Minimal Models I

Abstract

Chapters 5 and 6 dealt with general properties of two-dimensional conformal field theories. The present chapter is devoted to particularly simple conformal theories called minimal models. These theories are characterized by a Hilbert space made of a finite number of representations of the Virasoro algebra (Verma modules); in other words, the number of conformal families is finite. Such theories describe discrete statistical models (e.g., Ising, Potts, and so on) at their critical points. Their simplicity in principle allows for a complete solution (i.e., an explicit calculation of all the correlation functions). The discovery of minimal models and their identification with known statistical models at criticality constitutes the greatest application of conformal invariance so far. Since a detailed study of minimal models may rapidly become highly technical, we have split the discussion among two chapters (this one and the next). The present chapter first explains some general features of Verma modules (Sect. 7.1), and in particular the occurrence of states of zero norm, which must be quotiented out. In Sect. 7.2 the question of unitarity is discussed and the Kac determinant is introduced. In Sect. 7.3 a survey of the theory of minimal models is presented. In Sect. 7.4 various examples of the correspondence between minimal theories and statistical systems are described. The next chapter will be devoted to more technical issues and will provide proofs for some assertions of the present chapter.