Introduction to string theory and conformal field theory
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EMENTARY PARTICLES AND FIELDS Theory
Introduction to String Theory and Conformal Field Theory A. A. Belavin* and G. М. Tarnopolsky** Landau Institute for Theoretical Physics, Russian Academy of Sciences, Chernogolovka, Moscow oblast, 142432 Russia Received June 29, 2009
Abstract—A concise survey of noncritical string theory and two-dimensional conformal field theory is presented. A detailed derivation of a conformal anomaly and the definition and general properties of conformal field theory are given. Minimal string theory, which is a special version of the theory, is considered. Expressions for the string susceptibility and gravitational dimensions are derived. DOI: 10.1134/S1063778810050108
1. INTRODUCTION Relativistic-string theory still attracts much attention of theoretical physicists. The present survey is devoted primarily to noncritical string theory, as well as to conformal field theory and two-dimensional Liouville gravity. This article is organized as follows. Section 2 is devoted to formulating Polyakov’s covariant approach to string theory, calculating a conformal anomaly, and giving a general definition of conformal field theory. In Section 3, the vanishing of the stress–energy tensor in flat space (this is a key property of conformal field theory) is used along with other general principles of quantum field theory to deduce the properties of space of local fields and their correlation functions. This approach to conformal field theory is referred to as a conformal bootstrap, since all properties of the theory are derived by requiring the consistency of a small set of assumptions. The importance of conformal theory is associated predominantly with two factors. First, any version of string theory is a two-dimensional conformal field theory. Second, models of two-dimensional conformal field theory are various universality classes in the sense of Wilson’s renormalization-group philosophy—of the critical behavior of two-dimensional systems in statistical mechanics. In this respect, minimal models of conformal field theory are of particular interest. Some information about such models is also given in Section 3. In Section 4, which is the last one, we revisit noncritical string theory or two-dimensional Liouville gravity. The David–Distler–Kawai formulation, * **
which is alternative to Polyakov’s formulation, is described there. Expressions for the string susceptibility and gravitational dimensions are obtained. That version of two-dimensional Liouville gravity in which conformal matter is described by one of the minimal models is of interest in view of the fact that it provides a field-theory description of universality classes of critical systems on fluctuating two-dimensional surfaces realized in the theory of matrix models. 2. NONCRITICAL STRINGS AND CONFORMAL ANOMALY Constructing a quantum theory for a relativistic string will be the main issue of this section. The concept of a one-dimensional object (string) moving in D-dimensional spacetime is a natural generalization of the concept of a pointlike object (partic
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