Cubic Hodge integrals and integrable hierarchies of Volterra type

TL;DR

This paper links cubic Hodge integrals with integrable hierarchies of Volterra type, revealing new algebraic relations and reductions in the 2D Toda hierarchy for specific parameter values, especially positive rationals.

Contribution

It identifies a novel connection between cubic Hodge integrals and Volterra-type integrable hierarchies, providing a new perspective on their algebraic structure and reductions.

Findings

Reduced systems of 2D Toda hierarchy emerge at special parameter values.

Discrete series correspond to Volterra lattice and generalizations.

Provides new explanation for integrable structures of cubic Hodge integrals.

Abstract

A tau function of the 2D Toda hierarchy can be obtained from a generating function of the two-partition cubic Hodge integrals. The associated Lax operators turn out to satisfy an algebraic relation. This algebraic relation can be used to identify a reduced system of the 2D Toda hierarchy that emerges when the parameter $\tau$ of the cubic Hodge integrals takes a special value. Integrable hierarchies of the Volterra type are shown to be such reduced systems. They can be derived for positive rational values of $\tau$. In particular, the discrete series $\tau = 1,2,\ldots$ correspond to the Volterra lattice and its hungry generalizations. This provides a new explanation to the integrable structures of the cubic Hodge integrals observed by Dubrovin et al. in the perspectives of tau-symmetric integrable Hamiltonian PDEs.