Cubic Hodge integrals and integrable hierarchies of Volterra type

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 τ \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 τ \tau . In particular, the discrete series τ = 1 , 2 , … \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.