# Variations of BPS structure and a large rank limit

## TL;DR

This paper investigates how flat bundles from Donaldson-Thomas theory of Calabi-Yau threefolds behave as their rank becomes very large, connecting finite-dimensional structures to infinite-dimensional Riemann-Hilbert problems and Gromov-Witten invariants.

## Contribution

It establishes the convergence of flat sections of finite rank BPS structures to solutions of infinite-dimensional Riemann-Hilbert problems in the large rank limit, linking to Gopakumar-Vafa invariants.

## Key findings

- Flat sections converge to solutions of Riemann-Hilbert problems
- Expression for Gromov-Witten invariants in terms of hypergeometric equations
- Connection between finite and infinite-dimensional BPS structures

## Abstract

We study a class of flat bundles, of finite rank $N$, which arise naturally from the Donaldson-Thomas theory of a Calabi-Yau threefold $X$ via the notion of a variation of BPS structure. We prove that in a large $N$ limit their flat sections converge to the solutions to certain infinite dimensional Riemann-Hilbert problems recently found by Bridgeland. In particular this implies an expression for the positive degree, genus $0$ Gopakumar-Vafa contribution to the Gromov-Witten partition function of $X$ in terms of solutions to confluent hypergeometric differential equations.