# Mirror curve of orbifold Hurwitz numbers

## TL;DR

This paper demonstrates that for orbifold Hurwitz numbers, the mirror spectral curve and forms can be constructed from edge-contraction operations in genus 0, revealing a deep connection with Gromov-Witten theory.

## Contribution

It establishes a novel construction of mirror objects for orbifold Hurwitz numbers using edge-contraction operations in genus 0.

## Key findings

- Mirror spectral curve derived from edge-contraction operations.
- Parallelism with genus 0 Gromov-Witten invariants.
- Framework applicable to orbifold Hurwitz counting problems.

## Abstract

Edge-contraction operations form an effective tool in various graph enumeration problems, such as counting Grothendieck's dessins d'enfants and simple and double Hurwitz numbers. These counting problems can be solved by a mechanism known as topological recursion, which is a mirror B-model corresponding to these counting problems. We show that for the case of orbifold Hurwitz numbers, the mirror objects, i.e., the spectral curve and the differential forms on it, are constructed solely from the edge-contraction operations of the counting problem in genus $0$ and one marked point. This forms a parallelism with Gromov-Witten theory, where genus 0 Gromov-Witten invariants correspond to mirror B-model holomorphic geometry.