Introduction to Conformal Field Theory
This chapter is an introduction to conformal field theory (CFT) which is the basic tool for studying world-sheet properties of string theory. First, we discuss CFT defined on the complex plane, which is relevant for closed strings at tree level. As an app
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Introduction to Conformal Field Theory
Abstract This chapter is an introduction to conformal field theory (CFT) which is the basic tool for studying world-sheet properties of string theory. First, we discuss CFT defined on the complex plane, which is relevant for closed strings at tree level. As an application, we then discuss the CFT of a free boson, i.e. string theory in flat space. Finally, we discuss boundary conformal field theory relevant for describing open strings ending on D-branes and non-oriented strings in orientifolds.
4.1 General Introduction Conformal field theory deals with quantum field theories having conformal symmetry. The conformal group is the subgroup of those general coordinate transformations which preserve the angle between any two vectors. They leave the metric invariant up to a scale transformation. In distinction to higher dimensions the conformal algebra in two dimensions is infinite dimensional, as it is generated by analytic and anti-analytic vector fields. Associated with the infinity of generators is an infinity of conserved charges. That imposes important restrictions on the structure of two-dimensional conformally invariant theories. One class of physical systems which is described by conformal field theory (CFT) are two dimensional statistical systems showing the behavior of a second order phase transition at a critical temperature TC . At this particular point, the system has long range correlations, i.e. it has no particular length scale and is in fact conformally invariant. The representation theory of the conformal algebra places constraints on the critical exponents at TC . A simple such system is the twodimensional Ising model. We will, however, not cover this application of CFT. The second important application of conformal field theory is to string theory. We have seen in Chap. 2 that the string action in conformal gauge is invariant under conformal transformations with the infinite dimensional Virasoro algebra as symmetry algebra. The classical solutions of string theory are conformally invariant two dimensional field theories. A particular choice corresponds to a particular R. Blumenhagen et al., Basic Concepts of String Theory, Theoretical and Mathematical Physics, DOI 10.1007/978-3-642-29497-6 4, © Springer-Verlag Berlin Heidelberg 2013
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4 Introduction to Conformal Field Theory
vacuum which determines e.g. the number of space-time dimensions, the gauge group, etc. There are of course constraints that a conformal field theory has to satisfy in order to be an acceptable string vacuum. One obvious condition that we have already encountered is the vanishing of the conformal anomaly. Others, coming from modular invariance, spin-statistics etc. will be discussed in subsequent chapters. We can then use methods of conformal field theory to determine the string spectrum and to compute string scattering amplitudes. In a more general context, e.g. two-dimensional systems at the critical point, there is no need for the CFT to have c D 0. In fact, there c has a physic
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