Informal introduction to Conformal Field Theories in Statistical Mechanics

DANIEL BOYANOVSKY; CARLOS NAÓN

Connections among particle physics, nuclear physics, statistical physics and condensed matter

World Scientific Pub. Co.

Año: 1988; p. 367 - 409

We study some basic aspects of two-dimensional Conformal Field Theory which are currently applied to the análisis of Statistical Mechanics systems at the critical point. In Section 1 we present the fundamental concepts related to the infinite-dimensional conformal algebra in two dimensions. We introduce the primary fields and describe the radial quantization scheme and the basic operator product expansions (OPE). Section 2 contains a detailed discussion of the conformal properties of two-point correlation functions.We remark the special role of the projective transformations and compute the finite size corrections to the free energy and its relation to the conformal anomaly.  This allows us to clarify the transformation properties of the vacuum under general conformal transformations. In Section 3 we present a simple example  where some of the concepts introduced in previous sections are seen at work. Here we introduce the vertex operators and give a physical interpretation of the highest weight states (HWS). Finally, in Section 4, we sketch some of the main issues concerning the representations of the Virasoro Algebra and the minimal series of Friedan, Qiu and Shenker.