Hurwitz numbers, matrix models and enumerative geometry

TL;DR

This paper introduces a conjectural recursion for Hurwitz numbers across all genera, inspired by advances in topological string theory and mirror symmetry, connecting enumerative geometry with matrix models.

Contribution

It proposes a new recursion relation for Hurwitz numbers derived from topological string theory and mirror symmetry, extending previous methods to all genera.

Findings

Recursion relation for Hurwitz numbers at all genera

Connection between topological string theory and enumerative geometry

Limit of infinite framing yields Hurwitz theory conjecture

Abstract

We propose a new, conjectural recursion solution for Hurwitz numbers at all genera. This conjecture is based on recent progress in solving type B topological string theory on the mirrors of toric Calabi-Yau manifolds, which we briefly review to provide some background for our conjecture. We show in particular how this B-model solution, combined with mirror symmetry for the one-leg, framed topological vertex, leads to a recursion relation for Hodge integrals with three Hodge class insertions. Our conjecture in Hurwitz theory follows from this recursion for the framed vertex in the limit of infinite framing.