{
  "id": "1909.13095",
  "version": "v1",
  "published": "2019-09-28T13:39:13.000Z",
  "updated": "2019-09-28T13:39:13.000Z",
  "title": "Cubic Hodge integrals and integrable hierarchies of Volterra type",
  "authors": [
    "Kanehisa Takasaki"
  ],
  "comment": "latex2e, amsmath,amssymb,amsthm, 29pp, no figure",
  "categories": [
    "math-ph",
    "hep-th",
    "math.MP",
    "nlin.SI"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-09-28T13:39:13.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14N35",
      "37K10"
    ],
    "keywords": [
      "volterra type",
      "integrable hierarchies",
      "2d toda hierarchy",
      "two-partition cubic hodge integrals",
      "algebraic relation"
    ],
    "note": {
      "typesetting": "LaTeX",
      "pages": 29,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}