Hodge Integrals and Integrable Hierarchies

TL;DR

This paper demonstrates that certain generating series of Hodge integrals are tau-functions of integrable hierarchies and proposes a conjecture linking relative invariants to these hierarchies, verified in specific cases.

Contribution

It establishes a connection between Hodge integrals and integrable hierarchies and introduces a conjecture relating relative invariants to these structures.

Findings

Generating series of some Hodge integrals are tau-functions of KP or 2-Toda hierarchies.

Conjecture on the link between relative invariants and integrable hierarchies.

Verification of the conjecture in specific examples.

Abstract

We show that the generating series of some Hodge integrals involving one or two partitions are tau-functions of the KP hierarchy or the 2-Toda hierarchy respectively. We also formulate a conjecture on the connection between relative invariants and integrable hierarchies. The conjecture is verified in some examples.