CFT a la BPZ

This paper reports that during the past few years, two-dimensional conformal field theory has become a subject of great importance in string theory and statistical mechanics. The study of conformal field theory has also revealed deep new connections between mathematics and physics. A very sophisticated technology has been developed, based on relatively simple notions of two-dimensional field theory. In string theory, two-dimensional conformal field theories are the building blocks of classical string compactifications. Some configurations have very physical interpretations as honest-to-God compactifications of spacetime. Others have no spacetime interpretation at all. Both types, however, describe legal string backgrounds, where the corresponding string scattering amplitudes are given by two-dimensional conformally-invarient correlation functions. In two-dimensional statistical mechanics, conformal field theories appear in a different guise. They arise because the fluctuations of the dynamical variables are scale-invariant at second-order phase transitions. In two dimensions, scale invariance implies conformal invariance., so the interactions of the dynamical variable are described by a two-dimensional conformal field theory. This implies that all correlation functions are conformally invariant as well