The Arena of Conformal Models

Abstract

Abstract In this chapter we will study some significant minimal conformal models. As shown in Chapter 11, these models provide explicit examples of exactly solved quantum field theories: of these theories we know the operator content, the fusion rules of their fields, the corresponding structure constants, the correlation functions of the order parameters and, finally, their modular invariant partition function on a torus. Despite this large amount of knowledge, there is still an important open problem, namely the identification of the classes of universality they are describing. Is there a way to associate these exactly solved critical theories to the continuum limit of lattice statistical models? Unfortunately there is no direct method to answer this question: the identification of the various classes of universality can be achieved only by comparison of the critical exponents predicted by conformal field theory with the values obtained by the exact solution of the models defined on a lattice, further supporting this identification on the basis of the symmetry of the order parameters. This has been the approach followed, for instance, by Huse who identified a particular critical regime of the lattice RSOS models solved by Andrew, Baxter, and Forrester with the unitarity minimal models of conformal field theory. In this chapter, rather than going into a technical analysis of this identification, we prefer to analyze in detail the first minimal models (in the following denoted, in general cases, by p,qand qfor the unitary cases), in particular those corresponding to the Ising model, the tricritical Ising model and the Yang–Lee model. We will also discuss the three-state Potts model as an example of a statistical model associated to a partition function of the type (A, D), according to the notation introduced in Chapter 11. Finally, we will study the statistical models of geometric type (as, for instance, those that describe self-avoiding walks) and their formulation in terms of conformal minimal models.