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

Clément Hongler
2015
Article

Résumé

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.

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Conformal Field Theory at the Lattice Level: Discrete Complex Analysis and Virasoro Structure

Clément Hongler

Critical statistical mechanics and Conformal Field Theory (CFT) are conjecturally connected since the seminal work of Beliavin et al. (Nucl Phys B 241(2):333-380, 1984). Both exhibit exactly solvable structures in two dimensions. A long-standing question (Itoyama and Thacker in Phys Rev Lett 58:1395-1398, 1987) concerns 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 within lattice models of statistical mechanics. We give a positive answer to this question for the discrete Gaussian free field and for the Ising model, by connecting the structures of discrete complex analysis in 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 Gheissari et al. (Commun Math Phys 367(3):771-833, 2019) 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 (Chelkak et al. in preparation), our construction will complete the picture initiated in Hongler and Smirnov (Acta Math 211:191-225, 2013), Hongler (Conformal invariance of ising model correlations, 2012) and Chelkak et al. (Annals Math 181(3):1087-1138, 2015), where a number of conjectures relating specific Ising lattice fields and CFT correlations were proven.

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This thesis is devoted to the study of the local fields in the Ising model. The scaling limit of the critical Ising model is conjecturally described by Conformal Field Theory. The explicit predictions for the building blocks of the continuum theory (spin and energy density) have been rigorously established [HoSm13, CHI15]. We study how the field-theoretic description of these random fields extends beyond the critical regime of the model. Concretely, the thesis consists of two parts: The first part studies the behaviour of lattice local fields in the critical Ising model. A lattice local field is a function of a finite number of spins at microscopic distances from a given point. We study one-point functions of these fields (in particular, their asymptotics under scaling limit and conformal invariance). Our analysis, based on discrete complex analysis methods, results in explicit computations which are of interest in applications (e.g. [HKV17]). The second part considers the behaviour of the massive spin field. In the subcritical massive scaling limit regime first considered by Wu, McCoy, Tracy, and Barouch [WMTB76], we show that the correlations of the massive spin field in a bounded domain have a scaling limit. Furthermore, to this end we generalise the notions and methods of discrete complex analysis in the critical case to the massive regime, and give a new derivation of the formula for the two-point correlation in the full plane in terms of a Painlevé III transcendent.

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