On the generic existence of WKB spectral networks/Stokes graphs

I would like to thank Akishi Ikeda, Hiroshi Iritani, Tsukasa Ishibashi, Kohei Iwaki, and Hiroshi Ohta for discussions on several topics related to the subject of this paper. I also would like to thank the members of GHKK reading seminar (Tsukasa Ishibashi, Shunsuke Kano, Yuma Mizuno, Hironori Oya) where I learned the notion of scattering diagram. This work is supported by JSPS KAKENHI Grant Numbers JP22K13912 and JP20H01794.

Let $C$ be a compact Riemann surface and $D$ be a nonempty finite subset on $C$. Let $\mathcal{K}_{C}(*D)$ be the sheaf of meromorphic differentials with possible poles in $D$.

Let $\Phi$ be a spectral data. Let $z$ be a local coordinate of $C$. Then $\Phi$ can be locally written as

Consider the symbol $\sigma(\Phi):=\zeta^{K}-\sum_{i=1}^{K}\phi_{i}\zeta^{K-i}$. We set

where $\zeta$ is identified with the cotangent of $z$. This is independent of the choice of the local coordinate. The resulting 1-dimensional complex subvariety of $T^{*}C$ is called the spectral curve of $\Phi$.

We denote the defining projection $T^{*}C\rightarrow C$ by $\pi$.

Note also that the set of the branching values is finite by the compactness of $C$.

Let $z$ be a local coordinate of $C$ and $\zeta$ be the associated cotangent coordinate. Then the 1-form $\zeta dz$ on $T^{*}C$ does not depend on the choice of the local coordinate. The resulting global 1-form is called the Liouville form $\lambda$. The 2-form $\omega:=d\lambda$ gives a canonical symplectic structure of $T^{*}C$.

Let $L$ be a holomorphic Lagrangian submanifold (equivalently, 1-dimensional smooth complex submanifold).

We can recover $\Phi$ from $L_{\Phi}$ as follows. For each contractible open subset $U$ in $C{\backslash}D$, we have $\pi|^{-1}_{L}(U)=U_{1}\sqcup\cdots\sqcup U_{K}$ where each $U_{i}$ is isomorphic to $U$ through $\pi|_{L}$. For each $i\in[K]=\left\{1,...,K\right\}$, we have the 1-form on $U$ by pulling back $\lambda_{i}:=\lambda|_{U_{i}}$. Then we set

Then $\bigoplus_{i=1}^{K}\phi_{i}$ recovers $\Phi$.

For this reason, we will use $\Phi$ and $L$ interchangeably in the following sections.

For $\mathfrak{t}=(s_{1},s_{2})$, we set $\mathfrak{t}^{op}=(s_{2},s_{1})$.

The followings are basic facts.

We later use the following lemma:

In this section, we consider the case when $\Phi\in H^{0}(C,\mathcal{K}_{C}^{\otimes 2}(*D))$. In this case, the theory is classical. We refer to Strebel [Str84] and Bridgeland–Smith [BS15]. We use a little stronger version of the GMN condition in [BS15].

To deal with $\Phi$, it is convenient to use the knowledge from the theory of quadratic differentials. For this purpose, we introduce an associated quadratic differential as follows.

In the following, we generalize this example to higher order cases. Let $B(\Phi)\subset C$ be the set of branching values of $\Phi$. Take a contractible open subset $U$ in $C{\backslash}(D\cup B(\Phi))$. Then we have $\pi^{-1}(U)=U_{1}\sqcup\cdots\sqcup U_{K}$ where each $U_{i}$ is isomorphic to $U$ through $\pi$. For each $i\in[K]=\left\{1,...,K\right\}$, we have the 1-form on $U$ by pulling back $\lambda_{i}:=\lambda|_{U_{i}}$, which is again denoted by $\lambda_{i}$. Let $\Delta$ be the diagonal subset of $\operatorname{Sym}^{2}[K]$. For each $(ij)\in\operatorname{Sym}^{2}[K]{\backslash}\Delta$, we set

which is a quadratic differential on $U$. Consider the analytic continuation of $\lambda_{ij}$ over $C{\backslash}(D\cup B(\Phi))$, and we denote the resulting Riemann surface by $C_{ij}^{\prime}$.

Note that each sheet of $C_{ij}$ is parametrized by $(kl)\in\operatorname{Sym}^{2}[K]{\backslash}\Delta$. For each point of $B(\Phi)$, we can fill out $C_{ij}^{\prime}$ by one of the following:

1. 1.

   if a sheet is corresponding to $(kl)$ which is the type of the branch, then we fill out it by a disk. Then $\lambda_{ij}$ is extended by zero.

2. 2.

   if a sheet is corresponding to $(kl)$ which is not the type of the branch, then we fill out it by a branching disk. Then $\lambda_{ij}$ is extended.

We denote the resulting branched covering of $C{\backslash}D$ by $C_{ij}$. The following is obvious from the above construction.

As a result, we get a finite covering $\pi_{\widetilde{C}}\colon\widetilde{C}:=\bigcup_{(i,j)\in\operatorname{Sym}^{2}[K]{\backslash}\Delta}C_{ij}\rightarrow C{\backslash}D$ and a quadratic differential $\widetilde{\Phi}$ on $\widetilde{C}$.

For our purpose, either of construction works well. The following is also obvious.

To have tame behavior of global trajectories, we assume a higher-order analogue of GMN condition.

Here we give a sufficient condition to be strongly GMN. For this purpose, we recall the Hukuhara–Levelt–Turittin theorem.

For a point $z\in D$, take a local neighborhood, and consider a differential equation

associated to $\Phi$. Viewing this as a differential system, we define the formal type of $\Phi$ as the formal type of this equation.

A tree is a connected acyclic graph whose valency at each interior vertex is $\geq 3$. A rooted tree is a tree with a distinguished exterior vertex. For a rooted tree, the distinguished vertex is called the root vertex. The edge connected to the root vertex is called the root edge. A leaf vertex is an exterior vertex which is not the root vertex. A leaf edge is the edge connected to a leaf vertex. We always orient a rooted tree toward the root.

We first define the notion of a Stokes tree. We fix a meromorphic Lagrangian $L$ and $\theta$.

In §5, we state and prove a generic exisntence result of unobstructed angles.

We next assign a positive real number called BPS mass to each Stokes tree. For each preStokes curve $l$ of type $(ij)$, we set

For an open Stokes tree $\mathcal{T}$, we set

For the later use, for a point $p\in\mathcal{T}$ of an open Stokes tree, we set

where $l_{p}$ is the subset of edge starting from the ascendant vertex ending at $p$, and $\mathcal{T}_{p}$ is the subtree of $\mathcal{T}$ whose rooted edge is $l_{p}$.

For a closed Stokes tree $\mathcal{T}$, we set

The Stokes graph $\mathfrak{S}$ is equipped with an increasing filtration $\left\{\mathfrak{S}_{c}\right\}_{c\in\mathbb{R}_{>0}}$ where $\mathfrak{S}_{c}$ consists of those with mass less than $c$. We set

Suppose $L$ is strongly GMN. Suppose there are no saddle trajectories at $\theta$. Then any trajectory emanating from a branching value ends at $D$.

Note that any Stokes tree in the initial graph is mass zero.

Here is our main theorem.

We will prove this in the rest of this section.

We fix a strongly GMN Lagrangian $L$. We also fix $\theta\in S^{1}$.

For an ordered collision $p$ formed by $\mathcal{T}_{1},...,\mathcal{T}_{l}$, we set

and call it mass of the collision $p$.

Since we are interested in the generic behaviour of Stokes trees, we only treat deformable Stokes trees. We start with then notion of family of preStokes curves.

Now we can define families of open Stokes trees.

In the following, we will construct unobstructed Stokes diagram inductively. We explain the setup and key lemmas.

We start with $\mathfrak{S}(0)$, the initial unobstructed diagram, which is a family over a nonempty open subset $I\subset S^{1}$. We will choose an increasing sequence $E_{1}<E_{2}<\cdots$ such that $\lim_{n\rightarrow\infty}E_{n}=\infty$. We inductively set

when it is defined. Now we state the key lemma.

As a corollary of this lemma, we can prove our main theorem.

In this section, we prove the first part of Lemma 5.13. Since we have already proved for $n=1$ in Lemma 5.4, we prove the following induction.

In this section, we prove the latter part of Lemma 5.13. What we will show is that there exists $m_{\min{}}>0$ which does not depend on $n$ and satisfies $nm_{\min{}}\leq E_{\mathfrak{S}(n-1)}$. Then the desired statement follows.

We start with some preliminary notions. For a while, we consider about $\mathfrak{S}(0)$ paramatrized by $I_{0}$. In the following discussion, we will discuss uniformly over $\theta\in I_{0}$. For a branching value $v$ of type $(s_{1},s_{2})$, let $T$ be the union of the (three) preStokes curves emanating from $v$ of type $(s_{1},s_{2})$. If it is necessary we retake $I$ sufficiently small, so that we can take a compact neighborhood $U_{v}$ of $v$ uniformly over $I$ such that

1. 1.

   $U_{v}$ does not contain any other branching values, $U_{v}$ does not intersect with the Stokes trees of type $(i,s_{1}),(s_{2},i)$ in $\mathfrak{S}(0)$ for any $i$ by 4 of Definition 5.3.

2. 2.

   $U_{v}\cap T$ is topologically a trivalent tree with a single interior vertex and $U_{v}\cap T$ has three boundaries $z_{1},z_{2},z_{3}$.

We set

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 branching values and does not intersect with Stokes trees of $(i,1)$ or $(2,i)$ in $\mathfrak{S}(0)$. Consider the set $L_{v}$ of connected subsets of 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

Let $w$ be a branching value or an ordered collision of initial Stokes curves which is different from $v$. Let $L_{vw,i,j}$ be the set of preStokes curves of type $(ij)$ emanating from $w$ passing through $V_{v}$. We set

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

which is again a positive real number. Not that this number is taken uniformly over $I$.

We now set

Now we prove the following:

This completes the proof of Lemma 5.13.

As a variant of the above argument, we prove the following theorem.

We can run the same inductive argument to construct the Stokes graph in the presence of closed Stokes trees. Namely, we only add new preStokes curves to scatterble collisoins. As in the above proof, we obtain the following:

Let $\mathbb{K}$ be a field. Let $M$ be a real manifold and $\mathbb{R}_{t}$ be the real line with the standard coordinate $t$. We consider the discrete group of the real numbers $\mathbb{R}^{\delta}$ and let it act on $\mathbb{R}_{t}$ by addition.

We denote the derived category of equivariant $\mathbb{K}$-module sheaves by $\mathrm{Sh}^{\mathbb{R}^{\delta}}(M\times\mathbb{R}_{t})$. We define the subcategory $\mathrm{Sh}_{\tau\leq 0}^{\mathbb{R}^{\delta}}(M\times\mathbb{R}_{t})$ by the microsupport condition $\SS(\mathcal{E})\subset\{\tau\leq 0\}$ where $\tau$ is the cotangent coordinate of $\mathbb{R}_{t}$. We set

This is a version in [Kuw20] of the category introduced by Tamarkin [Tam18]. The feature of this version is the natural enrichment over $\Lambda_{0}$.

For an object $\mathcal{E}\in\mathrm{Sh}_{\tau>0}^{\mathbb{R}^{\delta}}(M\times\mathbb{R}_{t})$, the positive microsupport $\SS(\mathcal{E})\cap\{\tau>0\}$ is well-defined. We set

where $\rho\colon T^{*}M\times\left\{(t,\tau)\mathrel{}\middle|\mathrel{}\tau>0\right\}\rightarrow T^{*}M;(x,\xi,t,\tau)\mapsto(x,\xi/\tau)$.

Given a brane structure $\mathbf{b}$ of $L$, one can further speak about a sheaf quantization of $(L,\mathbf{b})$. Namely, associated to a brane structure of $\mathbf{b}$, we can associate a local system on $L$ to a sheaf quantization $\mathcal{E}$ of $L$. We say $\mathcal{E}$ is a sheaf quantization of $(L,\mathbf{b})$ if the local system is the rank 1 constant sheaf.

This conjecture was checked for integral Lagrangians in [KPS24].

Now let us assume $M$ be a Riemann surface $C$. Let $L$ be a holomorphic Lagrangian in $T^{*}C$. By the Solomon–Verbitsky theorem [SV19], one can consider $e^{2\pi i\theta}\cdot L$ as an object of $Fuk(T^{*}M)$ for generic $\theta$. Hence, if the above conjecture holds, one can deduce the following:

In the following, we prove this conjecture for spectral curves, without assuming Conjecture 6.2.

For the discussion in this section, we first recall the definition of brane structures.

Let $L$ be a strongly GMN Lagrangian submanifold.

In the following, we give a spectral-network-friendly form of a brane structure.

We first note that a spin structure gives a double covering $\widetilde{S}^{*}L$ of the co-circle bundle $S^{*}L$ of $L$. We pull back $\mathcal{L}$ to $\widetilde{S}^{*}L$, then push-forward to $S^{*}L$. We take the part $\mathcal{L}^{\sigma}$ where the fiber monodromy is $-1$. By this procedure, the set of local systems on $L$ and the set of local systems on $S^{*}L$ whose fiber monodromy is $-1$ are one-to-one, which depends on $\sigma$.

Consider an unobstructed angle $\theta$. Take $c>0$. Consider wall-crossing data $\alpha$ on $\mathfrak{S}_{c}$. We subtract the collision points of $\mathfrak{S}$ and the branching values and the poles from $C$ and denote it by $C^{\circ}_{\mathfrak{S}}$.

Let $\mathcal{T}$ be a Stokes tree with the rooted edge of type $(12)$. Let $\gamma$ be a path in $C^{\circ}_{\mathfrak{S}}$ crossing the root edge of $\mathcal{T}$ at one point. We set $p=\gamma(0),q=\gamma(1)$. We denote the lift of $p$ (resp. $q$) to the sheet $i$ by $p_{i}$ (resp. $q_{i}$). Let $\gamma_{i}$ be the lift of $\gamma$ to sheet (1). We denote the associated Gauss lift by $\widetilde{\gamma}_{i}\subset S^{*}L$.