A Riemann–Hilbert problem for uncoupled BPS structures

We study the Riemann–Hilbert problem attached to an uncoupled BPS structure proposed by Bridgeland in (“Riemann–Hilbert problems from Donaldson–Thomas theory I”). We show that it has “essentially” unique meromorphic solutions given by a product of Gamma functions. We reconstruct the corresponding connection.


Introduction
This paper studies the instance of Riemann-Hilbert problem proposed by Bridgeland in [4] for uncoupled BPS structures. It is stated in terms of complex-valued functions and it is solved in [4] for a fixed value of a certain parameter. We show that for any value of that parameter the solution is a pair of meromorphic functions expressed explicitly as a product of Gamma functions. An integral representation of the solution is used to reconstruct the corresponding connection.
The same class of Riemann-Hilbert problems was considered by Filippini, Garcia-Fernandez, Stoppa in [7], motivated by the physics work [10]. Their solution takes values in the automorphism group of an algebraic torus. The contexts of [4,7] are slightly different, and comparing the two articles might require some efforts. We show in which sense and to which extent the two problems (and the corresponding solutions) are related, and we propose a new way to express the solution. In turn, this is analogous to the "conformal limit" of coordinates for the moduli spaces of N = 2 four-dimensional gauge theories compactified on a circle, presented by Gaiotto in [9], the main difference being that we consider coordinates on a complex torus. Our discussion about the solutions will allow us to consider also the "quantized" version of this problem. This is the content of a work in progress [2].
Riemann-Hilbert problems are inverse problems in the theory of differential equations. They classically consist in seeking a piecewise holomorphic function on C * with values in a Lie group, with prescribed behaviour near the origin and jumping discontinuities along a real-codimension 1 boundary [8]. A BPS structure is an instance of the stability data defined by Kontsevich and Soibelman [12] and contains the information from the unrefined Donaldson-Thomas theory of dimension three Calabi-Yau categories. It defines naturally a Riemann-Hilbert problem with values in the automorphism group of an algebraic torus that, in some nice cases, can be traslated into a scalar problem [4]. Riemann-Hilbert problems for BPS structures are relevant in some attemps of defining a Frobenius manifold type structure from Donaldson-Thomas theory [3,15].
In the rest of the introduction we illustrate the content of the paper. BPS structures In the first section we briefly recall some notions about integral BPS structures ( , Z , ) and the associated twisted torus T. They are defined by a finite rank lattice with a pairing −, − , a homomorphism Z : → C and, a map : → Z. The twisted torus is the space with characters x γ , γ ∈ , acting on it as x γ (ξ ) = ξ(γ ). We restrict to a class of BPS structures called uncoupled. This is the analogue of the physics terminology of "mutually local" BPS structures. We impose moreover finiteness and convergence hypotheses.
The basic example of an uncoupled BPS structure is the "doubled A 1 BPS structure", defined by a lattice = Z · α ⊕ Z · α ∨ , a central charge Z ∈ Hom ( , C) with Z (α ∨ ) = 0, and a symmetric map : → Z with (±α) = 1 and vanishing otherwise. A Riemann-Hilbert problem In the second section a Riemann-Hilbert problem for uncoupled BPS structures is introduced. To the active rays are attached transforms S( γ ) of the torus T. Let ⊂ C * be the union of active rays. We are interested in finding a sectionally holomorphic map with discontinuities on each component γ of given by the composition with S( γ ) (jumping condition), asymptotic behaviour near the origin lim t→0 (t) • = exp(Z /t) Id, and algebraic behaviour at infinity. The uniqueness of the solution depends on the possibility of extending the restriction of to any sector bounded by two consecutive active rays over its edges. In the uncoupled case, the Aut(T)-valued Riemann-Hilbert problem can be turned into a scalar problem (that is with complex values) by fixing a point ξ ∈ T, evaluating in ξ and then applying it to a point β ∈ . We obtain the following diagram that allows for a complex-analytical approach.
Here is a holomorphicity sector for a solution and H is any open half-plane centred in a non-active ray r contained in . For non-active rays r , we seek for complex valued functions Y β,r , that can be compared in the common domain of definition. This is the approach of [4], where Y β,r are required to be holomorphic and never-vanishing. In fact the scalar problem as stated in [4] does not always admit solutions (Proposition 2.5 below), but it can be reformulated and solved in terms of meromorphic functions (Problem 2.6 below). This is the scalar counterpart of the Aut(T)-valued Riemann-Hilbert problem. It has "essentially" at most one solution, i.e. unique up to the choice of vanishing order of a finite number of points.
Solutions and Hamiltonian vector field The solution to the scalar Riemann-Hilbert problem and the corresponding connection are considered in Sects. 4 and 5, which are based on Sect. 3, where we develope the analytical background. We introduce the function It is a modification in two variables of the Gamma function and its relevant properties are listed in Lemma 3.1 and Theorem 3.2.
In Sect. 4 we prove that, for every ξ ∈ T, there exist non-trivial meromorphic functions Y β,r β,r solving the meromorphic Riemann-Hilbert problem for uncoupled BPS structures (Theorem 4.4). In the doubled A 1 case, for example, for any ξ ∈ T, Y α,r ≡ 1 and the solution is encoded in two meromorphic functions Y ± : C * \ ±i Z(α)R >0 → CP 1 , obtained by gluing together Y α ∨ ,r as r lies on one or the other side of ± . We have where θ := ln ξ(α), for a chosen branch of the logarithm. These obviously coincide with the result in [4] when θ = 0, and are very closed related to [9,Eq. 3.10]. The inverse problem is considered in Sect. 5 for a BPS structure with trivial pairing. "Doubling" the construction, one has that T has a symplectic structure. From {Y β,r } β,r we deduce and compute a connection ∇ on the trivial Aut(T)bundle over CP 1 , such that ∇ = 0. Say Analogous computations allow to define a similar connection for any uncoupled BPS structure.

BPS structures and notation
We briefly recall the notion of a BPS structure. The aim of this section is to fix the notation for the rest of the article. Most of the definitions recalled in the following are from [4], where it is possible to find a wider explanation of the mentioned objects. Definition 1.1. A BPS structure ( , Z , ) of rank n is the datum of a finite rank lattice Z ⊕n (the charge lattice) endowed with an intersection form, that is an integral, bilinear and skew-symmetric pairing −, − : × → Z, a homomorphism Z : → C, and a map of sets : → Q, such that (i) is symmetric, i.e. (−α) = (α) for all α ∈ , and (ii) there is a uniform constant C > 0 such that, for some fixed norm || · || in ⊗ R, |Z (α)| > C · ||α|| for all α with (α) = 0.
Z is called a central charge and is the BPS spectrum.
We denote by + the subset of where H + is the upper half-plane together with the negative real line

Definition 1.2.
An active class is a point γ ∈ such that (γ ) = 0. For every active class, we introduce an active ray γ := Z (γ )R >0 ⊂ C * . An active ray is sometimes referred to as a BPS ray. A ray r ⊂ C * which is not active is said to be generic.
In particular an uncoupled BPS structure is generic.
In this article we will mostly assume that a BPS structure is ray-finite, i.e. there are finitely many active rays, or finite, i.e. there are only finitely many active classes γ ∈ .

Twisted torus
The algebra C[ ] of formal elements x α , α ∈ , comes endowed with a commutative product · x α · x β = (−1) α,β x α+β , and Poisson Lie bracket [−, −] induced by the intersection form A central charge Z : → C acts on C[ ] as a derivation: The twisted torus is Elements of C[ ] act as characters on T: and Z extends to the twisted torus T via for every α ∈ , ξ ∈ T.
We can also interpret θ i as functions on the torus with non-trivial monodromy or make other choices of the branch of the complex logarithm: Sects. 4 and 5 would then require minor modifications.

Doubling construction
A BPS structure ( , Z , ) can be embedded into a richer structure, via doubling the construction [12,Sect. 2.6]. This is particularly useful when the intersection form −, − is degenerate. To this end, the lattice ⊕ ∨ , where ∨ := Hom( , Z), is considered. ⊕ ∨ is endowed with a non-degenerate skew-symmetric bilinear form denoted again by −, − and defined as follows A doubled BPS structure is obtained by extending the central charge Z and the BPS spectrum to ⊕ ∨ . We set We refer to ⊕ ∨ as the doubled lattice and to this procedure the "doubling procedure". With the choice (1.4) above, ( ⊕ ∨ , Z , ) is called a doubled BPS structure.
The twisted torus T associated with a doubled BPS structure inherits logarithmic coordinates . . , m, and comes equipped with the symplectic form ω = − m j=1 dθ j ∧ dθ ∨ j .

Riemann-Hilbert problems
A Riemann-Hilbert (RH) problem classically consists in finding maps from C * to a complex manifold with prescribed jumps across the supports of curves in C * . See for instance [8, Chapter 3] for a brief introduction to the topic. Suppose we are given a complex manifold M together with a complex Lie group G acting on it, the union of supports of curves in C * intersecting transversally at the origin, and a map S : → G. Solving the RH problem defined by S and with values in M means seeking a piecewise holomorphic function : C * \ → M such that for every t ∈ the limits ± (t) of from the opposite sides of exist and satisfy and has fixed constant limit lim t→0 (t) along any direction in C * \ .
Existence of a solution is not guaranteed in general. The problem in the scalar case (i.e. when M = C) was widely treated for instance in [14, Muskhelishvili, 1946] or [11, Gakhov, 1966], and solved for S(t) Hölder continuous on the contour apart from a finite number of points. The solution to a scalar Riemann-Hilbert problem is unique provided that its restriction to a holomorphicity sector can be continued to an invertible function on its closure¯ ⊂ CP 1 .

RH problems for finite BPS structures
A ray-finite, integral, convergent BPS structure ( , Z , ) induces naturally a RH problem with values in the automorphism group Aut(T) of the twisted torus. Heuristically, attached to any active ray there is a transform S( ) defined by pull-back We refer to [4,12] for the fundational issues about S( ) and the general definition.
where the action of Z on T is given in (1.2).
For any fixed point ξ ∈ T, such an Aut(T)-valued RH problem induces a problem with values in T simply by evaluating on ξ any automorphism of the torus.
Notice that the hypothesis of ray-finiteness of the structure ( , Z , ) is essential to define the problem in Definition 2.2, while BPS structures might present countably many active rays.

Scalar RH problems for uncoupled BPS structures
If moreover ( , Z , ) is generic and uncoupled, then for any choices of ξ the problem of Definition 2.2 can be turned into a scalar problem ([4, sections 4.2]) involving mapsŶ and defined by functions S : × C * → C, Remark 2. This does not applies to non-uncoupled BPS structures. It depends on the fact that S (β, t) is trivial when β is active. Uncoupledness also implies commutativity of S l 1 , S l 2 for any l 1 , l 2 .

Definition 2.3.
In analogy with the theory of differential equations, we call S (β, t) a Stokes factor of the problem. For any fixed β ∈ , we will call also S (t) a Stokes factor.
Let ( , Z , ) be an integral generic convergent uncoupled BPS structure and fix ξ ∈ T. For any ray l, let H l be the open half-plane centred in l For each non-active ray r ∈ C * and for every β ∈ , we seek a holomorphic function Y β,r : H r → C * such that the following conditions are satisfied.
R H 1 Suppose that two generic rays r 1 = r 2 form the boundary rays of a convex sector ⊂ C * taken in clockwise order, then for all t ∈ H r 1 ∩ H r 2 with 0 < |t| 1, Problem 2.4 has the advantage of involving complex functions, moreover it admits at most one solution [4]. A solution {Y β,r } β,r of 2.4 is related with the functionsŶ β in (2.1) via analytic continuation to half-planes of the restriction of Y β to the holomorphicity sectors. Problem 2.4 was solved in this formulation in [4] for the special fixed point ξ ≡ 1, but it is easily seen that it often does not admit solution. Proof. In the hypothesis of the Proposition, suppose for instance that Z(γ )/θ (γ ) is in the convex sector between i γ and γ . Suppose moreover that the Problem 2.4 admit solutions Y β,r : H r → C * for every non active ray r . Let β such that β, γ > 0. There are two distinct non-active rays r 1 and r 2 such that = H r 1 ∩H r 2 contains t = Z (γ )/θ (γ ) and γ . This implies that 1 − e θ(γ )−Z (γ )/t divides Y β,r 1 · Y β,r 2 −1 for every β ∈ , with Y β,r i never vanishing in . But t = Z (γ )/θ (γ ) ∈ is a zero of 1 − e θ(γ )−Z (γ )/t , yielding a contradiction.
In particular, if θ(γ ) ∈ R \ {0}, then Z (γ )/θ (γ ) lies in one of the active rays ± γ . Proposition 2.5 is not a counterexample to the existence of piecewise continuous solutions to 2.2 and we reformulate the scalar Riemann-Hilbert problem in terms of meromorphic functions. Problem 2.6. (Meromorphic RH problem) For every β ∈ and for each nonactive ray r , we seek a meromorphic function Y β,r : H r → CP 1 satisfying the following conditions: R H 0 Y β,r is holomorphic and C * -valued away from a finite number of zeroes or poles in position t = Z (γ ) θ(γ )+2kπi , γ ∈ , for some k ∈ Z; R H 1 suppose that two generic rays r 1 = r 2 form the boundary rays of a convex sector ⊂ C * taken in clockwise order, then and R H 2 , RH 3 hold as in 2.4, away from some t = Z (γ ) θ(γ )+2kπi , γ ∈ , k ∈ Z.
Notice that we keep the same notation for conditions in 2.6 as in 2.4, although the domain is different. Proof. Fix a vanishing order of a finite number of points. The proof goes as in [4,Lemma 4.9], with minor modifications. The argument is a standard application of the Liouville theorem, see also [8,Chapter 3] as an example.
Definition 2.8. We say that a solution is minimal if its finitely many critical points (zeroes or poles) associated with any γ (that is in position Z (γ ) θ(γ )+2kπi , k ∈ Z) are simple and lie on the same side of γ .

Complex analysis
In this section we introduce the complex multivalued function (x, y) := (x + y) · e y y x+y− 1 2 · √ 2π (3.1) defined for x ∈ C and y ∈ C * as a modification of the Gamma function. We study a number of properties (Lemma 3.1) and we provide an integral expression for (x, y) (Lemma 3.5) that justifies why it will define a basis of solutions to 2.6. To this end we also introduce the function S : C × C → C, symmetric with respect the exchange of the two variables, Multivaluedness of (3.1) depends on exp x + y − 1 2 ln y − ln y . For a chosen branch of the logarithm, it is a meromorphic function in two variable with poles prescribed by (x + y) at points (x, x + 2π ki), k ∈ Z ≤0 . Later, we will consider regarded as a family of meromorphic functions defined for y ∈ C * \ R >0 , parametrised by a choice of x = ln e x , kπ ≤ Im(x) < (k + 1)π , and with aligned poles at y = x + 2π ki, k a negative integer. Proof. For 1. recall the Euler reflection formula for the Gamma function For 2. use the property (z) = z (1 + z). 3. is clear as, for |y| < , −1 x (y) is bounded by a function that goes as a holomorphic function times y . 4. follows from the following formula for the logarithm of the shifted Gamma function [13, Chapter 1.1] valid for any N ∈ N, z, a ∈ C, One also deduces that the logarithm ln (x, y) has formal asymptotic expansion as y lies in any convex sector of C * not containing R <0 . (x, y) can be defined as the analytic continuation of an integral expression. For (θ, w) ∈ C 2 , 0 < Im(θ ) < 2π and Im(w) = 0, we consider the function It can be extended over Im(θ ) > 2π bŷ Lemmas 3.3 and 3.4 are aimed to prove the following Theorem.
The shift of 2πi in the Theorem above is essentially related with different choices of the branch of the logarithm.
The thesis follows from 2. in Lemma 3.1.
The functionX has the form of a classical solution to a Riemann-Hilbert problem.
Such an integral expression is the basis solution for an analogous Riemann-Hilbert problem considered in [7], Sect. 4.3. We can look atX as a piecewise function in 0 < | Im(θ )| < 2π , Im(w) = 0, satisfying the symmetrŷ and with discontinuities prescribed by S. This can be shown via a direct integral contour argument.

Lemma 3.5.
Denote byX(θ, w ± 0 ) the limits ofX(θ, w) as w approaches a point w 0 clock-wise and counter-clock-wise respectively. Then if w 0 ∈ R >0 , Proof. Assume w 0 ∈ R >0 . ComputingX(θ, w + 0 ) is equivalent to slightly deform the integral path R >0 clock-wise in the lower half-plane and evaluate the function in w 0 . We define the contour C: e −θ+s − 1 ds, (3.5) at the limit for , δ → 0. In (3.5), the contribution from C δ vanishes as δ → 0. 1 − s w 0 ∈ R <0 if s is real and has positive (resp. negative) imaginary part if s ∈ C + (resp. C − ). The contributions from C − and C + differ by 1 e −θ e s −1 ds, that is ln(1 − e θ−w 0 ), from which the thesis follows. The statement for v 0 ∈ R <0 can be deduced using (3.4).

Doubled A 1 BPS structure
We first consider the simplest case, that is the doubled A 1 BPS structure with two active classes ±α.
has at most algebraic growth when |t| 0. Call {Y β,r } a system of solutions to 2.6. For null-vectors β, Y β,r ≡ 1. For every non-active ray r occurring clock-wise between and − , take By Proposition 4.3, Y γ ∨ ,±r (t) satisfy R H 1 , R H 2 , R H 3 , and provide a solution to Problem 2.6 with only simple zeroes/poles. The shift θ → θ + 2kπi, k ∈ Z, produces another solution with shifted simple zeroes/poles. Notice moreover that, since the jumping factors S(± )(t) = S(±θ, ∓z/t), defined in (3.2), admits a factorisation in an infinite product S j (t), we have that, if {Y r } satisfies R H 0 -R H 3 , so {Y r · S j } does.

General case
Solutions to the Riemann-Hilbert problems in the finite uncoupled case are obtained by superimposing the solution in the doubled A 1 case along any "active direction". Let ( , Z , ) denote an integral uncoupled convergent BPS structure. For any generic ray r , define the set of active classes γ whose corresponding active ray lies "on the right" of r . We also define For β ∈ , a minimal solution Y β,r is given by the restriction to H r of The next Theorem 4.4 then follows.

Relation with the work [7]
In [6] studied a Riemann-Hilbert problem strictly related to those considered here is studied. Although conceptually different, the solutions in the uncoupled case are formally the same. In this section we briefly describe the relation between their approach and ours. The problem in [7] is stated for "positive BPS structures". It is viewed as the "conformal limit" of a Riemann-Hilbert problem considered in [10], which has a different asymptotic behaviour at infinity. By "positive BPS structure" we mean a triple ( , Z , ) satisfying Definition 1.1 apart from (i), together with a choice of a convex cone + ⊂ consisting in non-negative linear combinations of elements of a fixed basis for . Z ( + ) should lie in the strictly positive upper-half plane and (γ ) = 0 if γ / ∈ + . The solution is a piecewise continuous map with values in Aut C[ + ], and it is expressed as a sum of iterated integrals indexed by rooted trees whose vertices are labelled by elements of + . The proof involves 1) solving a fixed point integral problem, and 2) expanding terms of type log(1 − x γ ) formally as k≥1 x kγ k , in order to apply the Plemelj's theorem, which is a standard tool in the theory of Riemann-Hilbert problems. The arguments extend to a generic and convergent BPS structure, provided that there exists a strictly convex cone + ⊂ such that (γ ) = 0 for γ ∈ \ + ∪ − + , and Z ( + ) is contained in a strictly convex cone in C * . Let γ 1 , . . . , γ n be a basis for such + . In this case S( ]. Note also that, due to a different sign conventions, the central charge in [7] is −Z . In the case of an uncoupled BPS structure, integrals in the solution are not iterated and (t) |s=1 (x β ) is given by the sum One may deduce a solution to the (scalar) meromorphic Riemann-Hilbert Problem 2.6 for uncoupled BPS structures satisfying the conditions above by evaluating (t) |s=1 (x β ) at ξ ∈ T. Recall that x γ (ξ ) = ξ(γ ) ∈ C and h −2 β, γ = h −1 β, γ / h . Formally, reindexing and reordering the double sum γ ∈ \{0} h>0, γ /h∈ , we have that (t) |s=1 x β applied to ξ can be written as (4. 3) The formula (4.3) is a piecewise continuous map C * \ → C to be extended over the active rays to half-planes. If we now want to compare it with the minimal solution Y β,r (4.2) we should assume that the sum −

The Hamiltonian vector field for the doubled A 1 BPS structure
Riemann-Hilbert problems are related with the theory of irregular differential equations as inverse problems. Let U, V ∈ g = gl n (C) and be a meromorphic connection with irregular pole at the origin and logarithmic pole at infinity. For every direction r which is not a Stokes ray with a non-trivial Stokes factor in gl n (C), a fundamental solution lives in the half-plane centred in r , undergoing a discontinuity given by a Stokes factor as r crosses a Stokes ray [1]. If the solution Y to the corresponding RH problem can be inverted, we compute In this section we describe a similar picture. If T is a symplectic torus, we look at the solution to the Aut(T)-valued Riemann-Hilbert problem as fundamental solution to a meromorphic connection of type (5.1) with U, V in a different Lie algebra. More precisely, they are symplectic vector fields over the torus.

Remark 5.
A family of connections of the form (5.1) plays an important role in the theory of Frobenius manifolds. This points of view, in the context of BPS structures, is going to be developed and formalised by Bridgeland in the work in progress [5].
where Z is a vector field corresponding to the central charges, and Ham F is the Hamiltonian vector field of a function F : T → C depending on Z and on the BPS spectrum, due to the Stokes factors. The work of Bridgeland and Toledano-Laredo [6] suggests that the residue part of the connection associated to a BPS structure by mean of such a Riemann-Hilbert problem should be seen as the carrier of the information of the BPS spectrum. We define the function (γ ) Li 2 x γ on the twisted torus and we interpret it as a generating function for the -invariants. ( (t)(ξ )) (β) = e −Z (β)/t · Y β,r · ξ(β).
Proof of Theorem 5.1. We first compute the connection (5.2) for the doubled A 1 BPS structure of Example 4.1, from which we also borrow the notation. The proof generalises to the doubled of any uncoupled BPS structure. First observe that the solution Y ± (t) to Problem 4.2 satisfies 3) The very same computations can be performed in higher dimension. For any generic ray r , and j = 1, . . . , m, Eq. (5.3) generalises to where z k := Z (γ k ), a(γ ) j denotes the j-th component of γ with respect to the chosen basis, and r was defined in (4.1). Defininig the vector fields + const. Note that + ⊂ , defined in (1.1), as well as r , selects half of the points of \ {0}.
Acknowledgements. The author is grateful to Tom Bridgeland for many interesting discussions. Thanks are also due to Dylan Allegretti, Jacopo Stoppa (and the anonymous referee) for their comments on the preliminary version. The research leading to these results has received funding from the European research council.
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