# Riemann–Hilbert problems from Donaldson–Thomas theory

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## Abstract

We study a class of Riemann–Hilbert problems arising naturally in Donaldson–Thomas theory. In certain special cases we show that these problems have unique solutions which can be written explicitly as products of gamma functions. We briefly explain connections with Gromov–Witten theory and exact WKB analysis.

## 1 Introduction

In this paper we study a class of Riemann–Hilbert problems arising naturally in Donaldson–Thomas theory. They involve maps from the complex plane to an algebraic torus, with prescribed discontinuities along a collection of rays, and are closely related to the Riemann–Hilbert problems considered by Gaiotto et al. [14]; in physical terms we are considering the conformal limit of their story. The same problems have also been considered by Stoppa and his collaborators [1, 12]. One of our main results is that in the ‘uncoupled’ case the Riemann–Hilbert problem has a unique solution which can be written explicitly using products of gamma functions (Theorem 3.2). The inspiration for this comes from a calculation of Gaiotto [13].

We begin by introducing the notion of a BPS structure. This is a special case of Kontsevich and Soibelman’s notion of a stability structure [21]. In mathematical terms, it describes the output of unrefined Donaldson–Thomas theory applied to a three-dimensional Calabi–Yau category with a stability condition. There is also a notion of a variation of BPS structures over a complex manifold, which axiomatises the behaviour of Donaldson–Thomas invariants under changes of stability: the main ingredient is the Kontsevich–Soibelman wall-crossing formula.

To any BPS structure satisfying a natural growth condition we associate a Riemann–Hilbert problem. We go to some pains to set this up precisely. We then prove the existence of a unique solution in the uncoupled case referred to above. Given a variation of BPS structures over a complex manifold *M*, and a family of solutions to the corresponding Riemann–Hilbert problems, we can define a piecewise holomorphic function on *M* which we call the \(\tau \)-function. In the uncoupled case we give an explicit expression for this function using the Barnes *G*-function (Theorem 3.4).

Variations of BPS structures also arise in theoretical physics in the study of quantum field theories with \(N=2\) supersymmetry (see for example [14]). Our \(\tau \)-function then seems to be closely related to the partition function of the theory. Thus, as a rough slogan, one can think of the BPS invariants as encoding the Stokes phenomena which arise when Borel resumming the genus expansion of the free energy. As an example of this relationship, we compute in Section 8 the asymptotic expansion of \(\log (\tau )\) for the variation of BPS structures arising in topological string theory, and show that it reproduces the genus 0 part of the Gopakumar–Vafa expression for the Gromov–Witten generating function.

Another interesting class of BPS structures arise in theoretical physics from supersymmetric gauge theories of class *S*. In the case of gauge group \({\text {SU}}(2)\) these theories play a central role in the paper of Gaiotto et al. [15]. The corresponding BPS structures depend on a Riemann surface equipped with a meromorphic quadratic differential, and the BPS invariants encode counts of finite-length geodesics. These structures arise mathematically via the stability conditions studied by the author and Smith [8]. The work of Iwaki and Nakanishi [18] shows that the corresponding Riemann–Hilbert problems can be partially solved using the techniques of exact WKB analysis. We expect our \(\tau \)-function in this case to be closely related to the one computed by topological recursion [10].

The theory we attempt to develop here is purely mathematical. One potential advantage of our approach is its generality: the only input for the theory is a triangulated category satisfying the three-dimensional Calabi–Yau condition. When everything works, the output is a complex manifold—the space of stability conditions—equipped with an interesting piecewise holomorphic function: the tau function. Note that the theory is inherently global and non-perturbative: it does not use expansions about some chosen limit point in the space of stability conditions.

We should admit straight away that at present there many unanswered questions and unsolved technical problems with the theory. In particular, for general BPS structures we have no existence or uniqueness results for solutions to the Riemann–Hilbert problem. It is also not clear why the \(\tau \)-function as defined here should exist in the general uncoupled case. Nonetheless, the strong analogy with Stokes structures in the theory of differential equations, and the non-trivial answers obtained here (see also [5]) provide adequate mathematical motivation to study these problems further.

### 1.1 Plan of the paper

In Sect. 2 we introduce basic definitions concerning BPS structures. Section 3 contains a summary of the contents of the paper with technical details deferred to later sections. In Sect. 4 we discuss the Riemann–Hilbert problem associated to a BPS structure. In Sect. 5 we solve this problem in the uncoupled case using elementary properties of the gamma function. Sections 6 and 7 discuss the connections with Gromov–Witten invariants and exact WKB analysis referred to above. In “Appendix A” we give a rigorous definition of a variation of BPS structures following Kontsevich and Soibelman. “Appendix B” contains some simple analytic results involving partially-defined self-maps of algebraic tori.

## 2 BPS structures: initial definitions

In this section we introduce the abstract notion of a BPS structure and explain the corresponding picture of active rays and BPS automorphisms. In mathematics, these structures arise naturally as the output of generalized Donaldson–Thomas theory applied to a three-dimensional Calabi–Yau triangulated category with a stability condition. These ideas go back to Kontsevich and Soibelman [21, Section 2], building on work of Joyce (see [4] for a gentle review). The same structures also arise in theoretical physics in the study of quantum field theories with \(N=2\) supersymmetry [15, Section 1].

### 2.1 Definition and terminology

The following definition is a special case of the notion of stability data on a graded Lie algebra [21, Section 2.1]. It was also studied by Stoppa and his collaborators [1, Section 3], [12, Section 2].

### Definition 2.1

- (a)A finite-rank free abelian group \(\Gamma \cong \mathbb {Z}^{\oplus n}\), equipped with a skew-symmetric form$$\begin{aligned} \langle -,-\rangle :\Gamma \times \Gamma \rightarrow \mathbb {Z}, \end{aligned}$$
- (b)
A homomorphism of abelian groups \(Z:\Gamma \rightarrow \mathbb {C}\),

- (c)
A map of sets \(\Omega :\Gamma \rightarrow \mathbb {Q},\)

- (i)
Symmetry: \(\Omega (-\gamma )=\Omega (\gamma )\) for all \(\gamma \in \Gamma \),

- (ii)Support property: fixing a norm \(\Vert \cdot \Vert \) on the finite-dimensional vector space \(\Gamma \otimes _\mathbb {Z}\mathbb {R}\), there is a constant \(C>0\) such that$$\begin{aligned} \Omega (\gamma )\ne 0 \implies |Z(\gamma )|> C\cdot \Vert \gamma \Vert . \end{aligned}$$(1)

The lattice \(\Gamma \) will be called the charge lattice, and the form \(\langle -,-\rangle \) is the intersection form. The group homomorphism *Z* is called the central charge. The rational numbers \(\Omega (\gamma )\) are called BPS invariants. A class \(\gamma \in \Gamma \) will be called active if \(\Omega (\gamma )\ne 0\).

### 2.2 Donaldson–Thomas invariants

*m*in the lattice \(\Gamma \). The BPS and DT invariants are equivalent data: we can write

### 2.3 Poisson algebraic torus

### 2.4 Twisted torus

### 2.5 Ray diagram

Associated to any ray \(\ell \subset \mathbb {C}^*\) is a forma \(<H\).

### 2.6 Further terminology

In this section we gather some terminology for describing BPS structures of various special kinds.

### Definition 2.2

- (a)
finite, if there are only finitely many active classes \(\gamma \in \Gamma \);

- (b)
ray-finite, if for any ray \(\ell \subset \mathbb {C}^*\) there are only finitely many active classes \(\gamma \in \Gamma \) for which \(Z(\gamma )\in \ell \);

- (c)convergent, if for some \(R>0\)$$\begin{aligned} \big .\sum _{\gamma \in \Gamma } |\Omega (\gamma )|\cdot e^{-R|Z(\gamma )|}<\infty . \end{aligned}$$(9)

An equivalent condition to (9) already appears in the work of Gaiotto et al. [14, Appendix C]. The same condition also plays a prominent role in the work of Barbieri and Stoppa [1, Definition 3.5].

### Definition 2.3

- (a)
uncoupled, if for any two active classes \(\gamma _1,\gamma _2\in \Gamma \) one has \(\langle \gamma _1,\gamma _2\rangle =0\);

- (b)generic, if for any two active classes \(\gamma _1,\gamma _2\in \Gamma \) one has$$\begin{aligned} \mathbb {R}_{>0} \cdot Z(\gamma _1)=\mathbb {R}_{>0} \cdot Z(\gamma _2) \implies \langle \gamma _1, \gamma _2\rangle =0. \end{aligned}$$
- (c)
integral, if the BPS invariants \(\Omega (\gamma )\in \mathbb {Z}\) are all integers.

The uncoupled condition ensures that the Hamiltonian flows for any pair of functions on \(\mathbb {T}\) of the form \({\text {DT}}(\gamma )\cdot x_\gamma \) commute. This situation corresponds to the case of ‘mutually local corrections’ in [14]. Genericity is the weaker condition that all such flows for which \(Z(\gamma )\) lies on a given fixed ray \(\ell \subset \mathbb {C}^*\) should commute.

### 2.7 BPS automorphisms

- (i)
*Formal approach*If we are only interested in the elements \(\mathbb {S}(\ell )\) for rays \(\ell \subset \mathbb {C}^*\) lying in a fixed acute sector \(\Delta \subset \mathbb {C}^*\), then we can work with a variant of the algebra \(\mathbb {C}[\mathbb {T}]\) consisting of formal sums of the formsuch that for any \(H>0\) there are only finitely many terms with \(|Z(\gamma )|<H\). This is the approach we shall use in “Appendix A” to define variations of BPS structures: it has the advantage of not requiring any extra assumptions.$$\begin{aligned} \big .\sum _{Z(\gamma )\in \Delta } a_\gamma \cdot x_\gamma ,\quad a_\gamma \in \mathbb {C}, \end{aligned}$$ - (ii)
*Analytic approach*In “Appendix B”, we associate to each convex sector \(\Delta \subset \mathbb {C}^*\), and each real number \(R>0\), a non-empty analytic open subset \(U_\Delta (R)\subset \mathbb {T}\) defined to be the interior of the subsetWe then show that if the BPS structure is convergent, and \(R>0\) is sufficiently large, then for any active ray \(\ell \subset \Delta \), the formal series \({\text {DT}}(\ell )\) is absolutely convergent on \(U_\Delta (R)\subset \mathbb {T}\), and that the time 1 Hamiltonian flow of the resulting function defines a holomorphic embedding$$\begin{aligned} \big \{g\in \mathbb {T}: Z(\gamma )\in \Delta \text { and } \Omega (\gamma )\ne 0\implies |g(\gamma )|<\exp (-R\Vert \gamma \Vert )\big \}\subset \mathbb {T}. \end{aligned}$$We can then view this map as being a partially-defined automorphism of \(\mathbb {T}\). For a more precise statement see Proposition 4.1.$$\begin{aligned} \mathbb {S}(\ell ) :U_\Delta (R)\rightarrow \mathbb {T}. \end{aligned}$$ - (iii)
*Birational approach*In the case of a generic, integral and ray-finite BPS structure, the partially-defined automorphisms \(\mathbb {S}(\ell )\) discussed in (ii) extend to birational automorphisms of \(\mathbb {T}\); see Propostion 4.2. The induced pullback of twisted characters is expressed by the formulawhich is often taken as a definition (see e.g. [14, Section 2.2]).$$\begin{aligned} \mathbb {S}(\ell )^*(x_\beta )=x_\beta \cdot \prod _{Z(\gamma )\in \ell }(1-x_\gamma )^{\,\Omega (\gamma )\langle \beta ,\gamma \rangle } \end{aligned}$$(10)

### 2.8 Doubling construction

### 2.9 A basic example: the Kronecker quiver

## 3 Summary of the contents of the paper

In this section we give a rough summary of the contents of the rest of the paper. Precise statements and proofs can be found in later sections.

### 3.1 The Riemann–Hilbert problem

Given a convergent BPS structure \((\Gamma ,Z,\Omega )\) we will consider an associated Riemann–Hilbert problem. It depends on a choice of a point \(\xi \in \mathbb {T}\) which we call the constant term. We discuss the statement of this problem more carefully in Sect. 4; for now we just give the rough idea.

### Problem 3.1

- (a)As \(t\in \mathbb {C}^*\) crosses an active ray \(\ell \subset \mathbb {C}^*\) in the anti-clockwise direction, the function \(X(t)\) undergoes a discontinuous jump described by the formula$$\begin{aligned} X(t)\mapsto \mathbb {S}(\ell )(X(t)) . \end{aligned}$$
- (b)
As \(t\rightarrow 0\) one has \(\exp (Z/t)\cdot X(t)\rightarrow \xi \).

- (c)
As \(t\rightarrow \infty \) the element \(X(t)\) has at most polynomial growth.

The gist of condition (a) is that the function \(X\) should be holomorphic in the complement of the active rays, and for each active ray \(\ell \subset \mathbb {C}^*\) the analytic continuations of the two functions on either side should differ by composition with the corresponding automorphism \(\mathbb {S}(\ell )\).

### 3.2 Solution in the uncoupled case

In the case of a finite, integral, uncoupled BPS structure, and for certain choices of \(\xi \in \mathbb {T}\), the Riemann–Hilbert problem introduced above has a unique solution, which can be written explicitly in terms of products of modified gamma functions. The inspiration for this comes from work of Gaiotto [13, Section 3.1].

### Theorem 3.2

Note that the uncoupled assumption implies that the active classes \(\gamma \in \Gamma \) span a subgroup of \(\Gamma \) on which the form \(\langle -,-\rangle \) vanishes, and this ensures that there exist elements \(\xi \in \mathbb {T}\) satisfying \(\xi (\gamma )=1\) for all such classes. The proof of Theorem 3.2 is a good exercise in the basic properties of the gamma function. The details are given in Sect. 5.

### 3.3 Variations of BPS structure

The variation of BPS invariants in Donaldson–Thomas theory under changes in stability parameters is controlled by the Kontsevich–Soibelman wall-crossing formula. This forms the main ingredient in the following definition of a variation of BPS structures, which is a special case of the notion of a continuous family of stability structures from [21, Section 2.3]. Full details can be found in “Appendix A”; here we just give the rough idea.

### Definition 3.3

*M*consists of a collection of BPS structures \((\Gamma _p,Z_p,\Omega _p)\) indexed by the points \(p\in M\), such that

- (a)
The charge lattices \(\Gamma _p\) form a local system of abelian groups, and the intersection forms \(\langle -,-\rangle _p\) are covariantly constant.

- (b)
Given a covariantly constant family of elements \(\gamma _p\in \Gamma _p\), the central charges \(Z_p(\gamma _p)\in \mathbb {C}\) are holomorphic functions of \(p\in M\).

- (c)
The constant in the support property (1) can be chosen uniformly on compact subsets.

- (d)For each acute sector \(\Delta \subset \mathbb {C}^*\), the clockwise product over active rays in \(\Delta \)is covariantly constant as \(p\in M\) varies, providing the boundary rays of \(\Delta \) are never active.$$\begin{aligned} \mathbb {S}_p(\Delta )=\prod _{\ell \subset \Delta } \mathbb {S}_p(\ell ) \in {\text {Aut}}(\mathbb {T}_p), \end{aligned}$$(15)

*M*, and hence also on the associated bundle of twisted tori \(\mathbb {T}_p\). It will require some work to make rigorous sense of the wall-crossing formula, condition (d). We do this using formal completions in “Appendix A” following [21]. This needs no convergence assumptions and completely describes the behaviour of the BPS invariants as the point \(p\in M\) varies: once one knows all the invariants \(\Omega (\gamma )\) at some point of

*M*, they are determined at all other points.

### 3.4 Tau functions

Let us consider a variation of BPS structures \((\Gamma _p,Z_p,\Omega _p)\) over a complex manifold *M*. We call such a variation framed if the local system \((\Gamma _p)_{p\in M}\) is trivial, so that we can identify all the lattices \(\Gamma _p\) with a fixed lattice \(\Gamma \). We can always reduce to this case by passing to a cover of *M*, or by restricting to a neighbourhood of a given point \(p\in M\).

*M*.

*M*, and choose a basis \((\gamma _1,\ldots , \gamma _n)\subset \Gamma \) as above. For each point \(p\in M\) we can consider the Riemann–Hilbert problem associated to the BPS structure \((\Gamma ,Z_p,\Omega _p)\). The wall-crossing formula, Definition 3.3 (d), makes it reasonable to ask for a family of solutions to these problems which is a piecewise holomorphic function of \(p\in M\). Such a family of solutions is given by a piecewise holomorphic map

*t*, and which satisfies the equations

It is not clear at present why a \(\tau \)-function should exist in general, and the above definition should be thought of as being somewhat experimental. Nonetheless, in the uncoupled case we will see that \(\tau \)-functions do exist, and are closely related to various partition functions arising in quantum field theory. We hope to return to the general case in future publications.

### 3.5 Tau function in the uncoupled case

*M*. We will show that the family of solutions given by Theorem 3.2 has a corresponding \(\tau \)-function. To describe this function we first introduce the expression

*G*(

*x*) is the Barnes

*G*-function, and \(\zeta (s)\) the Riemann zeta function.

### Theorem 3.4

*M*. Then the function

*G*-function imply that \(\tau (Z,t)\) has an asymptotic expansion involving the Bernoulli numbers

### 3.6 Two classes of examples

There are two important classes of examples of BPS structures where we can say something about solutions to the Riemann–Hilbert problem. Both are also of interest in theoretical physics. They are treated a little more thoroughly in Sects. 6 and 7 , but there are many unanswered questions which we leave for future research.

*Topological strings*Let

*X*be a compact Calabi–Yau threefold. There is a variation of BPS structures over the complexified Kähler cone

*X*supported in dimension \(\le 1\). The BPS invariants are expected to coincide with the genus 0 Gopakumar–Vafa invariants (see [19, Conjecture 6.20]). Assuming this, we argue that the asymptotic expansion of the resulting \(\tau \)-function should be related to the Gromov–Witten partition function of

*X*. More precisely, it should reproduce the \(g\ge 2\) terms in those parts of the partition function arising from constant maps and genus 0 degenerate contributions:

*Theories of class*

*S*Our second example relates to the class of \(N=2\), \(d=4\) gauge theories known as theories of class

*S*. We consider only the case of gauge group \({\text {SU}}(2)\). To specify the theory we need to fix a genus \(g\ge 0\) and a collection of \(d\ge 1\) integers

*S*,

*q*) consisting of a Riemann surface

*S*of genus

*g*, and a meromorphic quadratic differential

*q*on

*S*which has simple zeroes, and poles of the given multiplicities \(m_i\). It is proved in [8] that this space also arises as a (discrete quotient of) the space of stability conditions on a triangulated category \(\mathcal {D}(g,m)\) having the three-dimensional Calabi–Yau property. Applying generalized Donaldson–Thomas theory then leads to a variation of BPS structures over \({\text {Quad}}(g,m)\), whose central charge is given by the periods of the differential

*q*, and whose BPS invariants are counts of finite-length trajectories. Work of Iwaki and Nakanishi [18] shows that the Riemann–Hilbert problem corresponding to a pair (

*S*,

*q*) is closely related to exact WKB analysis of the corresponding Schrödinger equation

*z*is some local co-ordinate in a fixed projective structure on

*S*, and \(\hbar \) should be identified with the variable

*t*in Problem 3.1. This story is the conformal limit of that described by Gaiotto et al. in the paper [15].

## 4 The BPS Riemann–Hilbert problem

In this section we discuss the Riemann–Hilbert problem defined by a convergent BPS structure. It is closely related to the Riemann–Hilbert problem considered by Gaiotto et al. [14], and has also been studied by Stoppa and his collaborators [1, 12].

### 4.1 Analytic BPS automorphisms

### Proposition 4.1

- (i)For each ray \(\ell \subset \Delta \), the power series \({\text {DT}}(\ell )\) is absolutely convergent on \(U_\Delta (R)\), and hence defines a holomorphic function$$\begin{aligned} {\text {DT}}(\ell ):U_\Delta (R)\rightarrow \mathbb {C}. \end{aligned}$$
- (ii)The time 1 Hamiltonian flow of the function \({\text {DT}}(\ell )\) with respect to the Poisson structure \(\{-,-\}\) on \(\mathbb {T}\) defines a holomorphic embedding$$\begin{aligned} \mathbb {S}(\ell ):U_\Delta (R)\rightarrow \mathbb {T}. \end{aligned}$$
- (iii)For each \(H>0\), the composition in clockwise ordercorresponding to the finitely many rays \(\ell _i\subset \Delta \) of height \(< H\) exists, and the pointwise limit$$\begin{aligned} \mathbb {S}_{<H}(\Delta ) = \mathbb {S}_{\ell _1} \circ \mathbb {S}_{\ell _2} \circ \cdots \circ \mathbb {S}_{\ell _k}, \end{aligned}$$is a well-defined holomorphic embedding.$$\begin{aligned} \mathbb {S}(\Delta )=\lim _{H\rightarrow \infty } \mathbb {S}_{<H}(\Delta ) :U_\Delta (R)\rightarrow \mathbb {T}\end{aligned}$$

### Proof

See “Appendix B”, Proposition B.3. \(\square \)

We think of the maps \(\mathbb {S}(\ell )\) constructed in Proposition 4.1 as giving partially-defined automorphisms of the twisted torus \(\mathbb {T}\). We will usually restrict attention to BPS structures which in the terminology of Sect. 2.6 are ray-finite, generic and integral. The map \(\mathbb {S}(\ell )\) can then be computed using the following result.

### Proposition 4.2

### Proof

See “Appendix B”, Proposition B.6. \(\square \)

Note that if the BPS structure \((\Gamma ,Z,\Omega )\) satisfies the stronger condition of being finite, then there are only finitely many active rays, so for any acute sector \(\Delta \subset \mathbb {C}^*\) the map \(\mathbb {S}(\Delta )\) of Proposition 4.1 also extends to a birational automorphism of \(\mathbb {T}\).

### 4.2 Statement of the problem

### Problem 4.3

- (RH1)
*Jumping.*Suppose that two non-active rays \({r}_1,{r}_2\subset \mathbb {C}^*\) form the boundary rays of a convex sector \(\Delta \subset \mathbb {C}^*\) taken in clockwise order. Thenfor all \(t\in \mathbb {H}_{{r}_1}\cap \mathbb {H}_{{r}_2}\) with \( 0<|t|\ll 1\).$$\begin{aligned} X_{{r}_1}(t)= \mathbb {S}(\Delta )( X_{{r}_2}(t)), \end{aligned}$$ - (RH2)
*Finite limit at 0*. For each non-active ray \({r}\subset \mathbb {C}^*\) and each class \(\gamma \in \Gamma \) we haveas \(t\rightarrow 0\) in the half-plane \(\mathbb {H}_{r}\).$$\begin{aligned} \exp (Z(\gamma )/t)\cdot X_{{r},\gamma }(t) \rightarrow \xi (\gamma ) \end{aligned}$$ - (RH3)
*Polynomial growth at*\(\infty \). For any class \(\gamma \in \Gamma \) and any non-active ray \({r}\subset \mathbb {C}^*\), there exists \(k>0\) such thatfor \(t\in \mathbb {H}_{r}\) with \(|t|\gg 0\).$$\begin{aligned} |t|^{-k}< |X_{{r},\gamma } (t)|<|t|^k, \end{aligned}$$

### Remark 4.4

### Remark 4.5

It follows from the condition (RH1) that if two non-active rays \({r}_1,{r}_2\) bound a convex sector containing no active rays, then the two functions \(X_{{r}_i}:\mathbb {H}_{{r}_i}\rightarrow \mathbb {T}\) required in Problem 4.3 glue together to give a holomorphic function on \(\mathbb {H}_{{r}_1}\cup \mathbb {H}_{{r}_2}\). It follows that if a non-active ray \({r}\subset \mathbb {C}^*\) is not a limit of active rays, then the corresponding function \(X_{r}\) extends analytically to a neighbourhood of the closure of \(\mathbb {H}_{r}\subset \mathbb {C}^*\).

### 4.3 Remarks on the formulation

The Riemann–Hilbert problem of the last subsection is the main subject of this paper. Unfortunately we have no general results giving existence or uniqueness of its solutions. Moreover one could easily imagine various small perturbations of the statement of Problem 4.3, and it will require further work to decide for sure exactly what the correct conditions should be. We make a few remarks on this here.

### Remarks 4.6

- (i)
From a heuristic point-of-view it is useful to consider a Riemann–Hilbert problem involving maps from \(\mathbb {C}^*\) into the group

*G*of Poisson automorphisms of the torus \(\mathbb {T}\). The above formulation is obtained by evaluating a*t*-dependent automorphism of \(\mathbb {T}\) at the chosen point \(\xi \in \mathbb {T}\). If we replace the infinite-dimensional group*G*with the finite-dimensional group \({\text {GL}}_n(\mathbb {C})\) the analogous Riemann–Hilbert problems are familiar in the theory of linear differential equations with irregular singularities, and play an important role in the theory of Frobenius manifolds [9, Lecture 4]. This connection between stability conditions and Stokes phenomena goes back to [7], and will be revisited in [6]. - (ii)
In Sect. 3.1 we gave a simplified formulation of the Riemann–Hilbert problem which considers a single function \(X:\mathbb {C}^*\rightarrow \mathbb {T}\) with prescribed discontinuities along active rays. This becomes a little tricky to make sense of when the active rays are dense in regions of \(\mathbb {C}^*\), so we prefer the formulation given in Problem 4.3, which is modelled on the standard approach in the finite-dimensional case. We can obtain a solution to Problem 3.1 from a solution to Problem 4.3 by defining \(X(t)=X_{\mathbb {R}_{>0}\cdot t} (t)\); Remark 4.5 shows that this defines a holomorphic function away from the closure of the union of the active rays. Note that Problem 4.3 imposes strictly stronger conditions on the resulting function \(X(t)\), because the conditions (RH2) and (RH3) are assumed to hold in half-planes.

- (iii)
We can weaken the conditions in Problem 4.3 in various ways, and until we have studied more examples in detail it is not possible to be sure exactly what is the correct formulation. For example, we could allow the functions \(X_{{r},\gamma }(t)\) to have poles on the half-plane \(\mathbb {H}_{r}\), or we could replace \(\mathbb {H}_{r}\) by a smaller convex sector of some fixed angle. We can also consider a variant of Problem 4.3 where we only assume that the map \(X_{r}\) is defined and holomorphic on the intersection of \(\mathbb {H}_{r}\) with some punctured disc \(\{t\in \mathbb {C}^*:|t|<r\}\), and drop condition (RH3) altogether. We shall refer to this last version as the weak Riemann–Hilbert problem associated to the BPS structure.

### 4.4 Symmetries of the problem

*Z*and

*t*.

### 4.5 Null vectors and uniqueness

Let \((Z,\Gamma ,\Omega )\) be a BPS structure, and denote by \(\mathbb {T}\) the corresponding twisted torus.

### Definition 4.7

An element \(\gamma \in \Gamma \) will be called null if it satisfies \(\langle \alpha ,\gamma \rangle =0\) for all active classes \(\alpha \in \Gamma \). A twisted character \(x_\gamma :\mathbb {T}\rightarrow \mathbb {C}^*\) corresonding to a null element \(\gamma \in \Gamma \) will be called a coefficient.

Note that the definition of the wall-crossing automorphisms \(\mathbb {S}(\ell )\) shows that they fix all coefficients: \(\mathbb {S}(\ell )^*(x_\gamma )=x_\gamma \). This leads to the following partial uniqueness result.

### Lemma 4.8

Let \((Z,\Gamma ,\Omega )\) be a convergent BPS structure and \(\gamma \in \Gamma \) a null element. Then for any solution to the Riemann–Hilbert problem, and any non-active rays \({r}\subset \mathbb {C}^*\), one has \(Y_{{r}, \gamma }(t)=1\) for all \(t\in \mathbb {H}_{r}\).

### Proof

Since coefficients are unchanged by wall-crossing, condition (RH1) shows that the functions \(Y_{{r},\gamma }(t)\) for different rays \({r}\subset \mathbb {C}^*\) piece together to give a single holomorphic function \(Y_\gamma :\mathbb {C}^*\rightarrow \mathbb {C}^*\). Since we can cover \(\mathbb {C}^*\) by half-planes \(\mathbb {H}_{{r}_i}\) corresponding to finitely many non-active rays \({r}_i\subset \mathbb {C}^*\), condition (RH2) shows that this function has a removable singularity at \(0\in \mathbb {C}\) with value \(Y_\gamma (0)=1\), and condition (RH3) shows that it has at worst polynomial growth at \(\infty \). It follows that \(Y_\gamma \) extends to a meromorphic function \(\mathbb {C}\mathbb {P}^1\rightarrow \mathbb {C}\mathbb {P}^1\) which has neither zeroes nor poles on \(\mathbb {C}\). This implies that \(Y_\gamma (t)\) is constant, which proves the result. \(\square \)

Recall the definition of an uncoupled BPS structure from Sect. 2.6.

### Lemma 4.9

Let \((Z,\Gamma ,\Omega )\) be a finite, uncoupled BPS structure. Then the associated Riemann–Hilbert problem has at most one solution.

### Proof

### Remark 4.10

In this way one sees that if a BPS structure has a non-degenerate form \(\langle -,-\rangle \), then solving the Riemann–Hilbert problem for the doubled BPS structure is precisely equivalent to solving the problem for the original BPS structure. However, when the form \(\langle -,-\rangle \) is degenerate a solution to the Riemann–Hilbert problem for the doubled BPS structure contains strictly more information than a solution to the original problem. We will see an example of this in Sect. 5.1 below.

### 4.6 Tau functions

*M*. For the relevant definitions the reader can either consult the summary in Sect. 3.3, or the full treatment in “Appendix A”. There is a holomorphic map

*M*.

*M*. The dual of the map (23) identifies \(\gamma _i\in \Gamma \) with \(dz_i\in T_p^*M\), and the Poisson structure has the Darboux form

*t*and translating to the identity \(1\in \mathbb {T}_+\) we get a map

*V*(

*t*) on

*M*depending on \(t\in \mathbb {C}^*\). A \(\tau \)-function for the family of solutions \(Y_{r}(p,t)\) is a piecewise holomorphic function

*V*(

*t*) is the Hamiltonian vector field of the function \((2\pi i)\cdot \log \tau _{r}\). In terms of the co-ordinates \(z_i=Z(\gamma _i)\) described above, the condition is that

*Z*and

*t*. In the case when the form \(\langle -,-\rangle \) is non-degenerate this is enough to determine the \(\tau \) function uniquely up to multiplication by an element of \(\mathbb {C}^*\).

## 5 Explicit solutions in the finite, uncoupled case

In this section we show how to solve the Riemann–Hilbert problem associated to a finite, uncoupled, integral BPS structure, and compute the \(\tau \)-function associated to a variation of such structures. The situation considered here corresponds to the case of ‘mutually local corrections’ in [14]. The inspiration for our solution comes from a calculation of Gaiotto [13, Section 3].

### 5.1 Doubled \(\hbox {A}_1\) example

- (i)
The lattice \(\Gamma =\mathbb {Z}\cdot \gamma \) has rank one, and thus \(\langle -,-\rangle =0\);

- (ii)
The central charge \(Z:\Gamma \rightarrow \mathbb {C}\) is determined by \(Z(\gamma )=z\in \mathbb {C}^*\);

- (iii)
The only non-vanishing BPS invariants are \(\Omega (\pm \gamma )=1\).

To define the Riemann–Hilbert problem for the doubled BPS structure we must first choose a constant term \(\xi _D\in \mathbb {T}_D\), where \(\mathbb {T}_D\) is the twisted torus corresponding to the lattice \(\Gamma _D\). For simplicity we take \(\xi _D(\gamma ^\vee )=1\), and write \(\xi =\xi _D(\gamma )\in \mathbb {C}^*\).

### Problem 5.1

- (i)There are relations$$\begin{aligned} Y_-(t)={\left\{ \begin{array}{ll} Y_+(t) \cdot \left( 1-\xi ^{+1}\cdot e^{-z/t}\right) &{} \text{ if } t\in \mathbb {H}_{\ell _+},\\ Y_+(t) \cdot \left( 1-\xi ^{-1}\cdot e^{+z/t}\right) &{} \text{ if } t\in \mathbb {H}_{\ell _-}. \end{array}\right. } \end{aligned}$$(25)
- (ii)
As \(t\rightarrow 0\) in \(\mathbb {C}^*{\setminus } i\ell _{\mp }\) we have \(Y_{\pm }(t)\rightarrow 1\).

- (iii)There exists \(k>0\) such thatas \(t\rightarrow \infty \) in \(\mathbb {C}^*{\setminus } i\ell _{\mp }\).$$\begin{aligned} |t|^{-k}< |Y_{\pm }(t)|<|t|^k \end{aligned}$$

To understand condition (i) note that if \(t\in \mathbb {H}_{\ell _+}\) then *t* lies in the domains of definition \(\mathbb {H}_{r}\) of the functions \(Y_{r}\) corresponding to sufficiently small deformations of the ray \(\ell _+\). Thus (21) applies to the ray \(\ell _+\) and we obtain the first of the relations (25). The second follows similarly from (21) applied to the opposite ray \(\ell _-\).

### 5.2 Solution in the doubled \(\hbox {A}_1\) case

*g*)th Bernoulli number. This expansion is valid as \(w\rightarrow \infty \) in the complement of a closed sector containing the ray \(\mathbb {R}_{<0}\). It implies in particular that \(\Lambda (w)\rightarrow 1\).

### Proposition 5.2

### Proof

### 5.3 The finite uncoupled case

In the case of a finite, uncoupled, integral BPS structure we can construct a unique solution to the Riemann–Hilbert problem by superposing the solutions from the previous section.

### Theorem 5.3

### Proof

*t*lie in \(\mathbb {H}_{r}\), so the argument of \(\Lambda \) does not lie in \(\mathbb {R}_{<0}\). The properties (RH2) and (RH3) then follow immediately as in the proof of Proposition 5.2. Consider two non-active rays \({r}_-\) and \({r}_+\) obtained by small perturbations, anti-clockwise and clockwise respectively, of an active ray \(\ell \). Then \(\ell \subset i\mathbb {H}_{{r}_+}\) whereas \(-\ell \subset i\mathbb {H}_{{r}_-}\). Assume that \(t\in \mathbb {H}_{{r}_-}\cap \mathbb {H}_{{r}_+}\). Then \(t\in \mathbb {H}_\ell \) and hence \(-Z(\gamma )/2\pi it\) lies in the upper half-plane whenever \(Z(\gamma )\in \ell \). Using (28) we therefore obtain

### 5.4 Tau function in the uncoupled case

*G*is the Barnes

*G*-function [2, 27, Appendix], and \(\zeta (s)\) is the Riemann zeta function. It defines a holomorphic and nowhere vanishing function on \(\mathbb {C}^*{\setminus } \mathbb {R}_{<0}\) which we specify uniquely by defining the factor \(w^{w^2/2}\) using the principal value of \(\log \). The asymptotic expansion of \(\Upsilon (w)\) is

### Lemma 5.4

### Proof

In the case of variations of BPS structures satisfying the conditions of Theorem 5.3 the following result gives a natural choice of \(\tau \)-function.

### Theorem 5.5

*M*. Given a ray \({r}\subset \mathbb {C}^*\), the function

### Proof

*Z*and

*t*. Choosing a basis \((\gamma _1,\ldots ,\gamma _n)\subset \Gamma \), and using the local co-ordinates \(z_i=Z(\gamma _i)\) on

*M*, we have

\(\square \)

### Remark 5.6

*x*by two properties: a difference equation, and the existence of an asymptotic expansion. Using the property \(G(w+1)=\Gamma (w)\cdot G(w)\) of the Barnes

*G*-function, and the expansion (31), it follows that we can take

## 6 Geometric case: Gromov–Witten invariants

In this section we consider a class of BPS structures related to closed topological string theory on a compact Calabi–Yau threefold. In mathematical terms they arise from stability conditions on the category of coherent sheaves supported in dimension \(\le 1\). These BPS structures are uncoupled but not finite. We will show that formally applying the expression (33) in this case reproduces the genus 0 degenerate contributions to the Gromov–Witten generating function. In [5] we give a more careful analysis for the special case of the resolved conifold.

### 6.1 Gopakumar–Vafa invariants

*X*be a smooth projective Calabi–Yau threefold. For the sake of notational simplicity we will assume that the group \(H_2(X,\mathbb {Z})\) is torsion-free. The Gromov–Witten potential of

*X*is a formal series

*g*Gromov–Witten invariant for stable maps of class \(\beta \in H_2(X,\mathbb {Z})\). Note that by definition these invariants are nonzero only for effective curve classes \(\beta \ge 0\). The symbols \(x^\beta \) are formal variables living in a suitable completion of the effective cone in the group ring of \(H_2(X,\mathbb {Z})\), and \(\lambda \) is a formal parameter corresponding to the string coupling.

### 6.2 Torsion sheaf BPS invariants

*X*supported in dimension \(\le 1\). Any sheaf \(E\in \mathcal {A}\) has a Chern character \({\text {ch}}(E)\in H^*(X,\mathbb {Z})\), which via Poincaré duality we can view as an element

*Z*defines a stability condition on the abelian category \(\mathcal {A}\).

For each class \(\gamma \in \Gamma \) there is an associated BPS invariant \(\Omega (\gamma )\in \mathbb {Q}\) first constructed by Joyce and Song ([19], see particularly Sections 6.3–6.4). They are defined using moduli stacks of semistable objects in \(\mathcal {A}\), and should not be confused with the ideal sheaf curve-counting invariants appearing in the famous MNOP conjectures [22]. Joyce and Song prove that the numbers \(\Omega (\gamma )\) are independent of the complexified Kähler class \(\omega _\mathbb {C}\). This is to be expected, since wall-crossing is trivial when the form \(\langle -,-\rangle \) vanishes: see Remark A.4 below.

*n*. We shall assume this in what follows. We emphasise that the higher genus Gopakumar–Vafa invariants are invisible from the point-of-view of the torsion sheaf invariants \(\Omega (\gamma )\).

### 6.3 Formal computation of the \(\tau \)-function

*X*, in which the BPS invariants \(\Omega (\gamma )\) are constant. Let us consider the family of double structures as defined in Sect. 2.8. We can identify

### Remarks 6.1

- (i)We can easily extend this to a miniversal variation if required, by first introducing an extra factor \(q\in \mathbb {C}^*\) rescaling the central charge
*Z*, and also adding a component \(Z^\vee :\Gamma ^\vee \rightarrow \mathbb {C}\) as in Sect. 2.8. The resulting central charge isNothing interesting is gained by doing this however.$$\begin{aligned} Z((\beta ,n),\lambda )=2\pi q(\beta \cdot \omega _\mathbb {C}-n)+Z^\vee (\lambda ). \end{aligned}$$ - (ii)
We view the doubled structures defined here as an approximation to the correct BPS structures, which should also incorporate BPS invariants corresponding to objects of the full derived category \(D^b{\text {Coh}}(X)\) supported in all dimensions. To define these rigorously would involve constructing stability conditions on \(D^b{\text {Coh}}(X)\), which for

*X*a general compact Calabi–Yau threefold is a well-known unsolved problem (see [3] and [25] for more on this).

^{1}The contribution from zero-dimensional sheaves is

### Remark 6.2

In the paper [5] we give a rigorous solution to the Riemann–Hilbert problem in the case when *X* is the resolved conifold. This involves writing down a non-perturbative function which has the above asymptotic expansion.

## 7 Quadratic differentials and exact WKB analysis

The only examples of \(\hbox {CY}_3\) categories where the full stability space is understood come from quivers with potential associated to triangulated surfaces. The associated stability spaces can be identified with moduli spaces of meromorphic quadratic differentials on Riemann surfaces [8], and the associated BPS invariants then count finite-length trajectories of these differentials. It turns out that the corresponding Riemann–Hilbert problems are closely related to the exact WKB analysis of time-independent Schrödinger equations. We give a brief and sketchy treatment of this connection here; we hope to return to this subject in future papers.

### 7.1 Quadratic differentials

*S*,

*q*), where

*S*is a compact Riemann surface of genus

*g*and

*q*a meromorphic quadratic differential on

*S*with simple zeroes, and poles of multiplicities \(m_i\). It is a complex orbifold of dimension

*S*,

*q*) is a double cover \(\pi :\hat{S}\rightarrow S\) branched at the zeroes and odd-order poles of

*q*. We denote by \(\hat{S}^\circ \subset \hat{S}\) the complement of the inverse image of the poles of

*q*, and define the hat-homology group

*S*,

*q*) we mean a path in

*S*along which \(\sqrt{q}\) has constant phase \(\theta \). A finite-length trajectory is of one of two types:

- (a)
a saddle connection connects two zeroes of the differential (not necessarily distinct);

- (b)
a closed trajectory: any such moves in an annulus of trajectories called a ring domain.

*q*by

### Claim 7.1

The data \((\Gamma , Z,\Omega )\) described above defines a miniversal variation of convergent BPS structures over the orbifold \({\text {Quad}}(g,m)\).

### Sketch proof

To check the wall-crossing formula one can use the results of [8] to view \({\text {Quad}}(g,m)\) as an open subset of a space of stability conditions on a \(\hbox {CY}_3\) triangulated category defined by a quiver with potential, and then apply the theory of wall-crossing for generalised DT invariants [19, 21].

The fact that the BPS structures are convergent should follow from the results of [8, Section 5]. The basic point is that the only non-finiteness in the BPS spectrum arises from finitely many ring domains. Each of these contributes an infinite collection of saddle connections with classes of the form \(\gamma +n\alpha \), where \(\alpha \) is the class of the ring domain. But for sufficiently large \(R>0\) these saddle connections give a finite contribution to the sum (9). \(\square \)

### 7.2 Voros symbols

The weak Riemann–Hilbert problem (see Remark 4.6(iii)) defined by the above BPS structures can be solved using exact WKB analysis of an associated Schrödinger equation. This was essentially proved by Iwaki and Nakanishi [18] following Gaiottoc et al. [15].

*S*,

*q*) as above. Let us also choose a projective structure on the Riemann surface

*S*.

^{2}We can then invariantly consider the holomorphic Schrödinger equation

*z*is a co-ordinate in the chosen projective structure. The WKB method involves substituting

*k*is even. One has \(\omega _0=\sqrt{q(z)} \, dz\).

*S*,

*q*), then the sum over \(g\ge 1\) in the above formal expressions can be Borel summed in the direction \({r}\). This results in Voros symbols which are holomorphic functions of \(t=\hbar \) defined in a neighbourhood of 0 in the half-plane \(\mathbb {H}_{r}\). They also compute the wall-crossing behaviour for these Borel sums as one varies the active ray \({r}\subset \mathbb {C}^*\).

### Claim 7.2

Given a quadratic differential (*S*, *q*) as above, the Borel sums of the cycle Voros symbols give a solution to the corresponding weak Riemann–Hilbert problem with \(t=\hbar \).

### Sketch proof

For the definition of the weak Riemann–Hilbert problem see Remark 4.6(iii). The claim should follow from the work of Iwaki and Nakanishi [18], although a careful proof would require additional continuity arguments to take care of differentials with multiple saddle connections. See particularly Theorem 2.18, Theorem 3.4 and formula (2.21). \(\square \)

It is interesting to ask whether given a suitable choice of base projective structure, the Voros symbols in fact give solutions to the full Riemann–Hilbert problem as stated in Sect. 4.2. Another interesting topic for further research is the connection with topological recursion [10], which is known to be closely related to exact WKB analysis. In particular, it is interesting to ask whether the \(\tau \)-function computed by topological recursion gives a \(\tau \)-function in the sense of this paper.

## Footnotes

## Notes

### Acknowledgements

The author is very grateful to Kohei Iwaki, Andy Neitzke, Ivan Smith and Balázs Szendrői for extremely helpful discussions during the long gestation period of this paper. This work was partially supported by an ERC Advanced grant. The calculations in Sect. 6 were carried out jointly with Kohei Iwaki during a visit to the (appropriately named) Bernoulli Centre in Lausanne. I would also like to thank Sven Meinhardt for a careful reading of the preprint version, and for several useful comments.

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