{
  "id": "2011.06292",
  "version": "v1",
  "published": "2020-11-12T10:08:43.000Z",
  "updated": "2020-11-12T10:08:43.000Z",
  "title": "Isomonodromic tau functions on a torus as Fredholm determinants, and charged partitions",
  "authors": [
    "Fabrizio Del Monte",
    "Harini Desiraju",
    "Pavlo Gavrylenko"
  ],
  "comment": "53 pages, 11 figures",
  "categories": [
    "math-ph",
    "hep-th",
    "math.CO",
    "math.MP",
    "nlin.SI"
  ],
  "abstract": "We prove that the isomonodromic tau function on a torus with Fuchsian singularities and generic monodromies in $GL(N,\\mathbb{C})$ can be written in terms of a Fredholm determinant of Cauchy-Plemelj operators. We further show that the minor expansion of this Fredholm determinant is described by a series labeled by charged partitions. As an example, we show that in the case of $SL(2,\\mathbb{C})$ this combinatorial expression takes the form of a dual Nekrasov-Okounkov partition function, or equivalently of a free fermion conformal block on the torus. Based on these results, we also propose a definition of the tau function of the Riemann-Hilbert problem on a torus with generic jump on the A-cycle.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-11-12T10:08:43.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "isomonodromic tau function",
      "fredholm determinant",
      "charged partitions",
      "free fermion conformal block",
      "dual nekrasov-okounkov partition function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 53,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}