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.
Highlights
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
Bridgeland mathematical terms, it describes the output of unrefined Donaldson–Thomas theory applied to a three-dimensional Calabi–Yau category with a stability condition
Summary
In this paper we study a class of Riemann–Hilbert problems arising naturally in Donaldson–Thomas theory. 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]. For general BPS structures we have no existence or uniqueness results for solutions to the Riemann–Hilbert problem It is not clear why the τ -function as defined here should exist in the general uncoupled case. “Appendix B” contains some simple analytic results involving partially-defined self-maps of algebraic tori
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