Double Hurwitz numbers: polynomiality, topological recursion and intersection theory

TL;DR

This paper proves that double Hurwitz numbers follow a polynomial structure and are governed by topological recursion, linking their enumeration to intersection theory and extending known results from single Hurwitz numbers.

Contribution

It confirms polynomiality and topological recursion for double Hurwitz numbers and proposes a new ELSV-like formula connecting them to intersection theory.

Findings

Double Hurwitz numbers satisfy polynomiality.

They are governed by topological recursion.

A preliminary ELSV-like formula is proposed.

Abstract

Double Hurwitz numbers enumerate branched covers of $\mathbb{CP}^1$ with prescribed ramification over two points and simple ramification elsewhere. In contrast to the single case, their underlying geometry is not well understood. In previous work by the second- and third-named authors, the double Hurwitz numbers were conjectured to satisfy a polynomiality structure and to be governed by the topological recursion, analogous to existing results concerning single Hurwitz numbers. In this paper, we resolve these conjectures by a careful analysis of the semi-infinite wedge representation for double Hurwitz numbers, by pushing further methods previously used for other Hurwitz problems. We deduce a preliminary version of an ELSV-like formula for double Hurwitz numbers, by deforming the Johnson-Pandharipande-Tseng formula for orbifold Hurwitz numbers and using properties of the topological recursion under variation of spectral curves. In the course of this analysis, we unveil certain vanishing properties of the Chiodo classes.