Conformal Field Theory

Abstract

Abstract From a lattice model one may obtain a field theory by taking a continuum or scaling limit; letting the lattice spacing ε → 0, whilst simultaneously approaching the critical temperature. As the scale or correlation length becomes infinite one obtains a scale invariant or conformal invariant theory. (Belavin et al.1980) suggested that the scale invariance at a critical point is enhanced to conformal invariance or using conformal invariance to understand scale invariance of special critical points. Conformal invariance is understood via representations of the Virasoro algebra, and a knowledge of the representations that appear tells us something about the statistical mechanical model, and the nature of the critical point. In this chapter we look at some of the basic principles of the theory, particularly with respect to those aspects of most interest to the operator algebraic approach to statistical mechanics and quantum field theory. We look in detail at the Ising model where many, but of course not all, the features are present and can be analysed explicitly. This means that we also emphasize the viewpoint of conformal field theory from critical statistical mechanical models, although there are other important aspects to the theory, particularly because conformal field theories describe the ground states of string theory.