# All-genus WDVV recursion, quivers, and BPS invariants

## TL;DR

This paper generalizes the genus 0 WDVV equation to all genera for local 3-folds over surfaces, linking Gromov--Witten invariants with Donaldson--Thomas invariants of quivers and expressing BPS invariants via quiver moduli.

## Contribution

It introduces an all-genus WDVV recursion for local surfaces and establishes a novel correspondence with refined Donaldson--Thomas invariants of acyclic quivers.

## Key findings

- Proved an all-genus WDVV equation for local 3-folds over surfaces.
- Connected Gromov--Witten invariants with quiver DT invariants.
- Expressed BPS invariants in terms of Betti numbers of quiver moduli.

## Abstract

Let $X$ be a smooth projective surface and $D$ a smooth rational ample divisor in $X$. We prove an all-genus generalization of the genus $0$ WDVV equation for primary Gromov--Witten invariants of the local 3-fold $\mathcal{O}_X(-D)$. The proof relies on a correspondence between all-genus Gromov--Witten invariants and refined Donaldson--Thomas invariants of acyclic quivers. In particular, the corresponding BPS invariants are expressed in terms of Betti numbers of moduli spaces of quiver representations.