_CC)\subset \musupp(\Psi(S_{L_c}))$. Any Voros edge cycle can be taken outside $\cC_p$'s, hence we can take Voros cycles in $L_c\bs T^*_CC$. Hence one can consider the images of Voros cycles in $\musupp(\Psi(\cE))$.

\begin{lemma}\label{prep2}
The projection of the images of Voros cycles in $\musupp(\Psi(\cE))$ is the projection of an edge cycle on the alternating surface.
\end{lemma}
\begin{proof}
A Voros cycle is depicted as the LHS of Figure~\ref{Cycles}, where the waved line is a specified branched cut. We put four gray dots on $L_c$ in the Middle for Figure~\ref{Cycles}. Then the Voros cycle is uniquely recovered as the connecting lines of the dots.

Applying the Hamiltonian isotopy, this is mapped to the RHS of Figure~\ref{Cycles}.
\end{proof}

\begin{figure}[htbp]
\begin{center}
\includegraphics[width=12.0cm]{pic6.png}
\end{center}
\caption{Left: Voros cycle, Middle: Deformed Voros cycle, Right: Edge cycle}
\label{Cycles}
\end{figure}

\begin{lemma}\label{prep3}
The microlocal monodromy of $p_*^\bR\circ \Psi(S_{L_c})$ along the edge cycle is the same as the microlocal monodromy of $\Psi(S_{L_c})$ along the image of the Voros cycle of under $\psi$.
\end{lemma}
\begin{proof}
This is obvious, since the equivariant push-forward identifies the microstalks.
\end{proof}

\begin{proof}[{Proof of Theorem~\ref{VorosisFG}}]
By Proposition~\ref{conicVoros}, the microlocal monodromy of $S_{L_c}$ along the Voros cycle is the Voros coordinate. Then the microlocal monodromy of $\Psi(S_{L_c})$ along the image of the Voros cycle under $\psi$ is the Voros coordinate. By Lemma~\ref{prep3}, it is the same as the microlocal monodromy of $p_*^\bR\circ \Psi(S_{L_c})$ along the edge cycle.

On the other hand, $p_*^\bR\circ \Psi(S_{L_c})$ is the same as $\Psi\circ p_*^\bR(S_{L_c})$ by Lemma~\ref{prep1}, which is the sheaf quantization of the alternating Lagrangian by Proposition~\ref{projection}. Hence the microlocal monodromy along the edge cycle is the Fock--Goncharov coordinate by Proposition~\ref{STWZisFG}.
\end{proof}

\begin{remark}
One may remove the assumption that the poles have order $\geq 3$, by using \cite[Proposition 7.5]{Allegretti}.
\end{remark}

\section{Spectral network and higher order differentials}
For higher ($\geq 3$) order differential equations with $\hbar$, exact WKB analysis is not so well-established, because of collisions of the Stokes curves. However, the algebraic procedure which we need to construct sheaf quantizations can be also carried out in this case under some hypothesis. The main idea is the use of BPS mass filtration~\cite{GMN} as a grading like in the theory of scattering diagrams~\cite{Affinestr, GS}.

\subsection{Stokes geometry of higher order connections}
Let us consider a rank $n$ meromorphic flat $\hbar$-connection $\cM$ with $n\geq 3$. We set $L:=\SS(\cM)$. Let $M$ be the pole divisor of $\cM$.

We would like to introduce Stokes geometry for higher order differentials.
\begin{definition}
A point $z$ in $C$ is a {\em usual turning point} if the spectral curve is branched over $z$.
\end{definition}

We fix $\hbarr\in \bC^\times_\hbar$. Locally, we name the branches by ${1,....,n}$. Then a turning point is a collision of some $i, j\in \{1,...,n\}$. We say this turning point is of type $\{i,j\}$. A Stokes curve emanating from an $\{i,j\}$-turning point $v$ is the set of $z\in C$ satisfying
\begin{equation}
    \Im\int_v^z\frac{\xi_i-\xi_j}{\hbarr}dz=0
\end{equation}
where $\xi_i$ means the restriction of $\lambda$ to the $i$-th branch of $L$. It is convenient to work with the normalized version $\zeta_i:=\xi_i/\hbarr$. Then we say this is an $\hbarr$-Stokes curve of type $(i,j)$ if $\Re\int_v^z(\zeta_i-\zeta_j)dz >0$. Otherwise (i.e., $\Re\int_v^z(\zeta_i-\zeta_j)dz<0$), is of type $(j,i)$.
\begin{definition}[Ordered collision]
\begin{enumerate}
    \item An intersection of two Stokes curves is said to be {\em ordered} if one of them is of type $(i,j)$ and the other is of type $(j,k)$.
    \item We say a point $p\in C$ is a {\em cyclic ordered collision} if there exists a set of Stokes curves $l_1, l_2,..., l_k$ passing through $p$ and the types of $l_1, l_2,..., l_k$ are of the form $(i_1, i_2), (i_2, i_3),..., (i_{k-1}, i_k)$.
\end{enumerate}
\end{definition}
Note that Stokes geometry only depends on the Stokes curve.

\begin{definition}\label{tameness0}
We say $L$ is {\em $\hbarr$-tame} if
\begin{enumerate}
    \item All the turning points are double branches.
    \item The set of turning points is finite.
    \item Any $\hbarr$-Stokes curve does not have an ordered collision on turning points.
    \item There are no cyclic ordered collision of $\hbarr$-Stokes curves.
    \item The complement of the union of $\hbarr$-Stokes curves is open and contractible.
    \item The set of ordered collisions of $\hbarr$-Stokes curves is discrete in $C\bs M$.
\end{enumerate}
\end{definition}

\subsection{Berk--Nevins--Roberts' example}
We would like to explain a baby version of the construction using Berk--Nevins--Roberts' example~\cite{BNR}, which treats the following equation
\begin{equation}
    \lb \hbar^3\frac{d^3}{dz^3}+3\hbar\frac{d}{dz}+2\sqrt{-1}z\rb\psi(z)=0
\end{equation}

\begin{figure}[htbp]\label{BNR1}
\begin{center}
\includegraphics[width=5.0cm]{pic7.png}
\end{center}
\caption{Ordinary Stokes curves of BNR equation}
\end{figure}
This example has the $\hbarr=1$-Stokes curve in Figure~\ref{BNR1}. There are only two turning points, the left and right nodes. Following the Voros connection formula, BNR observed nontrivial monodromies around the collision points, which should not occur. We explain it in our languages.

Let $v_1$ be the left turning point, $v_2$ be the right turning point, and $c$ be the upper collision point $c$. The Stokes curve emanating from $v_1$ (resp. $v_2$) passing through $c$ is of type $(21)$ (resp. $(32)$).

Let us put $\bigoplus_{c\in \bR} T_c\bigoplus_i\bK_{t\geq -\Re\int^z\zeta_i}$ on each connected component of the complement of the Stokes curves. We will use the Voros formula to glue up the sheaves as in the case of order 2. Around $c$, let us compose the gluing morphisms along a counterclockwise loop. We restrict the composition to $\zeta_3$-component, then we have
\begin{equation}
\begin{split}
    \bK_{t\geq -\Re\int^z_{c}\zeta_3}&\rightarrow T_{-\Re\int_c^{v_2}\zeta_3}\bK_{t\geq -\Re\int^z_{v_2}\zeta_3}\\
    &\rightarrow T_{-\Re\int_c^{v_2}\zeta_3}(\bK_{t\geq -\Re\int^z_{v_2}\zeta_3}\oplus \bK_{t\geq -\Re\int^z_{v_2}\zeta_2})\\
    &\rightarrow T_{-\Re\int_c^{v_2}\zeta_3}\bK_{t\geq -\Re\int^z_{v_2}\zeta_3}\oplus  T_{-\Re\int_c^{v_2}\zeta_3}  T_{-\Re\int_{v_2}^{v_1}\zeta_2}\bK_{t\geq -\Re\int^z_{v_1}\zeta_2}\\
    &\rightarrow T_{-\Re\int_c^{v_2}\zeta_3}\bK_{t\geq -\Re\int^z_{v_2}\zeta_3}\oplus  T_{-\Re\int_c^{v_2}\zeta_3}  T_{-\Re\int_{v_2}^{v_1}\zeta_2}(\bK_{t\geq -\Re\int^z_{v_1}\zeta_2}\oplus \bK_{t\geq -\Re\int^z_{v_1}\zeta_1})\\
    &\rightarrow \bK_{t\geq -\Re\int^z_{c}\zeta_3}\oplus  T_{-\Re\int_c^{v_2}\zeta_3}  T_{-\Re\int_{v_2}^{v_1}\zeta_2} T_{-\Re\int^c_{v_1}\zeta_1}\bK_{t\geq -\Re\int^z_{v_1}\zeta_1}.
\end{split}
\end{equation}
The projection to the second component of the last component  is nonzero, in particular, the composition is not $\id$.

This implies the following. Around the collision, we have four regions and one sheaf $S_i$ on each region. On an overlap of two (thickened) regions $D_i, D_j$, we have one sheaf $S_{ij}$ supported on $D_i\cap D_j$ and the gluing morphism
\begin{equation}
    S_{ij}\rightarrow S_i\oplus S_j.
\end{equation}
The cone of this morphism is what we want. Hence, we still have an object on the complement of the collision, however we cannot extend the object onto the collision because of the existence of the anomalous monodromy.

To resolve this paradox, BNR introduced additional Stokes curves emanating from the collision points (Figure~\ref{BNR2}).
\begin{figure}[htbp]\label{BNR2}
\begin{center}
\includegraphics[width=5.0cm]{pic8.png}
\end{center}
\caption{New Stokes curves of BNR equation}
\end{figure}

In our language, it says the following: Add the Stokes curve of $(31)$-type with the gluing morphism which cancels the morphism $ \bK_{t\geq -\Re\int^z_{c}\zeta_3}\rightarrow T_{\Re\int_{v_1}^c(\zeta_2-\zeta_1)+\Re\int_{v_2}^c(\zeta_3-\zeta_2)}\bK_{t\geq -\Re\int^z_{v_1}\zeta_1}$. Note that
\begin{equation}
    -\lb \Re\int_c^{v_2}\zeta_3+ \Re\int_{v_2}^{v_1}\zeta_2 +\Re\int^c_{v_1}\zeta_1\rb=\Re\int_{v_1}^c(\zeta_2-\zeta_1)+\Re\int_{v_2}^c(\zeta_3-\zeta_2)=:w(c).
\end{equation}
Then we can glue up them, fill up the collision, and get a sheaf quantization of the spectral curve around the collision. We denote the Yoneda module of the glued object by $S_L$.

Without the new Stokes curve, we can still do the following. Let $S$ be the Yoneda module of the glued-up object. By the construction, $S_L\otimes \Lambda_0/T^{w(c)}\Lambda_0$ is the same as $S$ outside the collision. In particular, one can extend $S$ onto the collision point. Hence we get an object in the category $\Mod^{\Lambda_0/T^{w(c)}\Lambda_0}(\Sh^{\Lambda_0}(C))$. The object $S_L$ in $\mu(T^*X, \lambda, B, \Lambda_0)$ and the object obtained by filled up the collision of $S$ ($=S_{L_c}\otimes \Lambda_0/T^{w(c)}\Lambda_0$) in $\Mod^{\Lambda_0/T^{w(c)}\Lambda_0}(\Sh^{\Lambda_0}(C))$ form an inverse system. We will use this inductive point of view in the general construction.

\begin{remark}
The Stokes curves are expected to correspond to holomorphic disks whose boundaries lying on the spectral curve and the cotangent fibers. Then the integrations like $\Re\int(\zeta_i-\zeta_j)$ are the areas of the holomorphic disks.
\end{remark}

\subsection{Spectral scattering diagram}
To package our construction, we would like to prepare some general notions and their consequences. The package is a variant of the notion of scattering diagram~\cite{Affinestr, GS}.

For a point $x\in C\bs M$, an $\hbarr$-preStokes curve of type $(i,j)$ emanating from $x$ is a curve consisting of the set of points satisfying $z\in C$ such that
\begin{equation}
\begin{split}
    \Im\int_x^z(\zeta_i-\zeta_j)dz=0, \Re\int_x^z(\zeta_i-\zeta_j)dz>0.
\end{split}
\end{equation}
Note that an $\hbarr$-preStokes curve is canonically oriented. The notion of ordered collisions and cyclic ordered collisions for pre-Stokes curves can be defined in the same way as for Stokes curves.

\begin{definition}
Suppose $L$ is $\hbarr$-tame. Let $B$ be a collection of $\hbarr$-preStokes curves containing the $\hbarr$-Stokes curves. The set $\Sing(B)$ is the set of turning points and ordered intersection points of $\hbarr$-preStokes curves in $B$.

We say $B$ is {\em tame} if the following hold:
\begin{enumerate}[(i)]\label{tamecollection}
    \item Any element of $B$ will end at $M$.
    \item The set $C\bs (M\cup \bigcup_{l\in B}l)$ is open and each connected component is contractible.
    \item Any ordered collisions are not cyclic.
    \item The set $\Sing(B)$ is discrete in $C\bs M$.
\end{enumerate}
\end{definition}
\begin{definition}
Let $B$ be a tame collection of $\hbarr$-preStokes curves. The union of $\hbarr$-preStokes curves forms a graph and we denote the graph by $G_B$.
\begin{enumerate}
    \item A connected component of $C\bs (M\cup \bigcup_{l\in B}l)$ is called a {\em Stokes region} of $B$.
    \item A {\em vertex} of $B$ is a vertex of $G_B$.
    \item For a Stokes region $D$, let $\overline{D}$ be the closure of a lift of $D$ in the universal covering of $C$. A vertex of $D$ is a point of $\overline{D}$ which is a lift of a vertex of $B$.
    \item An {\em edge} of $B$ is an edge of $G_B$.
\end{enumerate}
\end{definition}

\begin{definition}
Let $B$ be a tame collection of $\hbarr$-preStokes curves.
\begin{enumerate}
    \item A {\em spectral scattering diagram} $\frakS$ is a set of pairs $(l, \phi)$ and a set $\{A_e\}$ where $l$ is an $\hbarr$-preStokes curve in $B$, $\phi\in \Lambda_0$, and $A_e\in GL_n(\bC)$ is a diagonal matrix parametrized by edges of $G_B$
    \item A {\em pre-normalization datum} of $B$ consists of the following: a set of scalars $\{m_{(v_1, v_2),i}\}$ labeled by an ordered pair of vertices in a closure of a Stokes region and a sheet of $L$ with $m_{(v, v),i}=1$.
\end{enumerate}
\end{definition}
Let $v_1, v_2$ be vertices of $G_B$ and $\gamma$ be a path in ${G_B}$ starts from $v_1$ and end at $v_2$. Then $\gamma$ is a sequence of oriented edges $\{(v_1, v_2), (v_2, v_3), ..., (v_{k}, v_{k+1})\}$. We put
\begin{equation}
    m_{\gamma,i}:=m_{\gamma,i,B}:=\prod_{j=1}^k m_{(v_j, v_{j+1}),i}.
\end{equation}

\begin{definition}
A pre-normalization datum is a {\em normalization datum} if $m_\gamma=1$ for any loop $\gamma$ in $G_B$.
\end{definition}

\begin{remark}[The meaning of the data]
The normalization data will be given by normalization of WKB solutions, as the name suggests. The data $A_e$'s are gluing data of the trivializations of the meromorphic bundle underlying the given connection.
\end{remark}

Let $\frakS$ be a spectral scattering diagram with a normalization datum over $B$. For $\bfl=(l,\phi)\in \frakS$ of type $(i,j)$, we denote the starting point of $l$ by $v(l)$. For a point $c\in l\cap \Sing(\frakS)$, the interval $\gamma_c:=[v(l), c]$ is an oriented path. We set
\begin{equation}\label{monodromy}
    M_{\bfl}(c):=m_{\gamma_c,i}m_{\gamma_c, j}^{-1}\phi T^{\Re\int_{v(l)}^c(\zeta_i-\zeta_j)dz}.
\end{equation}

For $c\in \Sing(B)$, let $l_1,...., l_n$ be the counterclockwisely ordered collection of $\hbarr$-preStokes curves in $B$ passing through $c$. Let $(i_k, j_k)$ be the type of $l_k$. We denote the set of elements in $\frakS$ supported on $l_k$ by $\{\bfl_{kl}:=(l_k,\phi_{kl})\}_l$. The $l_k$ also gives an edge containing $c$. We also denote the corresponding edge by $l_k$ (by abuse of the notation). The monodromy automorphism at $c$ is defined by
\begin{equation}
    M(c):=M(\frakS, c):=\prod_k A_{l_k}\prod_l(\id+M_{\bfl_{kl}}(c)E_{i_kj_k}) \in GL_n(\Lambda_0)
\end{equation}
where the product with respect to $k$ is ordered and $E_{ij}$ is the matrix whose $(ij)$-entry is $1$ and the other entries are $0$. Note that if $l_i\subset l_j$ then the two $\hbarr$-preStokes curves should be of the same type or of separated types by Definition~\ref{tamecollection} (iii), hence the product is well-defined.

\begin{lemma}
If $i\neq j$ and $M(c)_{ij}$ is nonzero, $M(c)_{ji}=0$.
\end{lemma}
\begin{proof}
This is a consequence of the non-existence of cyclic ordered collisions.
\end{proof}

\begin{definition}
A spectral scattering diagram with a normalization datum is {\em consistent modulo $T^w$} if $M(\frakS, c)=\id$ modulo $T^w$ for any $c\in \Sing(\frakS):=\Sing(B)$.
\end{definition}

Let $\frakS$ be a consistent spectral scattering diagram with a normalization datum consistent modulo $T^w$. We would like to associate the gluing data.

For each Stokes region, let $\widetilde{D}$ be a neighborhood of $D$ in $C\bs (\Sing(B)\cup M)$ such that $\widetilde{D}\cap G_{B}$ is the Stokes edges of $D$.
For each Stokes region and a vertex $v$ of $D$, we put
\begin{equation}
    S^v_{L}|_{D}:=\bigoplus_c T_c\bigoplus_i\bK_{D^v_i}.
\end{equation}
where
\begin{equation}
    D^v_i:=\lc(x, t)\in \widetilde{D}\times \bR_t\relmid t\geq -\Re\int_v^z\zeta_idz\rc
\end{equation}
like in the quadratic case. When we go across an Stokes edge, we associate the following gluing-up morphism:

(Change of regions) For adjacent Stokes regions $D_1$ and $D_2$ with a common vertex $v$, let $\bfl_1,\bfl_2,...,\bfl_n$ be the subset of $\frakS$ supported on the separating edge. We will use the same notation as in (\ref{monodromy}).

An $\hbarr$-preStokes curve of type $(i,j)$ satisfies $\Re\int^x_v(\zeta_i-\zeta_j)dz >0$. Then we have
\begin{equation}
\End\lb \bK_{(\widetilde{D_1}\cap \widetilde{D_2})_i^v}\oplus \bK_{(\widetilde{D_1}\cap \widetilde{D_2})_j^v}\rb
        =\bK\cdot\id_{i}\oplus \bK\cdot \id_{j}\oplus \bK\cdot e_{ij}
\end{equation}
where
\begin{equation}
    e_{ij}\in \Hom\lb \bK_{(\widetilde{D_1}\cap \widetilde{D_2})_i^v}, \bK_{(\widetilde{D_1}\cap \widetilde{D_2})_j^v}\rb \text{ if $\Re\int^x_v(\zeta_i-\zeta_j)dz >0$}
\end{equation}
is a canonical nontrivial basis coming from the inclusion of the closed sets.
We set
\begin{equation}\label{regionchangemorphism}
    \varphi_{D_1, D_2}:= A_e\prod_{\bfl_k}(\id+M_{\bfl_{k}}(c)e_{i_kj_k})\colon S^v_{L}|_{D_1}|_{D_1\cap D_2}\rightarrow S^v_{L}|_{D_2}|_{D_1\cap D_2}
\end{equation}

\begin{proposition}\label{constructionfromscattering}
Let $\frakS$ be a consistent spectral scattering diagram modulo $T^w$ with a normalization datum. Then there exists a simple object with $\musupp=L$ in $\Mod^{\Lambda_0/T^w\Lambda_0}(\Sh^{\Lambda_0}(C))$.
\end{proposition}
\begin{proof}
For each point $v\in \Sing(B)$, take a small neighborhood $D_v$ such that $D_v\cap \Sing(B)=\{v\}$ and $D_v\cap G_B$ is connected.
For a Stokes region $D$, we set $D':=\widetilde{D}\cup \bigcup_{v\in \overline{D}}D_v$. Then $D'$ is a neighborhood of $D$ in $C\bs M$. We would like to glue up the sheaves
\begin{equation}
        \widetilde{S}^v_{L}|_{D}:=\bigoplus_c T_c\bigoplus_i\bK_{\widetilde{D}^v_i}.
\end{equation}
where
\begin{equation}
    \widetilde{D}^v_i:=\lc(x, t)\in D'\times \bR_t\relmid t\geq -\Re\int_v^z\zeta_idz\rc
\end{equation}.

For each Stokes region $D$ and a turning point $v\in \overline{D}$, we set
\begin{equation}
    U_{D, v}:=D'.
\end{equation}
The set $\{U_{D, v}\}_{D, v}$ is an open covering of $C\bs M$.
On each open subset $U_{D, v}$, we put $\widetilde{S}^v_{L}|_{D}$.

For each intersection, we would like to glue up the sheaves using isomorphisms defined above. Then it is enough to show that there is no obstruction. By the construction, only possibilities to have some obstructions will be on $\Sing(B)$. By the consistency of $\frakS$, the obstruction vanishes. This completes the proof.
\end{proof}

\begin{remark}
One can say that the notion of spectral scattering diagram is a 2d-version of usual scattering diagrams and a usual scattering diagram is 4d. Regarding with the aspect of 2d-4d wall-crossing, ultimately, one should consider the notion of ``2d-4d scattering diagram" where the consistency will be given by 2d-4d wall-crossing formulas.
\end{remark}

\subsection{An inductive step}
Let $\frakS$ be a spectral scattering diagram which is consistent modulo $T^w$. We would like to construct a new spectral scattering diagram.

The consistency modulo $T^w$ implies that $M(\frakS, c)=\id$ modulo $T^w$ for any $c\in \Sing(\frakS)$. We can write it as $\id+\sum_{i\neq j}c_{ij}T^{a_{ij}}E_{ij}$ and $a_{ij}\geq w$. Here $i\neq j$ comes from the non-existence of cyclic-ordered collision.
Let $l_{ij, c}$ be an $\hbarr$-preStokes curve of type $(i,j)$ emanating from $c$ if $c_{ij}\neq 0$ and consider the pair $\bfl_{ij,c}:=(l_{ij,c}, -c_{ij}T^{a_{ij}})$. We set $\frakS[c]:=\{\bfl_{ij, c}\}$ and $\frakS^+:=\frakS\cup \bigcup_{c\in \Sing(B)}\frakS[c]$. We assume the set $B^+:=B\cup \bigcup_{c,i,j, c_{ij}\neq 0}\{l_{c, ij}\}$ is again a tame collection of $\hbarr$-preStokes curves. For $A_e$'s, if an edge $e'$ of $B^+$ is a subset of an edge $e$ of $B$, we set $A_{e'}:=A_e$. Otherwise, $A_{e'}=\id$.
Then $\frakS^+$ is a spectral scattering diagram.

\begin{definition}
A normalization datum of $\frakS^+$ is an {\em extension} of the normalization datum of $\frakS$ if the following holds: For an edge $e$ of ${G_B}$,
\begin{equation}
    m_{e,i, B}=m_{\gamma_e,i, B^+}
\end{equation}
where $\gamma_e$ is $e$ viewed as a path in ${G_{B^+}}$.
\end{definition}
Suppose there exists an extension of the normalization datum of $B$. The following is obvious from the construction.
\begin{lemma}\label{compatibilityatcollision}
$M(\frakS^+, c)$ is consistent modulo $T^{2w}$.
\end{lemma}

To speak about the consistency of $\frakS^+$, we prepare some notions. For a pair $\bfl:=(l, \phi)$ of type $(i,j)$, its weight is defined by
\begin{equation}
    w(\bfl):=\sup\lc w\in \bR_{>0}\relmid \phi=0 \text{ modulo }T^w\rc,
\end{equation}
namely, the valuation of the Novikov ring.
For a point $y\in l$, the $\bfl$-weight of $y$ is defined by
\begin{equation}
    w_\bfl(y):=w(\bfl)+\Re\int_{v(l)}^y(\zeta_i-\zeta_j)dz.
\end{equation}
\begin{lemma}\label{weightofcollision}
Let $\bfl_1,...,\bfl_n$ be a subset of some spectral scattering diagram with a normalization data.
For a point $c\in \bigcap l_i$, $M(c)$ is $\id$ modulo $T^{\min_{i\neq j}\{w_{\bfl_i}(c)+w_{\bfl_j}(c)\}}$. More precisely, $(M(c))_{ij}, (i\neq j)$ is zero modulo $T^{w_{ij}}$ where
\begin{equation}
    w_{ij}:=\min\lc w_{\bfl_k}(c)+w_{\bfl_l}(c) \relmid k\neq l, \text{$\bfl_k$ is of type $(i, m)$ and $\bfl_l$ is of type $(m, j)$ for some $m$}\rc.
\end{equation}
\end{lemma}
\begin{proof}
This is clear from the definition of the monodromy.
\end{proof}

An element of $\bigcup_c\frakS[c]$ carries the weight greater than or equal to $w$. By Lemma~\ref{weightofcollision}, the new diagram $\frakS^+$ is also consistent modulo $T^w$.

In the following, we would like to show that $\frakS^+$ is consistent modulo $T^{w'}$ for some $w'$ with $w'>w$ for some special $\frakS$.

\subsection{A local study of a turning point}
We would like to prepare a local notion around a turning point. We assume that $L$ is $\hbarr$-tame.

Let $v$ be a turning point (i.e., a double branch). Without loss of generality, we consider the pair $(\zeta_1, \zeta_2)$ is branched. Let $T$ be the Stokes curve defined by $\Im(\int_v^x(\zeta_1-\zeta_2))=0$. Take a compact neighborhood $U_v$ of $x$ such that
\begin{enumerate}
    \item By 3 and 6 of Definition~\ref{tameness0}, we can take $U_v$ sufficiently small so that $U_v$ does not contain any other turning points and does not intersect with $(i,1), (2,i)$-Stokes curves for any $i$.
    \item $U_v\cap T$ is topologically a trivalent tree and  $U_v\cap T$ has three boundaries $z_1, z_2, z_3$.
\end{enumerate}

We set
\begin{equation}
  d(U_v):=\min_i\lc \Re\lb\int_v^{z_i}(\zeta_1-\zeta_2)dz\rb\rc>0
\end{equation}

We also take a small neighborhood $V_v$ of $\overline{U_v\cap T}$ such that the closure of $V_v$ does not contain any other turning points and does not intersect with $\hbarr$-Stokes curves of type $(i,1)$ or $(2,i)$. Consider the set $L_{v}$ of connected subsets of $\hbarr$-preStokes curves of type $(i,1)$ or $(2,i)$ such that each $l\in L_{v}$ starts from a point outside $V_v$ and ends at a point in $U_v\cap T$.
We set
\begin{equation}
    d(V_v):=\inf_{l\in L_{v}}\lc \begin{cases}
    \Re\lb\int_{l}(\zeta_i-\zeta_1)dz\rb&\text{ if $l$ is of type $(i,1)$}\\
    \Re\lb\int_{l}(\zeta_2-\zeta_i)dz\rb&\text{ if $l$ is of type $(2,i)$}
    \end{cases}
    \rc>0.
\end{equation}

Let $w$ be a turning point or an ordered collision of $\hbarr$-Stokes curves which is different from $v$. Let $L_{vw, i, j}$ be the set of $\hbarr$-Stokes curves of type $(ij)$ emanating from $w$ passing through $V_v$. We set
\begin{equation}
d_v:=\inf_{w\neq v}\inf_{i,j}\inf_{l\in L_{vw,i,j}}\inf_{x\in V_v}\lc \Re\lb\int_{w}^{x}(\zeta_i-\zeta_j)dz\rb\text{ if $l$ is of type $(ij)$ and the integral is along $l$}\rc.
\end{equation}
This is positive, since the set of turning points and the ordered collisions is discrete and $w$ is outside the closure of $V_v$.

We choose $U_v$ and $V_v$ for each turning point and set
\begin{equation}
    w_{min}:=\min_v\lc d(U_v), d(V_v), d_v/2\rc,
\end{equation}
which is again a positive real number.

\subsection{General construction}
We would like to construct a sequence of spectral scattering diagrams.

The $\hbarr$-initial spectral scattering diagram $\frakS_0$ associated to $L$ is the following:
\begin{enumerate}
    \item $B=:B_0$ is the set of $\hbarr$-Stokes curves emanating from the usual turning points.
    \item For an $\hbarr$-Stokes curve $l$ of type $(i,j)$, we consider the pair $(l, \phi)$ for some $\phi$ and $A$ which satisfies the following: For any turning point, we consider a version of the monodromy
    \begin{equation}
    \prod_k A_k\prod_l(\id+M_{\bfl_{kl}}(c)E_{i_kj_k}) \in GL_n(\Lambda_0)
\end{equation}
where the product runs only over the ray emanating from the turning point is equal to 1.
\end{enumerate}
We set
\begin{equation}
    \frakS_n:=\frakS_0^{+^n}, B_n:=B_0^{+^n}.
\end{equation}
We say an element of $\frakS_{n-1}'=\frakS_n\bs\frakS_{n-1}$ {\em an $n$-th Stokes curve}.

\begin{definition}
We say $L$ is {\em inductively $\hbarr$-tame}, if $L$ is $\hbarr$-tame and $B_n$ is tame (Definition~\ref{tamecollection}) for any $n\geq 0$.
\end{definition}
We expect the inductive $\hbarr$-tameness is satisfied by a generic $\hbar$-connection and a generic $\hbarr$ after imposing some conditions on the order of zeros and poles as in the quadratic case.

\begin{definition}
Suppose $\frakS_0$ is inductively $\hbarr$-tame. Then $\frakS_n$ is a spectral scattering diagram for any $n$. Let $N_n$ be a normalization data of $B_n$. A set of normalization data $\{N_n\}_{n\geq 0}$ is {\em inductive} if $N_i$ is an extension of $N_{i-1}$ for any $i\geq 1$.
\end{definition}

\begin{proposition}\label{inductiveconstruction}
Suppose there exists an inductive normalization datum and assume the inductive $\hbarr$-tameness.
\begin{enumerate}
    \item For $n\geq 0$, the weight of an $n$-th Stokes curve is greater than or equal to $nw_{min}$.
    \item The diagram $\frakS_n$ is consistent modulo $T^{(n+1)w_{min}}$.
\end{enumerate}
\end{proposition}
\begin{proof}
We will prove by induction. By the definition of $w_{min}$ and Lemma~\ref{weightofcollision}, the first Stokes curves have weights at least $w_{min}$. The inductive assumption is as follows: Any $m$-th Stokes curve ($m\leq n$) has weight at least $mw_{min}$ and the diagram $\frakS_n$ is consistent modulo $T^{(n+1)w_{min}}$.

Let $\bfl$ be an $(n+1)$-st Stokes curve. By the consistency of $\frakS_n$ modulo $T^{(n+1)w_{min}}$, the weight of $\bfl$ is greater than or equal to $(n+1)w_{min}$. This proves the first part.

We would like to prove the consistency of $\frakS_{n+1}$ modulo $T^{(n+2)w_{min}}$. Let $c$ be an ordered collision of $\bfl$ with another Stokes curve. If a collision consists of $m$-th Stokes curves with $m\geq 1$, Lemma~\ref{weightofcollision} implies that the monodromy is $\id$ modulo $T^{(n+1+m)w_{min}}$ and $n+1+m\geq 2$.

Hence we will consider the case when an $0$-th Stokes curve $\bfl_0$ also has an ordered collision with $\bfl$ on $c$. Let $v$ be the starting point of the $0$-th Stokes curve $\bfl_0$ and we will use the notation from 9.3. Hence $\bfl_0$ is of type $(1,2)$.

If the collision $c$ is not in $U_v$, the weight $w_{\bfl_0}(c)$ is greater than or equal to $d(U_v)\geq w_{min}$. Hence the monodromy around the collision is $\id$ modulo $(n+2)w_{min}$ by Lemma~\ref{weightofcollision}.

Hence we can assume that $c$ is in $U_v$. Suppose $\bfl$ emanates from a point in $c'\not\in V_v$. By Lemma~\ref{weightofcollision} and the definition of $w_{min}$, we have $w_{\bfl}(c)\geq (n+1)w_{min}+d(V_v)\geq (n+2)w_{min}$. By Lemma~\ref{weightofcollision}, this case is consistent.

Hence we can assume $\bfl$ emanates from a point in $c'\in V_v$.
\begin{claim}\label{claim}
Each element $\bfl'$ of $\frakS_{n}$ possibly passing through $c'$ satisfies $w_{\bfl}(c')\geq 2w_{min}$ except for $\bfl_0$.
\end{claim}
\begin{proof}[Proof of the claim]
For $m$-th Stokes curves with $m\geq 2$, it is obvious.
Any 0-th/first Stokes curve emanates from outside $V_v$. Hence $w_{\bfl'}(v)=d_v\geq 2w_{min}$. This completes the proof.
\end{proof}

By Claim~\ref{claim}, any ordered collision of an $n$-th Stokes curve with an $m$-th Stokes curve on $c'$ carries the weight $\geq (n+2)w_{min}$. Hence, for the production process of $\bfl$, we only have to consider an ordered collision with an $n$-th Stokes curve and $\bfl_0$. But, this collision only produces the Stokes curve $\bfl$ of type $(l, 2)$ or $(1,l)$ for some $l$, which cannot have an ordered collision with $\bfl_0$.
\end{proof}

\begin{corollary}
Suppose $L$ is inductively $\hbarr$-tame with an inductive normalization datum. Then there exists a sheaf quantization of $L$ with some brane data, which is an object of $\mu(T^*C, \Re(\lambda/\hbarr), \sqrt{-1}\Im(\lambda/\hbarr),\Lambda^\bK_0)$.
\end{corollary}
\begin{proof}
By Proposition~\ref{inductiveconstruction} and Proposition~\ref{constructionfromscattering}, we have a sheaf quantization of $L$ as $S_n\in \Mod(\Sh^{\Lambda_0/T^{(n+1)w_{min}}}(C))$ for each $n$. By the construction, this gives a locally Yoneda module in $\Mod(\Sh^{\Lambda_0}(C))$, which is a sheaf quantization of $L$.
\end{proof}

\subsection{Normalization data}
To conclude the discussion, we would like to present two inductive normalization data associated to the differential equation.

Also, in the higher order cases, WKB solutions have a similar expression (see e.g. \cite{Virtual}):
\begin{equation}
    \Psi_i(z,\hbar):=\exp\lb \int^z\frac{P^i_{-1}}{\hbar}\rb \exp\lb \int^z\sum_{j=0} P_{j}^i\hbar^j\rb
\end{equation}
where $i$ is a label of a sheet of the spectral covering and $P_j^i$ is uniquely determined from $P^i_{-1}$.

The first one is a normalization over $\bC((\hbar))$. For a sheet $i$ and vertices $v_1, v_2$, we set
\begin{equation}
    m=\exp\lb \int^{v_1}_{v_2}\sum_{i\geq 0}P_j^i\hbar^j\rb
\end{equation}
If one of $v_i$ is on a turning point, the integration is normalized as in the quadratic case. This obviously forms an inductive normalization datum.

We also have to choose an initial diagram. There exists a (conjecturally) canonical choice for this. Namely, under the turning point normalization, we take $\phi:=-1$ and $A$ is the gluing of a trivialization of the underlying meromorphic bundle such that we obtain a Schr\"odinger equation locally around each turning point. This choice comes from an observation that we can use the Voros connection formula around a simple turning point.

As a result, we have
\begin{theorem}
If $L$ is inductively $\hbarr$-tame, there exists a sheaf quantization of $L$ in $\mu(T^*C, \Re(\lambda/\hbarr), \sqrt{-1}\Im(\lambda/\hbarr),\Lambda^{\bC((\hbar))}_0)$ associated to $\cM$.
\end{theorem}

The second normalization is defined over $\bC$, however it is conjectural. We would like to state a summability conjecture. Philosophically speaking, a normalization datum is a ratio of solutions normalized at different points. However, for a Stokes region of $\frakS_n$, we cannot expect the Borel summability of the WKB solutions, since there might be $m$-th Stokes curves with $m> n$. Hence we will formulate as follows. The following story is only conjectural.

We assume $L$ is inductively $\hbarr$-tame.
Let $v_1, v_2$ be two vertices of a Stokes region $D$ of $\frakS_n$. We denote the corresponding normalized Borel transformed WKB-solutions by $\Psi^i_{B,v}(z, y)_{v_j}$ ($j=1,2$), which is expected to be well-defined and Laplace-transformable along appropriate paths.
\begin{conjecture}
For $z\in D$, $\Psi^i_{B, v}(z,y)_{v_j}$ does not have poles on the set \begin{equation}
        \lc y\relmid \arg y=\arg\hbarr, |y|\leq (n+1)w_{min}\rc.
    \end{equation}
\end{conjecture}
Let $\gamma$ be a path connecting $v_1$ and $v_2$. The Laplace transform of $\Psi^i_{B,v_j}(z, y)$ will have Stokes phenomena when there exists a pole of $\Psi^i_{B,v_1}(y)$ on $\bR_{>0}\cdot \hbarr$, but we can also holomorphically continue by deforming the path $\bR_{>0}\cdot \hbarr$.

\begin{conjecture}
\begin{enumerate}
    \item The deformed Laplace transform of $\Psi^i_{B, v_1}(y)$ and the usual Laplace transform $\Psi^i_{B, v_2}(y)$ are linearly dependent.
    \item We set the ratio of the deformed Laplace transforms by $m_{v_1,v_2}$. This gives an inductive normalization datum.
\end{enumerate}
\end{conjecture}

\begin{remark}
The first statement may be stated more elegantly by using the language of Ecalle's resurgence theory. A rough statement is: $\Psi^i_{B, v_1}(y)$ and $\Psi^i_{B, v_2}(y)$ are linearly dependent modulo the action of alien derivations supported on $y$ with $\Re y> (n+1)w_{min}$.
\end{remark}

\begin{theorem}
Suppose $L$ is inductively $\hbarr$-tame. If $\cM$ satisfies the above conjectures, there exists a sheaf quantization of $L$ in $\mu(T^*C, \Re(\lambda/\hbarr), \sqrt{-1}\Im(\lambda/\hbarr),\Lambda^{\bC}_0)$ associated to $\cM$.
\end{theorem}

We also have normalization data which do not come from $\cM$. It is simply that we set any $m_{\gamma,i}=1$ for any $\gamma$ and $i$. Then we conclude that
\begin{theorem}
Suppose $L$ is inductively $\hbarr$-tame. There exists a sheaf quantization of $L$ in $\mu(T^*C, \Re(\lambda/\hbarr), \sqrt{-1}\Im(\lambda/\hbarr),\Lambda^{\bC}_0)$.
\end{theorem}

\begin{comment}
\section{Riemann--Hilbert correspondence with $\hbar$}
In this section, we explain a conjectural functorial higher-dimensional version of the construction. Let $X$ be a complex manifold.

We will first consider the analytic side.
\begin{definition}
We say a holonomic $\cD_\hbar$-module $\cM$ is WKB-regular if it is torsion-free over $\hbar$, holonomic for any $\hbar$, and each irregularity type $f(\hbar, z)$ has the form $g(z)/\hbar$.
\end{definition}
\begin{remark}
This regularity is different from the notion of regularity developed by D'Agnolo--Guillermou--Schapira~\cite{DGS}.
\end{remark}

Note that the $\cD_\hbar$-modules associated to quadratic differentials satisfying Assumption~\ref{WKBregular} is WKB-regular by Proposition~\ref{constantformal}.

Consider the derived category of $\cD_\hbar$-modules over $X$ and denote it by $D^b(\cD_\hbar(X))$. We denote the full subcategory of cohomologically WKB-regular $\cD_\hbar$-modules and denote it by $D^b_{reg}(\cD_\hbar(X))$.

Next, we will define the geometric side. To state a precise conjecture, we have to modify the definition of the Novikov ring. We consider $\bR_{\geq 0}\times \bR/2\pi\bZ$ as a semigroup and take the group ring $R[\bR_{\geq 0}\times \bR/2\pi\bZ]$. We denote the indeterminate of the group ring by $T^{c+\sqrt{-1} d}$ for $(c, d)\in \bR_{\geq 0}\times \bR/2\pi \bZ$. Our new Novikov ring is defined by $\Lambda_0^R:=\lim_{\bR_{\geq 0}\ni a\rightarrow \infty}R[\bR_{\geq 0}\times \bR/2\pi \bZ]/T^{a}R[\bR_{\geq 0}\times \bR/2\pi \bZ]$.

For $\hbarr\in \bC^\times_\hbar$, we also define a copy $\Lambda_0^{R, \bC/|\hbarr|}$ of $\Lambda_0^R$ whose indeterminate is denoted by $T^{\frac{c+\sqrt{-1}d}{|\shbarr|}}$. For $\hbarr, \hbarr'\in \bC^\times_\hbar$ with $\arg\hbarr=\arg\hbarr'$ and $|\hbarr'|<|\hbarr|$, we consider the isomorphism
\begin{equation}\label{lambdaidentification}
\Lambda_0^{R, \bC/|\hbarr|}\rightarrow \Lambda_0^{R, \bC/|\hbarr'|}; \sum_{c, d} a_{c, d}T^{\frac{c+\sqrt{-1}d}{|\shbarr|}}\mapsto \sum_{c, d} a_{c, d}T^{\frac{c\cdot \frac{|\shbarr'|}{|\shbarr|}+\sqrt{-1}d\cdot \frac{|\shbarr'|}{|\shbarr|}}{|\shbarr'|}}.
\end{equation}

We modify the construction of sheaf quantizations: The gluing morphism of B-field will be encoded on $T^{\sqrt{-1}\Image\int \frac{\lambda}{\shbarr}}$.
The construction of sheaf quantization still works without no essential changes. We denote the triangulated hull of sheaf quantizations of holomorphic Lagrangian branes by $\mu^\bC(T^*X, \Re(\lambda/\hbarr), \sqrt{-1}\Im(\lambda/\hbarr), \Lambda_0)$.

For $\theta\in \bR/2\pi\bZ$ and $\hbarr\in e^{2\pi i\theta}\bR_{>0}\subset \bC^\times$, let $\cO_{\hbarr}$ be the ring of analytic functions defined over some sector containing $\hbarr$. We say an element $\sum_{c\geq c_0, d}a_c(\hbar)T^{\frac{c+\sqrt{-1}d}{|\shbarr|}}\in \Lambda_0^{\cO_{\shbarr},\bC/|\shbarr|}$ is convergent if the series $\sum_{c\geq c_0}a_c(\hbar)e^{-\frac{c+\sqrt{-1}d}{|\shbarr|}}$ is absolutely convergent for any $\hbar\in e^{2\pi i\theta}\bR_
{>0}$ with $|\hbar|\leq |\hbarr|$. We denote the subring of $\Lambda_0^{\cO_{\shbarr},\bC/|\shbarr|}$ consisting of convergent elements by $\Lambda_0^{conv,\hbarr}$. The ring $\Lambda_0^{conv,\shbarr}$ acts on $\Lambda^{\bC((\hbar)),\bC/|\shbarr|}_0$ by the asymptotic expansions along $\theta$. For $\hbarr, \hbarr'\in \bC^\times_\hbar$ with $|\hbarr'|<|\hbarr|$ and $\arg(\hbarr)=\arg(\hbarr')$, we have an inclusion $\iota_{\shbarr\shbarr'}\colon \Lambda_0^{conv,\shbarr}\hookrightarrow \Lambda_0^{conv,\shbarr'}$ induced by (\ref{lambdaidentification}).

We set $R_\theta:=\bR_{>0}e^{i\theta}\in \bC^\times_\hbar$.

\begin{conjecture}\label{mainconjecture}
There exists the following data:
\begin{enumerate}
    \item an exhaustive increasing filtration $\{\cD_{\leq |\shbarr|}\}_{\shbarr \in R_\theta}$ of triangulated categories of $D^b_{reg}(\cD_\hbar(X))$
    \item a family of categories $\{\cS_{\leq |\shbarr|}\}_{\shbarr\in R_{\theta}}$ such that
    \begin{enumerate}
        \item  $\cS_{\leq |\shbarr|}$ is defined over $\Lambda^{conv,\shbarr}_0$,
        \item For $\hbarr, \hbarr'\in \bC^\times_\hbar$ with $|\hbarr'|<|\hbarr|$ and $\arg(\hbarr)=\arg(\hbarr')$, we have an inclusion
        \begin{equation}
            \cS_{\leq |\shbarr|}\otimes_{\Lambda_0^{conv,\shbarr}} \Lambda_0^{conv,\shbarr'}\hookrightarrow \cS_{\leq |\shbarr'|}.
        \end{equation}
        where the tensor is taken with respect to $\iota_{\shbarr\shbarr'}$.
        \item  $\cS_{\leq |\shbarr|}\otimes_{\Lambda_0^{conv,\shbarr}}\Lambda^{\bC/|\shbarr|}_0$ is a subcategory of $\mu^\bC(T^*X,\Re(\lambda/\hbarr), \sqrt{-1}\Im(\lambda/\hbarr),\Lambda_0)$ where the tensor is taken with respect to the evaluation $\Lambda_0^{conv, \shbarr}\rightarrow \Lambda_0^{\bC, \bC/|\shbarr|}=:\Lambda_0^{\bC/|\shbarr|}$ at $\hbarr$,
    \end{enumerate}

    \item for each $\hbarr\in \cO_\hbarr$, two $\cO_\shbarr$-linear contravariant fully faithful functors
    \begin{equation}
        \mathrm{RH}^\hbarr_{\pm}\colon \cD_{\leq |\shbarr|}\otimes_{\bC[\hbar]}\cO_{\shbarr} \hookrightarrow \cS_{\leq |\shbarr|}\otimes_{\Lambda_0^{conv,\shbarr}}\cO_{\shbarr}
    \end{equation}
\end{enumerate}
such that they satisfy the followings:
\begin{enumerate}
    \item (functor relation) For $\hbarr, \hbarr'$ with $|\hbarr'|<|\hbarr|$, the restriction of $\mathrm{RH}^{\shbarr'}_{\pm}$ to $\cD_{\leq |\shbarr|}\otimes_{\bC[\hbar]}\cO_{\shbarr}$ factors through the subcategory
    \begin{equation}
    \begin{split}
    \cS_{\leq |\shbarr|}\otimes_{\Lambda_0^{conv,\shbarr}} \cO_{\shbarr}&\hookrightarrow
     \cS_{\leq |\shbarr|}\otimes_{\Lambda_0^{conv,\shbarr}} \cO_{\shbarr'}\\
     &=\cS_{\leq |\shbarr|}\otimes_{\Lambda_0^{conv,\shbarr}}\Lambda_0^{conv,\shbarr'}\otimes_{\Lambda_0^{conv,\shbarr'}}\cO_{\shbarr'}\\
     &\hookrightarrow \cS_{\leq |\shbarr'|}\otimes_{\Lambda_0^{conv,\shbarr}}\cO_{\shbarr'}
    \end{split}
    \end{equation}
    and coincides with $\mathrm{RH}^{\hbarr}_{\pm}$.
    \item (relation to exact WKB) On the object-level, $\mathrm{RH}_{\pm}^\hbarr$ recovers our sheaf quantization.
    \item (relation to D'Agnolo--Kashiwara) For each $\hbarr$, there exists a functor $E^\hbarr\colon \cS_{\leq |\shbarr|}\otimes_{\Lambda_0^{conv, \shbarr}} \bC\rightarrow E^b(\mathrm{I}\bC_X)$ such that $E^\hbarr\circ(\mathrm{RH}^\hbarr_{\pm}\otimes \bC)\cong \mathrm{Sol}^E|_{\cD_{\leq |\shbarr|}\otimes \bC}$.
    \item (t-structure) A regular $\cD_\hbar$-module $\cM$ concentrated in degree $0$ is sent to a pure object by $\mathrm{RH}_\pm^\hbarr$ if it is defined and $\supp(\cM/\hbar \cM)=\musupp(\mathrm{RH}_\pm^\hbarr(\cM))$.
\end{enumerate}
\end{conjecture}

\end{comment}

\footnotesize
\bibliographystyle{alpha}
\bibliography{bibs.bib}

\noindent
Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka, 560-0043, tatsuki.kuwagaki.a.gmail.com

\end{document}