A Generating Function for Fatgraphs

TL;DR

This paper develops a generating function for fatgraphs with specified properties, providing a compact expression and a matrix integral representation that enables semi-classical computations and explicit formulas for genus zero trees.

Contribution

It introduces a new generating function for fatgraphs with specified valences, offering a compact expression and a matrix integral approach for advanced computations.

Findings

Derived a compact expression for the generating function of fatgraphs.

Established a matrix integral representation enabling semi-classical analysis.

Obtained a closed formula for genus zero, connected trees.

Abstract

We study a generating function for the sum over fatgraphs with specified valences of vertices and faces, inversely weighted by the order of their symmetry group. A compact expression is found for general (i.e. non necessarily connected) fatgraphs. This expression admits a matrix integral representation which enables to perform semi--classical computations, leading in particular to a closed formula corresponding to (genus zero, connected) trees.