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 $\tau = 1,2,\ldots$ 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.
- Pub Date:
- September 2019
- Mathematical Physics;
- High Energy Physics - Theory;
- Nonlinear Sciences - Exactly Solvable and Integrable Systems;
- latex2e, amsmath,amssymb,amsthm, 29pp, no figure