Virtual Euler characteristics via topological recursion

TL;DR

This paper develops a novel approach using topological recursion to compute virtual Euler characteristics for various classes of maps and ensembles, including nonorientable surfaces, and explores related recursion relations.

Contribution

It introduces a Seiberg--Witten-like framework within topological recursion to calculate virtual Euler characteristics for multiple map types and ensembles, extending previous methods.

Findings

Derived virtual Euler characteristics for uni- and multicellular maps.

Established recursion relations for the Legendre ensemble.

Extended topological recursion techniques to nonorientable surfaces.

Abstract

We use Seiberg--Witten-like relations in the topological recursion framework to obtain virtual Euler characteristics for uni- and multicellular maps for ensembles of classic orthogonal polynomials and for ensembles related to nonorientable surfaces. We also discuss Harer--Zagier-type recursion relations for 1-point correlation function for the Legendre ensemble.