Abstract: To connect probabilistic lattice models to conformal field theories (CFT), recent works have introduced a novel definition of local fields of the lattice models. In this picture, local fields are probabilistically concrete: they are built from random variables of the model. The key insight is that discrete complex analysis ideas allow to equip the space of local fields with the main algebraic structure of a CFT: a representation of the Virasoro algebra.

In this talk, we will illustrate this approach to CFT with a concrete model: the discrete gaussian free field (DGFF). We construct the space of local fields of polynomials in the gradient of the DGFF and, as a first main result, we prove that it is isomorphic to the Fock space of the free boson CFT discussed in the previous talk by Kalle Kytölä. This constitutes the first instance of a one-to-one correspondence between the local fields of a lattice model and those of a CFT. Our second main result concerns the scaling limits of correlation functions. We show that, when correlation functions of local fields of the DGFF are renormalized properly —in a way dictated by the algebraic structure—, they converge to the correlation functions of the free boson CFT.