{
  "id": "1807.00085",
  "version": "v1",
  "published": "2018-06-29T23:37:34.000Z",
  "updated": "2018-06-29T23:37:34.000Z",
  "title": "Hurwitz numbers and integrable hierarchy of Volterra type",
  "authors": [
    "Kanehisa Takasaki"
  ],
  "comment": "latex2e, amsmath,amssymb,amsthm, 12 pages, no figure",
  "categories": [
    "math-ph",
    "hep-th",
    "math.MP",
    "math.QA",
    "nlin.SI"
  ],
  "abstract": "A generating function of the single Hurwitz numbers of the Riemann sphere $\\mathbb{CP}^1$ is a tau function of the lattice KP hierarchy. The associated Lax operator $L$ turns out to be expressed as $L = e^{\\mathfrak{L}}$, where $\\mathfrak{L}$ is a difference-differential operator of the form $\\mathfrak{L} = \\partial_s - ve^{-\\partial_s}$. $\\mathfrak{L}$ satisfies a set of Lax equations that form a continuum version of the Bogoyavlensky-Itoh (aka hungry Lotka-Volterra) hierarchies. Emergence of this underlying integrable structure is further explained in the language of generalized string equations for the Lax and Orlov-Schulman operators of the 2D Toda hierarchy. This leads to logarithmic string equations, which are confirmed with the help of a factorization problem of operators.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-06-29T23:37:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14N10",
      "37K10"
    ],
    "keywords": [
      "volterra type",
      "integrable hierarchy",
      "2d toda hierarchy",
      "single hurwitz numbers",
      "aka hungry lotka-volterra"
    ],
    "note": {
      "typesetting": "LaTeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}