Speaker
Kalle Kytola, Aalto
Abstract
Since 1980’s, physicists have known that scaling limits of probabilistic lattice models at criticality should be described by conformal field theories (CFT). Since 2000’s, mathematicians have indeed proved a number of notable conformal invariance results for scaling limits of such lattice models — but still (arguably) without obtaining a fully fledged CFT as the scaling limit.
The most fundamental data of a CFT is its space of local fields: first because this coincides with the state space of the theory (by the state-field correspondence) and therefore specifies the spectrum of the theory (the eigenvalues of the energy operator), and second because correlation functions of the local fields are the observable quantities of the theory. A natural analogue of local fields in probabilistic lattice models is random variables built locally from the basic degrees of freedom of the model. In certain models, discrete complex analysis can be used to equip spaces of lattice model local fields with the main algebraic structure of CFTs: a representation of the Virasoro algebra. This opens up the possibility of full and structured correspondence between the space of local fields of a CFT and of its lattice discretization. In this talk we address this for the model of discrete Gaussian Free Field (dGFF). We show that the space of local fields of the gradient of the dGFF is isomorphic to a Fock space of a free bosonic CFT. Moreover, we show that scaling limits of correlations of the dGFF local fields converge in the scaling limit to correlation functions of that CFT, when appropriately renormalized according to the eigenvalues of the Virasoro generators corresponding to the energy in CFT.
The talk is based on joint works with Clément Hongler and Fredrik Viklund