Topological recursion for the Poincare polynomial of the combinatorial moduli space of curves

TL;DR

This paper demonstrates that the Poincare polynomial of the moduli space of curves satisfies a topological recursion, linking it to the enumeration of Grothendieck's dessins d'enfants through Laplace transform.

Contribution

It establishes a topological recursion formula for the Poincare polynomial, connecting algebraic geometry with combinatorial enumeration of dessins d'enfants.

Findings

Poincare polynomial satisfies Eynard-Orantin topological recursion.

The polynomial is the Laplace transform of dessins d'enfants count.

Recursion determines Poincare polynomial from initial data.

Abstract

We show that the Poincare polynomial associated with the orbifold cell decomposition of the moduli space of smooth algebraic curves with distinct marked points satisfies a topological recursion formula of the Eynard-Orantin type. The recursion uniquely determines the Poincare polynomials from the initial data. Our key discovery is that the Poincare polynomial is the Laplace transform of the number of Grothendieck's dessins d'enfants.