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

TL;DR

This paper establishes a direct link between lattice models in statistical mechanics and Conformal Field Theory by connecting discrete complex analysis with Virasoro algebra representations, advancing the understanding of their continuum limits.

Contribution

It demonstrates the presence of Virasoro algebra structures within lattice models like the Gaussian free field and Ising model, bridging lattice local fields with CFT correlation functions.

Findings

Virasoro structures are identified in lattice models.

Lattice local fields are connected to CFT local fields.

The work advances the correspondence between lattice and continuum theories.

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] 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 [GHP19] 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 [CHI19], 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.