An irregular spectral curve for the generation of bipartite maps in topological recursion

TL;DR

This paper introduces a novel irregular spectral curve for topological recursion that efficiently generates bipartite maps of any genus and boundary length, extending previous work on ordinary maps and dessins d'enfant.

Contribution

It derives a new irregular spectral curve for bipartite maps, linking topological recursion with advanced map enumeration techniques.

Findings

Derived an efficient generating function method for bipartite maps.

Identified a new irregular spectral curve related to dessins d'enfant.

Extended topological recursion framework to bipartite map enumeration.

Abstract

We derive an efficient way to obtain generating functions of bipartite maps of arbitrary genus and boundary length using a spectral curve as initial data for the framework of topological recursion. Based on an earlier result of Chapuy and Fang counting these maps and having a structural proximity to topological recursion, we deduce the corresponding spectral curve which has a strong relation to the spectral curve giving rise to generating functions of ordinary maps. In contrast to ordinary maps, the spectral curve is an irregular one in the sense of Do and Norbury. It generalises the irregular curve for the enumeration of Grothendieck's dessins d'enfant.