{
  "id": "2401.06694",
  "version": "v1",
  "published": "2024-01-12T16:59:48.000Z",
  "updated": "2024-01-12T16:59:48.000Z",
  "title": "Topological recursion and variations of spectral curves for twisted Higgs bundles",
  "authors": [
    "Christopher Mahadeo",
    "Steven Rayan"
  ],
  "comment": "53 pages, 5 figures",
  "categories": [
    "math.AG",
    "math-ph",
    "math.CV",
    "math.DG",
    "math.MP",
    "math.SG"
  ],
  "abstract": "Prior works relating meromorphic Higgs bundles to topological recursion, in particular those of Dumitrescu-Mulase, have considered non-singular models that allow the recursion to be carried out on a smooth Riemann surface. We start from an $\\mathcal{L}$-twisted Higgs bundle for some fixed holomorphic line bundle $\\mathcal{L}$ on the surface. We decorate the Higgs bundle with the choice of a section $s$ of $K^*\\otimes\\mathcal{L}$, where $K$ is the canonical line bundle, and then encode this data as a $b$-structure on the base Riemann surface which lifts to the associated Hitchin spectral curve. We then propose a so-called twisted topological recursion on the spectral curve, after which the corresponding Eynard-Orantin differentials live in a twisted cotangent bundle. This formulation retains, and interacts explicitly with, the singular structure of the original meromorphic setting -- equivalently, the zero divisor of $s$ -- while performing the recursion. Finally, we show that the $g=0$ twisted Eynard-Orantin differentials compute the Taylor expansion of the period matrix of the spectral curve, mirroring a result of Baraglia-Huang for ordinary Higgs bundles and topological recursion. Starting from the spectral curve as a polynomial form in an affine coordinate rather than a Higgs bundle, our result implies that, under certain conditions on $s$, the expansion is independent of the ambient space $\\mbox{Tot}(\\mathcal{L})$ in which the curve is interpreted to reside.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-01-12T16:59:48.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14D20",
      "70H06",
      "14A21"
    ],
    "keywords": [
      "spectral curve",
      "twisted higgs bundle",
      "topological recursion",
      "eynard-orantin differentials",
      "works relating meromorphic higgs bundles"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 53,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}