{
  "id": "1505.00259",
  "version": "v1",
  "published": "2015-05-01T19:43:39.000Z",
  "updated": "2015-05-01T19:43:39.000Z",
  "title": "Isomonodromic $τ$-functions and $W_N$ conformal blocks",
  "authors": [
    "P. Gavrylenko"
  ],
  "comment": "20 pages,7 figures",
  "categories": [
    "hep-th",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We study the solution of the Schlesinger system for the 4-point $\\mathfrak{sl}_N$ isomonodromy problem and conjecture an expression for the isomonodromic $\\tau$-function in terms of 2d conformal field theory beyond the known $N=2$ Painlev\\'e VI case. We show that this relation can be used as an alternative definition of conformal blocks for the $W_N$ algebra and argue that the infinite number of arbitrary constants arising in the algebraic construction of $W_N$ conformal block can be expressed in terms of only a finite set of parameters of the monodromy data of rank $N$ Fuchsian system with three regular singular points. We check this definition explicitly for the known conformal blocks of the $W_3$ algebra and demonstrate its consistency with the conjectured form of the structure constants.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-05-01T19:43:39.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "conformal block",
      "isomonodromic",
      "2d conformal field theory",
      "regular singular points",
      "painleve vi case"
    ],
    "publication": {
      "doi": "10.1007/JHEP09(2015)167",
      "journal": "Journal of High Energy Physics",
      "year": 2015,
      "month": "Sep",
      "volume": 2015,
      "pages": 167
    },
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015JHEP...09..167G",
      "inspire": 1365828
    }
  }
}