We discuss two results about the Ising model, which capture the main algebraic structure in CFTs. The first concerns bulk CFT, with independent holomorphic and antiholomorphic chiral algebras, while the second concerns boundary CFT, with just one chiral algebra suitably coupling the holomorphic with the antiholomorphic. By the first result, bulk local fields of the Ising model can be defined as probabilistic objects (locally defined random variables modulo equivalence in correlations) on the lattice level, and they carry exact Virasoro algebra representations of central charge c=1/2 for both chiralities. By the second result, three point boundary correlation functions for the Ising model with locally monochromatic boundary conditions in a slit-strip geometry have scaling limits which allow to recover all structure constants of the vertex operator algebra (VOA) of the boundary CFT. The proofs of both results rely on discrete holomorphicity techniques, and in particularly on finding suitable families of discrete holomorphic functions that take the role of monomials in the algebraic axiomatization of CFTs which builds on formal series.
The talk is based on joint works with Taha Ameen (Urbana-Champaign), Clément Hongler (EPFL, Lausanne), Shinji Koshida (Aalto), SC Park (KIAS, Seoul), David Radnell (Aalto), Fredrik Viklund (KTH, Stockholm).