Hodge integrals and tau-symmetric integrable hierarchies of Hamiltonian evolutionary PDEs

TL;DR

This paper constructs a new class of integrable hierarchies associated with Frobenius manifolds, linking them to Gromov-Witten invariants and Hodge classes, and explores their deformation of classical hierarchies like KdV.

Contribution

It introduces Hodge integrable hierarchies for semisimple Frobenius manifolds and relates them to Gromov-Witten theory and tau-functions, proposing a universal hierarchy conjecture.

Findings

Hierarchy generates intersection numbers of Gromov-Witten classes.

For 1D case, hierarchy deforms KdV with infinite parameters.

Conjecture: hierarchy is universal among scalar Hamiltonian integrable hierarchies.

Abstract

For an arbitrary semisimple Frobenius manifold we construct Hodge integrable hierarchy of Hamiltonian partial differential equations. In the particular case of quantum cohomology the tau-function of a solution to the hierarchy generates the intersection numbers of the Gromov--Witten classes and their descendents along with the characteristic classes of Hodge bundles on the moduli spaces of stable maps. For the one-dimensional Frobenius manifold the Hodge hierarchy is a deformation of the Korteweg--de Vries hierarchy depending on an infinite number of parameters. Conjecturally this hierarchy is a universal object in the class of scalar Hamiltonian integrable hierarchies possessing tau-functions.