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 $\tau$ 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 $\tau =1,2,\dots$ 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.

latex2e, amsmath,amssymb,amsthm, 29pp, no figure; (v2) final version for contribution to Boris Dubrovin memorial volume, American Mathematical Society

References in corpus (3)

• The Extended Bigraded Toda hierarchy
• A Conjecture on Hodge Integrals
• Orbifold melting crystal models and reductions of Toda hierarchy

Cited by in corpus (3)

• KP integrability of triple Hodge integrals. I. From Givental group to hierarchy symmetries
• Reduction of the 2D Toda Hierarchy and Linear Hodge Integrals
• Integrable structures of specialized hypergeometric tau functions