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arXiv · 1909.13095

Cubic Hodge integrals and integrable hierarchies of Volterra type

Kanehisa Takasaki
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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 $\tau = 1,2,\ldots$ 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.

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Kanehisa Takasaki. 2019-09-28. Cubic Hodge integrals and integrable hierarchies of Volterra type. https://doi.org/10.1090/pspum/103.1/01844

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