Kalle Kytölä: Conformal field theory on lattice: from discrete complex analysis to Virasoro algebra

Conjecturally, critical statistical mechanics in two dimensions can be described by conformal field theories (CFT). The CFT description has in particular lead to exact and correct (albeit mostly non-rigorous) predictions of critical exponents and scaling limit correlation functions in many models and to detailed predictions about the fractal geometry of their scaling limits. The main ingredient of CFT is the Virasoro algebra, accounting for the effect of infinitesimal conformal transformations on local fields. In this talk we show that an exact Virasoro algebra action exists on the probabilistic local fields of two discrete models: the discrete Gaussian free field and the critical Ising model on the square lattice. This is quite surprising, since the models on lattices have no conformal symmetry. An explanation lies in the exact solvability of the lattice models formulated in terms of discrete complex analysis.
The talk is based on joint work [arXiv:1307.4104] with Clément Hongler (EPFL, Lausanne) and Fredrik Viklund (KTH, Stockholm).