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

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.