An intersection-theoretic proof of the Harer-Zagier formula

TL;DR

This paper introduces an intersection-theoretic approach to derive a formula for the Euler characteristic of the moduli space of smooth curves, utilizing Hodge integrals and providing a new proof of the Harer-Zagier formula.

Contribution

It offers a novel intersection-theoretic formula for the Euler characteristic, connecting Hodge integrals with the Chern class of the log tangent bundle, and extends properties of Omega-classes.

Findings

Derived a new formula for the Euler characteristic of moduli spaces

Connected Omega-classes with the Chern class of the log tangent bundle

Provided an extensive list of properties of Omega-classes

Abstract

We provide an intersection-theoretic formula for the Euler characteristic of the moduli space of smooth curves. This formula reads purely in terms of Hodge integrals and, as a corollary, the standard calculus of tautological classes gives a new short proof of the Harer-Zagier formula. Our result is based on the Gauss-Bonnet formula, and on the observation that a certain parametrisation of the $\Omega$-class - the Chern class of the universal $r$-th root of the twisted log canonical bundle - provides the Chern class of the log tangent bundle to the moduli space of smooth curves. Being $\Omega$-classes by now employed in many enumerative problems, mostly recently found and at times surprisingly different from each other, we dedicate some work to produce an extensive list of their general properties: extending existing ones, finding new ones, and writing down some only known to the experts.