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Riemann-Hilbert problems from Donaldson-Thomas theory

Tom Bridgeland

TL;DR

The paper develops a nonperturbative framework linking Donaldson-Thomas theory to Riemann-Hilbert problems via BPS structures, proving a unique, gamma-function–based solution in the uncoupled finite case. It introduces tau-functions that encode wall-crossing and connect to Gromov-Witten theory and topological string invariants, offering explicit constructions and asymptotics. It also bridges to exact WKB methods through quadratic differentials, suggesting deep connections between stability conditions, Stokes phenomena, and nonperturbative partition functions in string/gauge theories.

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.

Riemann-Hilbert problems from Donaldson-Thomas theory

TL;DR

The paper develops a nonperturbative framework linking Donaldson-Thomas theory to Riemann-Hilbert problems via BPS structures, proving a unique, gamma-function–based solution in the uncoupled finite case. It introduces tau-functions that encode wall-crossing and connect to Gromov-Witten theory and topological string invariants, offering explicit constructions and asymptotics. It also bridges to exact WKB methods through quadratic differentials, suggesting deep connections between stability conditions, Stokes phenomena, and nonperturbative partition functions in string/gauge theories.

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.

Paper Structure

This paper contains 51 sections, 17 theorems, 196 equations, 1 figure.

Table of Contents

  1. Introduction
  2. BPS structures: initial definitions
  3. Definition and terminology
  4. Donaldson-Thomas invariants
  5. Poisson algebraic torus
  6. Twisted torus
  7. Ray diagram
  8. Further terminology
  9. BPS automorphisms
  10. Doubling construction
  11. A basic example: the Kronecker quiver
  12. Summary of the contents of the paper
  13. The Riemann-Hilbert problem
  14. Solution in the uncoupled case
  15. Variations of BPS structure
  16. ...and 36 more sections

Key Result

Theorem 3.1

Let $(Z,\Gamma,\Omega)$ be a finite, uncoupled, integral BPS structure. Suppose that $\xi\in \bT$ satisfies $\xi(\gamma)=1$ for all active classes $\gamma\in \Gamma$. Then Problem an has the unique solution where the product is over the finitely many active classes $\gamma\in \Gamma$ with $\operatorname{Im} Z(\gamma)/t>0$.

Figures (1)

  • Figure 1: The ray diagram associated to a BPS structure.

Theorems & Definitions (47)

  • Definition 1
  • Definition 2
  • Definition 3
  • Theorem 3.1
  • Definition 4
  • Theorem 3.2
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • ...and 37 more