A Riemann-Hilbert problem for uncoupled BPS structures

We study the Riemann-Hilbert problem attached to an uncoupled BPS structure proposed by Bridgeland in [3]. We show that it has"essentially"unique meromorphic solutions given by a product of Gamma functions. We reconstruct the corresponding connection.

Here ∆ is a holorphicity sector for a solution Ψ and H ∆ is any open half-plane centred in a non-active ray r contained in ∆. This is the approach of [3], where Y β,r are required to be holomorphic and never-vanishing. We show that the scalar problem as stated in [3] does not always admit solutions (Proposition 2.5), but it can be reformulated in terms of meromorphic functions (Problem 2.8). This is the scalar counter-part 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.
In Section 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 (Theorem 4.4). These are given by a modification in two variables of the Gamma function (equation (3.1)). The inverse problem is considered in Section 5, where the connection ∇ = d − Z t 2 − Ham F t dt, such that ∇Y β,r = 0, is computed (Theorem 5.1). The Hamiltonian function F is vaguely related with an enumerative generating function. Sections 4 and 5 are based on Section 3, where we introduce the function Λ x (y) := y x+y− 1 2 · √ 2π Γ(x + y) · e y , of which we list some relevant properties (Lemma 3.1 and Theorem 3.2). Consider the doubled A 1 example. For any ξ ∈ T, the solution is encoded in two meromorphic functions x ± : C * \ ∓iZ(α)R >0 , with where θ := ln ξ(α), for a chosen branch of the logarithm. These obviously coincide with the result in [3] when θ = 0, and are very closed related to [8,Eq. 3.10]. The Hamiltonian function is F (θ) = − θ 2 4πi + θ 2 .

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 [3], where it is possible to find a wider explanation of the mentioned objects.
Z is called a central charge and Ω is the BPS spectrum.
We denote by Γ + Ω the subset of Γ where H + is the upper half-plane together with the negative real line

Definition 1.2.
An active class is a point γ ∈ Γ such that Ω(γ) = 0. For every active class, we introduce an active ray ℓ 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 these notes 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 ∈ Γ.

Twisted torus
The algebra C[Γ] of formal elements x α , α ∈ Γ, comes endowed with a commutative product · and Poisson Lie bracket [−, −] induced by the intersection form The twisted torus is Elements of C[Γ] act as characters on T: and Z extends to the twisted torus T via It is useful to introduce the map on T It satisfies Given any basis {γ 1 , . . . , γ n } of Γ, the twisted torus T is then endowed with coordinates and logarithmic coordinates We could make other choices of the branch of the complex logarithm: section 4 and 5 would then require minor modifications.

Doubling construction
A BPS structure (Γ, Z, Ω) 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 Γ D := Γ ⊕ Γ ∨ , where Γ ∨ := Hom(Γ, Z), is introduced. Γ D is endowed with a non-degenerate skew-symmetric bilinear form A doubled BPS structure is obtained by extending the central charge Z and the BPS spectrum Ω to Γ D . We set Definition 1.5. We refer to Γ D as the doubled lattice and to this procedure the "doubling procedure". With the choice (1.4) above, (Γ D , Z, Ω) is called a doubled BPS structure. The twisted torus T D associated with Γ D is called the doubled torus.
In 4.2 we mention a "positive BPS structure". Let Γ + be a convex cone in Γ consisting in positive linear combinations of elements of a fixed basis for Γ. By "positive BPS structure" we mean a triple (Γ, Z, Ω) satisfying Definition 1.1 apart from (i), together with a choice of Γ + ⊂ Γ, and such that Z(Γ + ) ⊂ H + , Ω(γ) = 0 if γ ∈ Γ + .

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 [7, 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 Σ of non-intersecting supports of curves in C * and a map S : Σ → G. Solving the RH problem defined by S and with values in M means seeking a piecewise holomorphic function Ψ : C * \ Σ → M such that for every t ∈ Σ the limits Ψ ± (t) of Ψ from the opposite sides of Σ exist and satisfy Ψ + (t) = S(t)Ψ − (t), and fixed constant limit lim t→0 Ψ(t) along any direction in C * \ Σ.
Existence of a solution is not guaranteed in general. The problem in the scalar case (M = C) was widely treated for instance in [14, Muskhelishvili, 1946] or [10, Gakhov, 1966], and solved for S(t) Hölder continuous on the contour Σ 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 (Γ, Z, Ω) induces naturally a RH problem with values in the automorphism group Aut(T) of the twisted torus. Heuristically, attached to any active ray ℓ there is a transform S(ℓ) defined by pull-back in C[Γ] We refer to [12,3] for the fundational issues about S(ℓ) and the general definition. However, it is worth mentioning that S(ℓ) can be viewed as the is absolutely convergent, and the time 1 Hamiltonian flow of this map is the holomorphic map S(ℓ) : U ℓ → T.
Let Σ ⊂ C * be the union of active rays Definition 2.1. The Aut(T)-valued RH problem attached to (Γ, Z, Ω) consists in finding a piecewise holomorphic map Ψ : C * \ Σ → Aut(T) with discontinuities on each component ℓ γ of Σ given by the composition with S(ℓ γ ) (jumping condition), and with asymptotic behaviour near the origin (asymptotic condition) where the action of Z on T is given in (1.1).
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 (Γ, Z, Ω) 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 (Γ, Z, Ω) is generic and uncoupled, then for any choices of ξ the problem of Definition 2.2 can be turned into a scalar problem ([3, sections 4.2]) involving mapŝ and defined by functions S ℓ : Γ × C * → C, Remark 1. This does not applies to non-uncoupled BPS structures.

Definition 2.3.
In analogy with the theory of differential equations, we call S ℓ (β, t) a Stokes factor of the problem. For any fixed β ∈ Γ, we will call also S ℓ (t) a Stokes factor.
Let (Γ, Z, Ω) 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

Problem 2.4 ([3, Problem 4.3]).
For each non-active ray r ∈ C * and for every β ∈ Γ, we seek a holomorphic function Y β,r : H r → C * such that the following conditions are satisfied. RH 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, [3]. 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 ofŶ β to the holomorphicity sectors. Problem 2.4 was solved in this formulation in [3] 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 strictly convex sector between iℓ γ and ℓ γ . Suppose moreover that the Problem 2.4 admit solutions Y β,r : H r → C * for every non active ray r. Take two non active rays r 1 and r 2 such that r 1 is a small deformation of iℓ γ and r 2 = −ir 1 , and say ∆ = H r 1 We can narrow the domain H r of definition of Y β,r without losing the uniqueness of the solution, but the fact that ξ takes values in C * does not ensure that Y β,r never vanishes. Definition 2.6. For any ray r ⊂ C * and given 0 < ǫ ≤ π 2 , we denote by ∆ ǫ r the sector of angle 2ǫ centred in r Fix 0 < ǫ ≤ π 2 and modify Problem 2.4. For every β ∈ Γ and for each non-active ray r seek for a holomorphic function Y β,r :  RH 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 RH 2 , RH 3 hold as in 2.4, away from some Notice that we keep the same notation for conditions in 2.8 as in 2.4, although the domain is different.

Complex analysis
In this section we introduce the complex multivalued function defined for x ∈ C and y ∈ C * as a modification of the inverse Gamma function. We study a number of properties (Lemma 3.1) and we provide an integral expression for Λ (x, y) (Lemma 3.5) that justify why it will define a basis of solutions to 2.8. 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) depeds on exp x + y − 1 2 ln y − ln y . For a chosen branch of the logarithm, it is a holomorphic function in two variable with zeroes prescribed by Γ(x + y) −1 at points (x, x + 2πki), k ∈ Z ≤0 . Later, we will consider Λ x (y) := Λ (x, y) , regarded as a family of holomorphic 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 zeroes at y = x + 2πki, k a negative integer.
The thesis follows from Lemma 3.1, 2.
The functionX has the form of a classical solution to a Riemann-Hilbert problem and in the 4.2 it will be compared with [6]. 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.

Explicit solution
It is clear that it is enough to understand the solution for the double of a one-dimensional structure in order to understand the solution to any finite uncoupled BPS structure. So, we first consider the simplest case, that is the A 1 BPS structure with two active classes ±α.
x − and x + are respectively a holomorphic function with zeroes at points z θ−2mπi , m ∈ N, and a meromorphic function with poles at z θ+2mπi , m ∈ N \ {0}. Points z θ)+2kπi , k ∈ Z, lie in a circle divided in two halves by ℓ or −ℓ, and cluster at the origin. Every half-plane H ±r , r = ±ℓ, contains then at most a finite number of those points. Call {Y β,r } a system of solutions to 2.8. For null-vectors β, set Y β,r ≡ 1. For every non-active ray r occurring clock-wise betwen ℓ and −ℓ, take By Proposition 4.3, Y γ ∨ ,±r (t) satisfy RH 1 , RH 2 , RH 3 , and provide a solution to Problem 2.8. The shift θ → θ+2kπi, k ∈ Z, produces another solution with shifted zeroes/poles. Multiplying by any factor of S(θ, −z/t) we obtain again a solution to the problem. Say ǫ the width of the maximal holomorphicity sector for Y γ ∨ ,r (t). Then the restriction of Y γ ∨ ,r (t) to ∆ ǫ ′ r solve the holomorphic Riemann-Hilbert problem for any 0 < ǫ ′ < ǫ. If for instance r ⊂ H + , the function Y γ ∨ ,r is holomorphic and C * -valued in H r ∩ H + and can be analytically continued beyond ±ℓ as long as no critical points are met. At the limit for r arbitrary close to ±ℓ this means that it can be analytically continued at most for t such that arg z t < min {arg (θ + 2kπi)}. Solutions to the Riemann-Hilbert problems in the finite uncoupled case are obtained by superimposing the solution in the A 1 case along any "active direction". For any generic ray r, define Γ Ω r := γ ∈ Γ : Ω(γ) = 0 and 0 < arg v Z(γ) < π, ∀v ∈ r , the set of active classes γ whose corresponding active ray lies "on the left" of r. For β ∈ Γ D , a solution Y β,r is given by the restriction to H r of The next Theorem 4.4 then follows.

A formal approach, [6]
Filippini, Garcia-Fernandez, Stoppa in [6] studied and solved a problem strictly related to those mentioned in Section 2.1, for "positive BPS structures". The problem is stated in terms of a piecewise continuous map Ψ with values in Aut C[Γ + ], but it can easily be translated in the language of definition 2.1. It is viewed as the "conformal limit" of an analogous Riemann-Hilbert problem already stated in [9], differing for the asymptotic behaviour of the solutions at the origin and at infinity. The solution Ψ is expressed as a sum over connected rooted trees with vertices decorated by elements of Γ of iterated integrals. 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 theorem, which is a standard tool in the theory of Riemann-Hilbert problems.
The authors also show that their arguments extend to a generic and convergent BPS structure (Definition 1.1), provided that there exists a strictly convex cone Γ + ⊂ Γ such that Ω(γ) = 0 for γ ∈ Γ \ Γ + ∪ −Γ + , and Ω(γ) = Ω(−γ) for every γ ∈ Γ + , Z(Γ + ) is contained in a strictly convex cone in C * . In this case S(ℓ) and Ψ are replaced by maps S s (ℓ) and . In the case of (the doubled of) an uncoupled BPS structure, Ψ(t) |s=1 (x β ) is given by where DT(γ) := h∈N\{0} γ/h∈Γ 1 h 2 Ω(γ/h). One may deduce a solution to the (scalar) meromorphic Riemann-Hilbert problem 2.8 for uncoupled BPS structures via evaluating with some caution Ψ s=1 (t) at ξ ∈ T D : ξ(x γ ) = ξ(γ). Notice that DT(γ)x γ e Z(γ) = hγ ′ =γ 1 h 2 Ω(γ ′ )x hγ ′ e hZ(γ ′ ) . For every β ∈ Γ D , we have that is a piecewise continuous function C * \ Σ → C given by to be extended over the active rays to half-planes. We can re-arrange the sum in (4.2) in such a way that we recognise the presence of critical points required by the Stokes factors. Assume that only primitive vectors have non-vanishing BPS spectrum and this is equal to one. Then (4.2) "matches" with (4.1) if, ignoring convergence issues, we write This coincides with (4.1) after changing of variable s → Z(γ)/s, and integration by part.

The Hamiltonian vector field
Riemann-Hilbert problems are related with the theory of irregular differential equations as inverse problems. Let U, V ∈ gl n (C) and be an 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]. In best cases, the solution Y to the corresponding RH problem can be inverted to compute A = dY · Y −1 . A connection of the form (5.1) plays an important röle in the theory of Frobenius manifolds. This points of view, in the context of BPS structures, si going to be developed and formalised in the work in progress [4].
In the case under consideration, we want to define the connection on the Aut(T)-principal bundle corresponding to a framed variation of our BPS data (see [3,Section 3.3]), having generalised monodromy given by (ℓ γ , S(ℓ γ )) Ω(γ) =0 . This has the form where • Z ∈ Hom(Γ, C) is the central charges, acting on the twisted torus via (1.1); • 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.