{
  "id": "1908.09677",
  "version": "v1",
  "published": "2019-08-26T13:37:21.000Z",
  "updated": "2019-08-26T13:37:21.000Z",
  "title": "An analytic version of the Langlands correspondence for complex curves",
  "authors": [
    "Pavel Etingof",
    "Edward Frenkel",
    "David Kazhdan"
  ],
  "comment": "69 pages",
  "categories": [
    "math.AG",
    "hep-th",
    "math.AP",
    "math.FA",
    "math.RT"
  ],
  "abstract": "The geometric Langlands correspondence for complex curves is traditionally formulated in terms of sheaves rather than functions. Recently, Langlands asked whether it is possible to construct a function-theoretic version. In this paper we use the algebra of commuting global differential operators (quantum Hitchin Hamiltonians and their complex conjugates) on the moduli space of G-bundles of a complex algebraic curve to develop such a theory. We conjecture a canonical self-adjoint extension of the symmetric part of this algebra acting on an appropriate Hilbert space and link its spectrum with the set of opers for the Langlands dual group of G satisfying a certain reality condition, as predicted earlier by Teschner. We prove this conjecture for G=GL(1) and in the simplest non-abelian case.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-08-26T13:37:21.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "complex curves",
      "analytic version",
      "langlands dual group",
      "commuting global differential operators",
      "appropriate hilbert space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 69,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}