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

2025/07/09 by Simon Barazer, Barazer, Simon
Mathematics · #Algebraic Geometry and Number Theory #Combinatorics (math.CO) #FOS: Mathematics #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.2507.07308

openalex publication_date 2025/07/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 dessins d'enfants.

Citations

• Cutting directed ribbon graphs and recursion for volumes of the combinatorial moduli spaces
• A short overview of the "Topological recursion"
• String and dilaton equations for counting lattice points in the moduli space of curves
• Invariants of algebraic curves and topological expansion
• Random Matrices

Related

• Cutting directed ribbon graphs and recursion for volumes of the combinatorial moduli spaces
• A combinatorial model for the moduli of bordered Riemann surfaces and a compactification
• A dual point of view on the ribbon graph decomposition of moduli space
• Recursion between Mumford volumes of moduli spaces
• Topological recursion for symplectic volumes of moduli spaces of curves