Modular Invariance and Two-Dimensional Critical Systems

Abstract

1. Critical systems or massless field theories are highly sensitive to geometric effects. The idea that one can implement a local (as opposed to global) scale invariance follows from the work of Belavin Polyakov and Zamolodchikov coming after numerous studies on finite size scaling. These authors applied the techniques developed in the framework of string theories to the study of two dimensional statistical systems, with a prominent role played by the energy momentum tensor, the generator of coordinate transformations. A natural role is played by the infinitesimal conformal transformations generating an infinite Lie algebra, the Virasoro algebra with a “quantum” anomaly -the so called central charge- for which a rich representation theory exists.