# Towards an orbifold generalization of Zvonkine's $r$-ELSV formula

## TL;DR

This paper advances the understanding of Zvonkine's $r$-ELSV formula by proving its quasi-polynomiality and proposing an orbifold generalization, linking Hurwitz numbers with moduli spaces of $r$-spin structures.

## Contribution

It proves the quasi-polynomiality property of Zvonkine's $r$-ELSV formula and introduces an orbifold generalization with supporting evidence.

## Key findings

- Proved quasi-polynomiality of Zvonkine's $r$-ELSV formula.
- Proposed and validated an orbifold generalization.
- Confirmed the spectral curve data reproduces unstable cases.

## Abstract

We perform a key step towards the proof of Zvonkine's conjectural $r$-ELSV formula that relates Hurwitz numbers with completed $(r+1)$-cycles to the geometry of the moduli spaces of the $r$-spin structures on curves: we prove the quasi-polynomiality property prescribed by Zvonkine's conjecture. Moreover, we propose an orbifold generalization of Zvonkine's conjecture and prove the quasi-polynomiality property in this case as well. In addition to that, we study the $(0,1)$- and $(0,2)$-functions in this generalized case and we show that these unstable cases are correctly reproduced by the spectral curve initial data.