Combinatorics of loop equations for branched covers of sphere

TL;DR

This paper proves the spectral curve topological recursion and quantum spectral curve equations for bi-colored maps, providing a combinatorial approach, and extends the ideas to 4-colored maps.

Contribution

It introduces a purely combinatorial proof of spectral curve topological recursion for bi-colored maps and extends the framework to 4-colored maps.

Findings

Proved spectral curve topological recursion for bi-colored maps

Established quantum spectral curve equation for the problem

Outlined extension to 4-colored maps

Abstract

We prove, in a purely combinatorial way, the spectral curve topological recursion for the problem of enumeration of bi-colored maps, which are dual objects to dessins d'enfant. Furthermore, we give a proof of the quantum spectral curve equation for this problem. Then we consider the generalized case of 4-colored maps and outline the idea of the proof of the corresponding spectral curve topological recursion.