All-genus WDVV recursion, quivers, and BPS invariants

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.