Fock space of local fields of the discrete GFF and its scaling limit bosonic CFT

Abstract: To connect conformal field theories (CFT) to probabilistic lattice models, recent works [HKV22, Ada23] have introduced a novel definition of local fields of the lattice models. Local fields in this picture are probabilistically concrete: they are built from random variables in the model. The key insight is that discrete complex analysis ideas allow to equip the space of local fields with the main structure of a CFT: a representation of the Virasoro algebra. In this article, for the first time, we fully analyze the structure of the space of local fields of a lattice model as a representation, and use this to establish a one-to-one correspondence between the local fields of a lattice model and those of a conformal field theory. The CFT we consider is probabilistically realized in terms of the gradient of the Gaussian Free Field (GFF). Its space of local fields is just a bosonic Fock space for two chiral symmetry algebras. The corresponding lattice model is the discrete Gaussian Free Field. Our first main result is that the space of local fields of polynomials in the gradient of the discrete GFF is isomorphic to the Fock space. These local fields make sense with both Dirichlet and Neumann boundary conditions. Our second main result is that with the appropriate renormalization, correlation functions of local fields of the discrete GFF converge in the scaling limit to the correlation functions of the CFT. The renormalization needed is, conceptually correctly, according to the eigenvalue of the Virasoro generator $L_0 + \bar{L}_0$ on the local field.