The Kontsevich constants for the volume of the moduli of curves and topological recursion

TL;DR

This paper introduces a topological recursion formula for the Euclidean volume of the combinatorial moduli space of curves, providing a new proof of the Kontsevich constants relating Euclidean and symplectic volumes.

Contribution

It develops an Eynard-Orantin type topological recursion based on edge removal in ribbon graphs, offering a novel proof of the Kontsevich constants.

Findings

Derived a topological recursion formula for Euclidean volumes

Provided a new proof of the Kontsevich constants

Connected combinatorial and geometric volume ratios

Abstract

We give an Eynard-Orantin type topological recursion formula for the canonical Euclidean volume of the combinatorial moduli space of pointed smooth algebraic curves. The recursion comes from the edge removal operation on the space of ribbon graphs. As an application we obtain a new proof of the Kontsevich constants for the ratio of the Euclidean and the symplectic volumes of the moduli space of curves.