On enumeration of $b$-angulations of surfaces from an integrability perspective

TL;DR

This paper explores the enumeration of polygonal angulations on surfaces of fixed genus using integrability techniques, revealing new structural insights and conjectural relations.

Contribution

It introduces new structural results for specific angulation cases based on Toda integrability and connects to the Gharakhloo--Latimer conjecture via the Hodge--GUE correspondence.

Findings

Established structural results for $b=3$ and $b=4$ angulations.

Derived a fine structure in the $b=2 u$ case through the Hodge--GUE correspondence.

Implied a conjectural statement of Gharakhloo--Latimer.

Abstract

In this paper, we study generating series enumerating polygonal angulations of closed oriented surfaces of fixed genus, focusing on $b$-angulations with $b = 3$ or $b = 2\nu$, $\nu \geq 2$. Based on Toda integrability, we establish new structural results in the cases $b = 3$ and $b = 4$. Furthermore, via the Hodge--GUE correspondence, we derive a fine structure in the $b = 2\nu$ case, which implies a conjectural statement of Gharakhloo--Latimer.