Reduction of the 2D Toda Hierarchy and Linear Hodge Integrals

TL;DR

This paper constructs a reduction of the 2D Toda hierarchy resulting in a tau-symmetric Hamiltonian integrable hierarchy that connects linear Hodge integrals with other well-known integrable systems.

Contribution

It introduces a novel reduction of the 2D Toda hierarchy that links linear Hodge integrals to the intermediate long wave and fractional Volterra hierarchies.

Findings

Established a new integrable hierarchy controlling linear Hodge integrals.

Connected the hierarchy's flows to the intermediate long wave hierarchy.

Linked the remaining flows to a limit of the fractional Volterra hierarchy.

Abstract

We construct a certain reduction of the 2D Toda hierarchy and obtain a tau-symmetric Hamiltonian integrable hierarchy. This reduced integrable hierarchy controls the linear Hodge integrals in the way that one part of its flows yields the intermediate long wave hierarchy, and the remaining flows coincide with a certain limit of the flows of the fractional Volterra hierarchy which controls the special cubic Hodge integrals.