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Some Aspects of Conformal Field Theories on the Plane and Higher Genus Riemann Surfaces

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Some Aspects of Conformal Field Theories on the Plane and Higher Genus Riemann Surfaces Pramg.na - J. Phys., Vol. 35, No. 3, September 1990, pp. 205-286. © Printed in India. Some aspects of conformal field theories on the plane and higher genus Riemann surfaces ASHOKE SEN Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India MS received 6 June 1990 Al~traet. We review some aspects of conformal field theories on the plane as well as on higher genus Riemann surfaces. Keywords. Conformal field theory; Riemann surfaces. PACS No. 11.10 1. Introduction Conformally invariant two dimensional field theories (CFT) have become the subject of intense investigation in recent years. In this article I shall try to give a general introduction' to conformal field theory with special emphasis on a particular class of conformal field theories, known as rational conformal field theories (RCFT). I shall begin by discussing the reasons for the recent upsurge of interest in these theories, and then discuss the various properties of these theories in some detail. One of the two main applications of two dimensional conformal field theories is that they describe the critical behavior of many known two dimensional statistical mechanical models. In order to understand this connection we must first understand the meaning of conformal invariance. In any dimension, conformal invariance refers to a group of coordinate transformations which leave the angle between any two intersecting lines fixed. Obviously the Poincare group, consisting of translations and rotations have this property, and hence they form part of the conformal group. Another transformation which has this property is the scale transformation. In general the conformal group has other elements also which we shall discuss later, but for the purpose of understanding the connection to the critical behavior of statistical models, the above properties are enough. In order to understand the behavior of a statistical model at the critical point, let us consider a simple system, the Ising model, which consists of a lattice with classical spin placed at each site, which can point in either the up or the down direction. At very high temperature, the system is completely disordered in the absence of a magnetic field. This may be expressed by saying that the conditional probability that the spin at site i is up given that the spin at the site j is up reduces to the unconditional probability (1/2) as the distance between the sites increases. In fact this approach takes place in an exponential manner, if the distance between the sites is labelled by !, then the above conditional probability is of the form 1/2 + (9 exp (- Ill), where 205 206 Ashoke Sen is a length parameter known as the correlation length. (Mathematically this is measured by the expectation value of the product of the spins at the sites i and j, known as the correlation function. It is the correlation function that decays exponentially with distance.) On the other hand at very low temperature this conditional probability approaches a value larger than 1/2 as l---, ~, which is reflected in the fact that the two spin correlation function approaches a finite non-zero value as I--, ~. In particular, at zero temperature the conditional probability approaches the value 1, if the spin at the site i is up, then the spin at the site j is also up, however large is the distance between the two sites. These two different phases of the system are separated by a special point on the temperature axis, known as the critical temperature T c. It is clear that in order that these two behaviors match at To, the correlation length must become infinite as we approach Tc from T > T¢, and the asymptotic value of the correlation function must approach zero as we approach Tc from T < To. In fact, one finds that at T = Tc the correlation function falls off with distance according to power law rather than exponentially. Now for T > T¢, ~ sets the length scale of the problem, and the behavior of the system is quite different at length scales smaller than ~ and larger than ~. Thus the system is not scale invariant. On the other hand, since at T= T~, ~ becomes infinite, there is no length parameter which sets the scale, and the system looks the same at all scales provided we consider distances large compared to the lattice spacing. The limit where we look at distances large compared to the lattice spacing is known as the continuum limit, and in this limit the system is described by a scale invariant two dimensional field theory. Furthermore, in two dimensions scale invariant theories are often also conformally invariant, and hence, for a wide class of statistical models, the behavior of the system at the critical point is described by conformally invariant two dimensional field theories. Statistical models at phase transition are often characterised by what is known as critical exponents. They describe the behavior of various quantities which become singular at the critical point. For example, if we consider the lsing model at zero magnetic field and below the critical temperature, then the spontaneous magnetisation M goes to zero as T~ T~ according to the relation M ,-~ (T~- T)~, where ~ is some fixed number and gives one of the critical exponents. Similarly one can define many other critical exponents. Conformal field theory provides us with a simple and powerful means of calculating the critical exponents as well as all the correlation functions of the theory at the critical point. Another reason conformal field theories have become interesting in recent years, is because of their application to string theory. Originally string theory was formulated in fiat twentysix dimensional space-time for bosonic string theory, and flat ten dimensional space-time for fermionic string theory. This formulation was based on a two dimensional field theory of twentysix free scalar fields (or ten free scalar fields and ten majorana spin 1/2 fields). It has now been realised that one can also formulate string theory by replacing the free field theory by a two dimensional ¢onformally invariant field theory with central charge 26 (or a two dimensional supersymmetric conformal field theory with central charge 15). If we take this conformal field theory to be a direct sum of a free field theory of d scalar fields and a conformal field theory of central charge 26 - d, it may be interpreted as a string theory in d dimensions with the extra dimensions compactified. Thus two dimensional conformal field theories provide a way of constructing string theories in less than Conformal field theories on higher genus Riemann surfaces 207 twentysix dimensions. Furthermore, one can show that string theories based on two dimensional conformal field theories other than free field theories correspond to non-trivial classical solutions of the string theory formulated in fiat twentysix dimensional background. Thus one expects that the study of conformal field theories will provide new insight into solutions of the string equations of motion, and hence the dynamics of string theory. One can also show that tree level string amplitudes may be expressed in terms of various correlation functions in the corresponding conformal field theory on the plane, whereas string loop amplitudes may be expressed in terms of correlation functions in this conformal field theory on higher genus Riemann surfaces. With this brief introduction to the motivation for the study of conformal field theories, we shall now discuss various properties of CFT's, both on the plane, as well as on higher genus Riemann surfaces. 2. Review of conformal field theory on the plane 2.1 Representations of the Virasoro algebra In the heart of the study of all conformal field theories in two dimensions is the infinite dimensional conformal algebra, or the Virasoro algebra, generated by the set of generators L,,, Lm(- go < m < + c~), which generate transformations of the form z~z+emz "+I, and ~--*~+g,.z"'+l respectively, where z, :~ denote the complex coordinates of the two dimensional surface (Belavin et al 1984)*. The algebra is of the form: I- Lm, L.] = (m -- n) Lm +. + ]~(mc 3 - m) g,. +.,o c 3 EL.,, L.] = (m -- n) Lr~+. + ~(m -- m)6m+..o (2.1) ELm, L.] =0 where c is a c-number known as the central charge of the Virasoro algebra. Since any two dimensional conformal field theory, by definition, has the conformal algebra as a (sub-)algebra of the full symmetry algebra, the first attempt towards systematic study of conformally invariant theories in two dimensions was made by classifying all possible representations of the Virasoro algebra. In this paper I shall restrict my attention to a special class of representations of the conformal algebra, namely the unitary representations. This means that we demand all the states in the representation to have positive norm. Although some critical phenomena may be described by conformal field theories with non-unitary representations, only unitary representations are allowed in conformal field theories which provide classical ground states of the string theory. Also, since the algebra generated by L. and Lm commute with each * In the analysis of conformal field theory it turns out to be more convenient to treat z and ~ to be independent variables for most part of the analysis, and only near the end set them to be complex conjugates of each other. This will become clear during the later part of the paper. 208 Ashoke Sen other, we shall concentrate on classification of the representations of only one of the algebras (generated by Ln's, say); possible representations of the algebra generated by Ln will be identical. The states of the system will form representations of both the algebras. The study of representations of the Virasoro algebra starts in the same way as the study of representations of an ordinary Lie algebra, say an SU(2) algebra.
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