Lectures on the ELSV formula

TL;DR

This paper explains the ELSV formula, which connects Hurwitz numbers and Hodge integrals, and details its proof via virtual localization on moduli spaces of relative stable maps, based on lectures from a summer school.

Contribution

It provides an accessible exposition of the ELSV formula and its proof using virtual localization techniques on moduli spaces of relative stable maps.

Findings

Clarifies the relationship between Hurwitz numbers and Hodge integrals.

Details the proof of the ELSV formula using virtual localization.

Provides educational insights from summer school lectures.

Abstract

The ELSV formula, first proved by Ekedahl, Lando, Shapiro, and Vainshtein, relates Hurwitz numbers to Hodge integrals. Graber and Vakil gave another proof of the ELSV formula by virtual localization on moduli spaces of stable maps to the projective line, and also explained how to simplify their proof using moduli spaces of relative stable maps to the projective line relative to a point. In this expository article, we explain what the ELSV formula is and how to prove it by virtual localization on moduli spaces of relative stable maps, following Graber-Vakil. This note is based on lectures given by the author at Summer School on "Geometry of Teichmuller Spaces and Moduli Spaces of Curves" at Center of Mathematical Sciences, Zhejiang University, July 14--20, 2008.