{
  "id": "1307.4104",
  "version": "v2",
  "published": "2013-07-15T21:07:24.000Z",
  "updated": "2017-03-27T13:12:19.000Z",
  "title": "Conformal Field Theory at the Lattice Level: Discrete Complex Analysis and Virasoro Structure",
  "authors": [
    "Clément Hongler",
    "Fredrik Johansson Viklund",
    "Kalle Kytölä"
  ],
  "comment": "48 pages, 14 figures",
  "categories": [
    "math-ph",
    "math.MP",
    "math.PR"
  ],
  "abstract": "Critical statistical mechanics and Conformal Field Theory (CFT) are conjecturally connected since the seminal work of Beliavin, Polyakov, and Zamolodchikov [BPZ84a]. Both exhibit exactly solvable structures in two dimensions. A long-standing question [ItTh87] asks whether there is a direct link between these structures, that is, whether the Virasoro algebra representations of CFT, the distinctive feature of CFT in two dimensions, can be found at the lattice level. We give a positive answer to this question for the discrete Gaussian free field and for the Ising model, by connecting the discrete complex analysis structures of the lattice models with the Virasoro symmetry that is expected to describe their scaling limits. This allows for a tight connection of a number of objects from the lattice model world and the field theory one. In particular, our results link the CFT local fields with lattice local fields introduced in [GHP13] and the probabilistic formulation of the lattice model with the continuum correlation functions. Our construction is a decisive step towards establishing the conjectured correspondence between the correlation functions of the CFT fields and those of the lattice local fields. In particular, together with the upcoming [CHI17], our construction will complete the picture initiated in [HoSm13, Hon12, CHI15], where a number of conjectures relating specific Ising lattice fields and CFT correlations were proven.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-07-15T21:07:24.000Z",
      "title": "Lattice Representations of the Virasoro Algebra I: Discrete Gaussian Free Field",
      "abstract": "Most two-dimensional massless field theories carry representations of the Virasoro algebra as consequences of their conformal symmetry. Recently, conformal symmetry has been rigorously established for scaling limits of lattice models by means of discrete complex analysis, which efficiently expresses the integrability of these models. In this paper we study the discrete Gaussian free field on the square grid. We show that the lattice integrability of this model gives explicit representations of the Virasoro algebra acting on the Gibbs measures of the model. Thus, somewhat surprisingly, the algebraic structure of Conformal Field Theory describing the scaling limit of the model is already present on lattice level.",
      "comment": "26 pages, 3 figures",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2017-03-27T13:12:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "17B68",
      "30G25",
      "82B20",
      "82B23",
      "60G60",
      "60G15"
    ],
    "keywords": [
      "discrete gaussian free field",
      "virasoro algebra",
      "field theory",
      "lattice representations",
      "field theories carry representations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 48,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1307.4104H"
    }
  }
}