On the Goulden-Jackson-Vakil conjecture for double Hurwitz numbers

TL;DR

This paper provides three new formulas expressing double Hurwitz numbers as intersection numbers on various moduli spaces, supporting the Goulden-Jackson-Vakil conjecture and revealing deeper structures in tautological classes.

Contribution

It introduces three novel intersection formulas for double Hurwitz numbers involving different moduli spaces, advancing understanding of their geometric and combinatorial properties.

Findings

Formulas relate double Hurwitz numbers to intersection theory on moduli spaces.

Reveals relations between tautological classes and Chiodo classes.

Hints at deeper geometric structures underlying Hurwitz numbers.

Abstract

Goulden, Jackson and Vakil observed a polynomial structure underlying one-part double Hurwitz numbers, which enumerate branched covers of $\mathbb{CP}^1$ with prescribed ramification profile over $\infty$, a unique preimage over 0, and simple branching elsewhere. This led them to conjecture the existence of moduli spaces and tautological classes whose intersection theory produces an analogue of the celebrated ELSV formula for single Hurwitz numbers. In this paper, we present three formulas that express one-part double Hurwitz numbers as intersection numbers on certain moduli spaces. The first involves Hodge classes on moduli spaces of stable maps to classifying spaces; the second involves Chiodo classes on moduli spaces of spin curves; and the third involves tautological classes on moduli spaces of stable curves. We proceed to discuss the merits of these formulas against a list of desired properties enunciated by Goulden, Jackson and Vakil. Our formulas lead to non-trivial relations between tautological intersection numbers on moduli spaces of stable curves and hints at further structure underlying Chiodo classes. The paper concludes with generalisations of our results to the context of spin Hurwitz numbers.