{
  "id": "1911.03476",
  "version": "v1",
  "published": "2019-11-08T18:57:07.000Z",
  "updated": "2019-11-08T18:57:07.000Z",
  "title": "All-order differential equations for one-loop closed-string integrals and modular graph forms",
  "authors": [
    "Jan E. Gerken",
    "Axel Kleinschmidt",
    "Oliver Schlotterer"
  ],
  "comment": "54+24 pages",
  "categories": [
    "hep-th",
    "math.NT"
  ],
  "abstract": "We investigate generating functions for the integrals over world-sheet tori appearing in closed-string one-loop amplitudes of bosonic, heterotic and type-II theories. These closed-string integrals are shown to obey homogeneous and linear differential equations in the modular parameter of the torus. We spell out the first-order Cauchy-Riemann and second-order Laplace equations for the generating functions for any number of external states. The low-energy expansion of such torus integrals introduces infinite families of non-holomorphic modular forms known as modular graph forms. Our results generate homogeneous first- and second-order differential equations for arbitrary such modular graph forms and can be viewed as a step towards all-order low-energy expansions of closed-string integrals.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-11-08T18:57:07.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "modular graph forms",
      "all-order differential equations",
      "one-loop closed-string integrals",
      "second-order differential equations",
      "non-holomorphic modular forms"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}