  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.
