ILW equation for the Hodge integrals revisited

TL;DR

This paper provides a shorter proof that the generating series of linear Hodge integrals satisfies the ILW hierarchy, leveraging an explicit formula for one-point integrals discovered independently by multiple researchers.

Contribution

It offers a more concise proof of the ILW equation's relation to Hodge integrals using a known explicit formula for one-point integrals.

Findings

The generating series satisfies the ILW hierarchy.

A shorter proof is achieved compared to previous work.

The proof utilizes an explicit formula for one-point integrals.

Abstract

In a previous paper we proved that after a simple transformation the generating series of the linear Hodge integrals on the moduli space of stable curves satisfies the hierarchy of the Intermediate Long Wave equation. In this paper we present a much shorter proof of this fact. Our new proof is based on an explicit formula for the one-point linear Hodge integrals that was found independently by Faber, Pandharipande and Ekedahl, Lando, Shapiro, Vainshtein.