Equivalence of ELSV and Bouchard-Mari\~no conjectures for $r$-spin Hurwitz numbers

TL;DR

This paper introduces two conjectures relating $r$-spin Hurwitz numbers to Gromov-Witten invariants and topological recursion, proving their equivalence and providing supporting evidence.

Contribution

It formulates the $r$-ELSV formula and the $r$-BM conjecture for $r$-spin Hurwitz numbers and proves their equivalence, extending classical results to the $r$-spin setting.

Findings

Proposes the $r$-ELSV formula linking Hurwitz numbers to $r$-spin structures.

Introduces the $r$-BM conjecture relating Hurwitz numbers to topological recursion.

Shows the equivalence of the $r$-ELSV formula and the $r$-BM conjecture.

Abstract

We propose two conjectures on Huwritz numbers with completed $(r+1)$-cycles, or, equivalently, on certain relative Gromov-Witten invariants of the projective line. The conjectures are analogs of the ELSV formula and of the Bouchard-Mari\~no conjecture for ordinary Hurwitz numbers. Our $r$-ELSV formula is an equality between a Hurwitz number and an integral over the space of $r$-spin structures, that is, the space of stable curves with an $r$th root of the canonical bundle. Our $r$-BM conjecture is the statement that $n$-point functions for Hurwitz numbers satisfy the topological recursion associated with the spectral curve $x = -y^r + \log y$ in the sense of Chekhov, Eynard, and Orantin. We show that the $r$-ELSV formula and the $r$-BM conjecture are equivalent to each other and provide some evidence for both.