# A note on BPS structures and Gopakumar-Vafa invariants

## TL;DR

This paper links BPS structures derived from sheaf-theoretic Gopakumar-Vafa invariants to Gromov-Witten partition functions, showing a formal power series correspondence under certain conjectures, extending previous genus 0 results.

## Contribution

It establishes a formal power series correspondence between BPS structures and Gromov-Witten invariants, generalizing prior genus 0 findings to all genera under conjectural assumptions.

## Key findings

- Reproduces Gromov-Witten partition functions from BPS structures
- Extends Bridgeland and Iwaki's genus 0 results to all genera
- Provides a formal framework connecting sheaf-theoretic invariants and GW theory

## Abstract

We regard the work of Maulik and Toda, proposing a sheaf-theoretic approach to Gopakumar-Vafa invariants, as defining a BPS structure, that is, a collection of BPS invariants together with a central charge. Assuming their conjectures, we show that a canonical flat section of the flat connection corresponding to this BPS structure, at the level of formal power series, reproduces the Gromov-Witten partition function for all genera, up to some error terms in genus 0 and 1. This generalises a result of Bridgeland and Iwaki for the contribution from genus 0 Gopakumar-Vafa invariants.