{
  "id": "1608.02568",
  "version": "v1",
  "published": "2016-08-08T19:38:16.000Z",
  "updated": "2016-08-08T19:38:16.000Z",
  "title": "Backlund transformation of Painleve III($D_8$) tau function",
  "authors": [
    "M. A. Bershtein",
    "A. I. Shchechkin"
  ],
  "comment": "28 pages",
  "categories": [
    "math-ph",
    "hep-th",
    "math.MP",
    "math.RT",
    "nlin.SI"
  ],
  "abstract": "We study explicit formula (suggested by Gamayun, Iorgov, Lisovyy) for Painlev\\'e III ($D_8$) $\\tau$~function in terms of Virasoro conformal blocks with central charge $1$. The Painleve equation has two types of bilinear forms, we call them Toda-like and Okamoto-like. We obtain these equations from the representation theory using an embedding of direct sum of two Virasoro algebra in certain superalgebra. These two types of bilinear forms correspond to Neveu-Schwarz sector and Ramond sector of this algebra. We also obtain $\\tau$ functions of algebraic solutions of Painlev\\'e III ($D_8$) from the special representations of the Virasoro algebra of highest weight $(n+1/4)^2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-08-08T19:38:16.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "tau function",
      "backlund transformation",
      "virasoro algebra",
      "bilinear forms correspond",
      "virasoro conformal blocks"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}