Volumes of moduli spaces of directed ribbon graphs and Cut-and-Join operators

TL;DR

This paper explores the algebraic structure of directed ribbon graphs to compute moduli space volumes recursively, introduces integral operators satisfying Cut-and-Join equations, and connects these to generating series for dessins d'enfants.

Contribution

It introduces a novel algebraic framework for directed ribbon graphs, enabling recursive volume computations and linking to Cut-and-Join operators and dessins d'enfants.

Findings

Recursive formulas for moduli space volumes

Cut-and-Join operators satisfy specific equations

Generating series for dessins d'enfants derived

Abstract

In this paper, we investigate the algebraic structure underlying the acyclic decomposition. This decomposition applies to directed metric ribbon graphs and enables the recursive computation of the volumes of their moduli spaces. Building on this, we define integral operators with these volumes and show that they satisfy a Cut-and-Join type equation. Furthermore, we demonstrate that a suitable specialization of these operators gives rise to a generating series for \textit{dessins d'enfants}.