Polynomiality of orbifold Hurwitz numbers, spectral curve, and a new proof of the Johnson-Pandharipande-Tseng formula

TL;DR

This paper provides a new combinatorial proof of the quasi-polynomiality of orbifold Hurwitz numbers, demonstrating their derivation via spectral curve topological recursion and connecting to the Johnson-Pandharipande-Tseng formula.

Contribution

It introduces a direct combinatorial proof of quasi-polynomiality and applies spectral curve topological recursion to orbifold Hurwitz numbers, offering a new perspective on their structure.

Findings

Spectral curve topological recursion applies to orbifold Hurwitz numbers.

A new combinatorial proof of quasi-polynomiality is provided.

The orbifold Hurwitz numbers relate to the Johnson-Pandharipande-Tseng formula.

Abstract

In this paper we present an example of a derivation of an ELSV-type formula using the methods of topological recursion. Namely, for orbifold Hurwitz numbers we give a new proof of the spectral curve topological recursion, in the sense of Chekhov, Eynard, and Orantin, where the main new step compared to the existing proofs is a direct combinatorial proof of their quasi-polynomiality. Spectral curve topological recursion leads to a formula for the orbifold Hurwitz numbers in terms of the intersection theory of the moduli space of curves, which, in this case, appears to coincide with a special case of the Johnson-Pandharipande-Tseng formula.