{
  "id": "2006.15652",
  "version": "v1",
  "published": "2020-06-28T16:51:03.000Z",
  "updated": "2020-06-28T16:51:03.000Z",
  "title": "Irregular conformal blocks, Painlevé III and the blow-up equations",
  "authors": [
    "Pavlo Gavrylenko",
    "Andrei Marshakov",
    "Artem Stoyan"
  ],
  "categories": [
    "math-ph",
    "hep-th",
    "math.MP",
    "nlin.SI"
  ],
  "abstract": "We study the relation of irregular conformal blocks with the Painlev\\'e III$_3$ equation. The functional representation for the quasiclassical irregular block is shown to be consistent with the BPZ equations of conformal field theory and the Hamilton-Jacobi approach to Painlev\\'e III$_3$. It leads immediately to a limiting case of the blow-up equations for dual Nekrasov partition function of 4d pure supersymmetric gauge theory, which can be even treated as a defining system of equations for both $c=1$ and $c\\to\\infty$ conformal blocks. We extend this analysis to the domain of strong-coupling regime where original definition of conformal blocks and Nekrasov functions is not known and apply the results to spectral problem of the Matheiu equations. Finally, we propose a construction of irregular conformal blocks in the strong coupling region by quantization of Painlev\\'e III$_3$ equation, and obtain in this way a general expression, reproducing $c=1$ and quasiclassical $c\\to\\infty$ results as its particular cases. We have also found explicit integral representations for $c=1$ and $c=-2$ irregular blocks at infinity for some special points.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-06-28T16:51:03.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "irregular conformal blocks",
      "blow-up equations",
      "4d pure supersymmetric gauge theory",
      "dual nekrasov partition function",
      "irregular block"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}