  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.
