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 twopartition 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 tausymmetric integrable Hamiltonian PDEs.
 Publication:

arXiv eprints
 Pub Date:
 September 2019
 arXiv:
 arXiv:1909.13095
 Bibcode:
 2019arXiv190913095T
 Keywords:

 Mathematical Physics;
 High Energy Physics  Theory;
 Nonlinear Sciences  Exactly Solvable and Integrable Systems;
 14N35;
 37K10
 EPrint:
 latex2e, amsmath,amssymb,amsthm, 29pp, no figure