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.

arXiv commentlatex2e, amsmath,amssymb,amsthm, 29pp, no figure; (v2) final version for contribution to Boris Dubrovin memorial volume, American Mathematical Society

References in corpus (22)

• The Topological Vertex
• Integrable Hierarchies and Dispersionless Limit
• Hurwitz numbers and intersections on moduli spaces of curves
• Framed knots at large N
• Hodge integrals and invariants of the unknot
• Free fermions and tau-functions
• The Extended Bigraded Toda hierarchy
• Generating functions for colored 3D Young diagrams and the Donaldson-Thomas invariants of orbifolds
• Toda hierarchies and their applications
• A Conjecture on Hodge Integrals
• Fractional Volterra Hierarchy
• Modified melting crystal model and Ablowitz-Ladik hierarchy
• Deformations of the KdV hierarchy and related soliton equations
• Hodge-GUE correspondence and the discrete KdV equation
• Open string amplitudes of closed topological vertex
• Orbifold melting crystal models and reductions of Toda hierarchy
• Generalized Ablowitz-Ladik hierarchy in topological string theory
• Hurwitz numbers and integrable hierarchy of Volterra type
• Simple Lax Description of the ILW Hierarchy
• Modified melting crystal model and Ablowitz-Ladik hierarchy
• Three-partition Hodge integrals and the topological vertex
• On cubic Hodge integrals and random matrices

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