Lattice Representations of the Virasoro Algebra I : Discrete Gaussian Free Field

Most two-dimensional massless field theories carry represe ntations of the Virasoro algebra as consequences of their conformal symmetry. Recently, conformal symmetry has been rigorously established for scaling limit s of lattice models by means of discrete complex analysis, which efficiently expresses th e 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, some what surprisingly, the algebraic structure of Conformal Field Theory describing the scaling limit of the model is already present on lattice level.