Topological recursion for irregular spectral curves

TL;DR

This paper explores topological recursion on irregular spectral curves, deriving new recursive formulas to count dessins d'enfant and analyze asymptotic behaviors in enumerative geometry.

Contribution

It introduces a novel three-term recursion for counting dessins d'enfant with one face on irregular spectral curves.

Findings

Derived topological recursion for irregular spectral curves.

Calculated all one-point invariants for the curve xy^2=1.

Established connections to asymptotic enumerative problems.

Abstract

We study topological recursion on the irregular spectral curve $xy^2-xy+1=0$, which produces a weighted count of dessins d'enfant. This analysis is then applied to topological recursion on the spectral curve $xy^2=1$, which takes the place of the Airy curve $x=y^2$ to describe asymptotic behaviour of enumerative problems associated to irregular spectral curves. In particular, we calculate all one-point invariants of the spectral curve $xy^2=1$ via a new three-term recursion for the number of dessins d'enfant with one face.