An intersection-theoretic proof of the Harer–Zagier fomula

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 Ω-class -the Chern class of the universal rth root of the twisted log canonical bundle -provides the Chern class of the log tangent bundle to the moduli space of smooth curves. These Ω-classes have been recently employed in a great variety of enumerative problems. We produce a list of their properties, proving new ones, collecting the properties already in the literature or only known to the experts, and extending some of them.

On some hyperelliptic Hurwitz–Hodge integrals

“…(3) By Okounkov [12] and Okounkov and Pandharipande [21], we have that Hurwitz numbers can be efficiently written as vacuum expectation of operators in the Fock space, which in this case form a handy algebra closed under commutation relations:…”

“…Remark 2•3. By looking at the formula above is it easy to deduce a few properties of the classes , see [12] for a more exhausive list. For instance, [x]…”