A Riemann–Hilbert problem for uncoupled BPS structures

Barbieri, Anna

doi:10.1007/s00229-019-01115-y

Cited by 15 publications

(24 citation statements)

References 15 publications

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“…We conclude by discussing two natural limits of the adjoint form of the solution, which both relate to the classical Riemann-Hilbert problem studied in [1,6]. In one of these limits we find the Hamiltonian generating function for the classical solution.…”

Section: Introductionmentioning

confidence: 93%

“…Properties (a), (b) and (d) are clear from expression (18) and well-known properties of the gamma function. For (c) we can use the homogeneity property to reduce to the case a = 1, when the given relation is a simple consequence of the Euler reflection formula for the gamma function: see Lemma 3.1 in [1] for more details. For (e) we can reduce to the case a = 1 using the homogeneity properties (15) and (19), when the claim is a form of the Stirling expansion.…”

Section: Multiple Gamma Functionsmentioning

confidence: 99%

“…These problems involve piecewise holomorphic maps from the complex plane to the group of automorphisms of a Poisson algebraic torus, with discontinuities along a collection of rays prescribed by the DT invariants. Such problems appeared in the physics literature in the work of Gaiotto, Moore and Neitzke [13,14], and have since been considered by mathematicians [1,6,7,9]. The works listed above are mostly concerned with Riemann-Hilbert problems defined using unrefined DT invariants.…”

Section: Introductionmentioning

confidence: 99%

“…Solution to the doubled A 1 problem. Due to a symmetry of the problem, a solution to the above quantum Riemann-Hilbert problem can be deduced from the solution to the corresponding classical problem studied in [1]. Unfortunately we can only prove uniqueness properties for this solution under some strong additional hypotheses (see Remark 3.7).…”

Section: Introductionmentioning

confidence: 99%

“…Two interesting limits. There are two limits to the solution of Theorem 1.5 which it is interesting to consider, and which relate to the classical Riemann-Hilbert problem studied in [1,6].…”

Section: Introductionmentioning

confidence: 99%

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A Quantized Riemann–Hilbert Problem in Donaldson–Thomas Theory

Barbieri

Bridgeland

Stoppa

2020

International Mathematics Research Notices

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Section: Introductionmentioning

confidence: 93%

Section: Multiple Gamma Functionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A Quantized Riemann–Hilbert Problem in Donaldson–Thomas Theory

Barbieri

Bridgeland

Stoppa

2020

International Mathematics Research Notices

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On the monodromy of the deformed cubic oscillator

Bridgeland

Masoero²

2022

Math. Ann.

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We study a second-order linear differential equation known as the deformed cubic oscillator, whose isomonodromic deformations are controlled by the first Painlevé equation. We use the generalised monodromy map for this equation to give solutions to the Riemann-Hilbert problems of (Bridgeland in Invent Math 216(1):69–124, 2019) arising from the Donaldson-Thomas theory of the A$$_2$$ 2 quiver. These are the first known solutions to such problems beyond the uncoupled case. The appendix by Davide Masoero contains a WKB analysis of the asymptotics of the monodromy map.

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Topological Recursion and Uncoupled BPS Structures II: Voros Symbols and the $$\tau $$-Function

Iwaki

Kidwai

2023

Commun. Math. Phys.

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We continue our study of the correspondence between BPS structures and topological recursion in the uncoupled case, this time from the viewpoint of quantum curves. For spectral curves of hypergeometric type, we show the Borel-resummed Voros symbols of the corresponding quantum curves solve Bridgeland’s “BPS Riemann–Hilbert problem”. In particular, they satisfy the required jump property in agreement with the generalized definition of BPS indices $$\Omega $$ Ω in our previous work. Furthermore, we observe the Voros coefficients define a closed one-form on the parameter space, and show that (log of) Bridgeland’s $$\tau $$ τ -function encoding the solution is none other than the corresponding potential, up to a constant. When the quantization parameter is set to a special value, this agrees with the Borel sum of the topological recursion partition function $$Z_\textrm{TR}$$ Z TR , up to a simple factor.

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A Riemann–Hilbert problem for uncoupled BPS structures

Abstract: We study the Riemann-Hilbert problem attached to an uncoupled BPS structure proposed by Bridgeland in [4]. We show that it has "essentially" unique meromorphic solutions given by a product of Gamma functions. We reconstruct the corresponding connection.

Cited by 15 publications

References 15 publications

A Quantized Riemann–Hilbert Problem in Donaldson–Thomas Theory

A Quantized Riemann–Hilbert Problem in Donaldson–Thomas Theory

On the monodromy of the deformed cubic oscillator

Topological Recursion and Uncoupled BPS Structures II: Voros Symbols and the $$\tau $$-Function

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