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Quantum Groups and Algebraic Geometry in Conformal Field Theory

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Quantum Groups and Algebraic Geometry in Conformal Field Theory QUANTUM GROUPS AND ALGEBRAIC GEOMETRY IN CONFORMAL FIELD THEORY DlU'KKERU EI.INKWIJK BV - UTRECHT QUANTUM GROUPS AND ALGEBRAIC GEOMETRY IN CONFORMAL FIELD THEORY QUANTUMGROEPEN EN ALGEBRAISCHE MEETKUNDE IN CONFORME VELDENTHEORIE (mrt em samcnrattint] in hit Stdirlands) PROEFSCHRIFT TER VERKRIJGING VAN DE GRAAD VAN DOCTOR AAN DE RIJKSUNIVERSITEIT TE UTRECHT. OP GEZAG VAN DE RECTOR MAGNIFICUS. TROF. DR. J.A. VAN GINKEI., INGEVOLGE HET BESLUIT VAN HET COLLEGE VAN DE- CANEN IN HET OPENBAAR TE VERDEDIGEN OP DINSDAG 19 SEPTEMBER 1989 DES NAMIDDAGS TE 2.30 UUR DOOR Theodericus Johannes Henrichs Smit GEBOREN OP 8 APRIL 1962 TE DEN HAAG PROMOTORES: PROF. DR. B. DE WIT PROF. DR. M. HAZEWINKEL "-*1 Dit proefschrift kwam tot stand met "•••; financiele hulp van de stichting voor Fundamenteel Onderzoek der Materie (F.O.M.) Aan mijn ouders Aan Saskia Contents Introduction and summary 3 1.1 Conformal invariance and the conformal bootstrap 11 1.1.1 Conformal symmetry and correlation functions 11 1.1.2 The conformal bootstrap program 23 1.2 Axiomatic conformal field theory 31 1.3 The emergence of a Hopf algebra 4G The modular geometry of string theory 56 2.1 The partition function on moduli space 06 2.2 Determinant line bundles 63 2.2.1 Complex line bundles and divisors on a Riemann surface . (i3 2.2.2 Cauchy-Riemann operators (iT 2.2.3 Metrical properties of determinants of Cauchy-Ricmann oper- ators 6!) 2.3 The Mumford form on moduli space 77 2.3.1 The Quillen metric on determinant line bundles 77 2.3.2 The Grothendieck-Riemann-Roch theorem and the Mumford form 82 2.3.3 An algebraic formula for the partition function 87 2.3.4 The partition function on compactified moduli space 91 2.4 The Quillen metric as a modular geometry 95 The quantum group structure of W algebras 103 3.1 Introduction 103 3.2 Classical W algebras 106 3.2.1 The bi-Poisson structure for the Virasoro algebra 107 3.2.2 The geometrical construction of W algebras 113 3.2.3 The relation with Toda field theory and the classical Yang- Baxter equation 121 3.3 Quantization of the Poisson brackets of W algebras 124 3.3.1 The Hilbert space completion of a quantum group 125 3.3.2 Quantization of the Poisson brackets for IV algebras 136 il Contents 4 A Hopf algebra structure in orbifold models 147 4.1 Introduction 117 4.2 Orbifold partition functions are Thompson-series 152 4.2.1 Partition functions in genus-one 152 4.2.2 Thompson-series and Heckc operators 1.55 4.2.3 Generalized Thompson-series and orbifold partition functions . lfil -1.3 The Hopf algebra associated with the fusion algebra 168 4.3.1 The Hopf algebra structure of A 169 4.3.2 Outlook 178 4.4 Appendix ISO Bibliography 184 Samenvatting 192 Dankwoord 196 Curriculum Vitae 197 Chapter 1 Introduction and Summary of Results During the past few years, two-dimensional confonnally invariant quantum field theory has become of importance in string theory and statistical mechanics. Aside from its physical applications, this theory revealed interesting connections between mathematics and physics, which lead to a formulation of quantum field theory based on algebraic geometrical principles. Conformal field theory turned out to be relevant in models in two-dimensional statistical mechanics exhibiting a second order phase transition [26]. At such a phase transition the typical configurations have fluctuations on all length scales, so that the field theory describing the model at a critical point is expected to be invariant under scale transformations. In fact, it turns out that the field theory is invariant under a larger set of symmetry transformations which form the so-called conformal group. Unlike in three or higher dimensions the conformal group is infinite dimensional in two dimensions. This fact imposes significant restrictions on two-dimensional conformally invariant theories. Ultimately, one might hope that these restrictions are strong enough to obtain a complete classification of two-dimensional critical systems. Two-dimensional conformally invariant field theories are also central to string theory. This is a theory in which it is assumed that elementary particles are not point-like but rather behave as one dimensional objects, called strings. Strings can propagate in different vibration modes, and interact by joining and splitting. In one of these possible modes the string behaves as a massless spin-two state, so that one concludes that string theory is in fact a theory of gravity. In perturbation theory, the amplitudes of a string theory can be expressed as sums over all possible two-dimensional world-sheets of various topology, swept out by a string in space-time. In a Euclidean formulation the world-sheet is described as a Riemann surface of a given genus. The terms in the perturbation series only depend on the conformal equivalence class of the two-dimensional metric on the Riemann surface. These classes are labelled by the so-called moduli of the surface. Let us make this more precise. The metrics that can be realized on a two dimensional Eurlidean compact surface fall into equivalence classes of the diffeomorphism group Chapter I. Introduction and summary and the group of scale transformations acting on the infinite dimensional space of all metrics. These two groups arc the symmetry groups in string theory, so it seems natural to require that the perturbation series have the same symmetries. This can bo conveniently achieved by a so-called gauge fixing procedure. Ideally, one would like to gauge fix both groups simultaneously. However, this is impossible as there does not exist a measure on the space of metrics that is invariant under the both scale transformations and the diffeomorphism group. One way to deal with this problem is to choose the measure to be invariant under the diffeomorphism group. One then tries to compute the dependence on the conformal factor of the metric, which describes scale transformations. This tan be done easily on a Riemann surface of genus zero, for a given conformal factor, in which case all metrics are equivalent, i.e. they are pure gauge. The condition of scale in variance is then expressed as a differential equation for the conformal factor. For a generic Riemann surface this analysis is much more complicated. One now has to distinguish between metrics that can be identified by an infinitesimal diffeomorphism, and metrics that are only equivalent under a global diffeomorphism which cannot be continuously deformed to the identity. Consequently, one now has to consider the space of equivalence classes of diffeomorphisms, which is called the Teichmuller space. On this space we have the action of the mapping class group. which is formed by the global diffeomorphisms of a Riemann surface. As we will show in detail in chapter two, the condition of scale invariance is equivalent to the requirement that the theory be invariant under the action of the mapping class group. The scale invariant theory is thus defined on the orbit space obtained from the Teichmuller space by dividing out the mapping class group. This orbit space is known as the moduli space of the Riemann surface. For compact Riemann surfaces it is a finite dimensional space. In this way the condition of scale invariance reduces to the problem of finding a real valued measure on the moduli space. We will compute such a measure, which we refer to as the Polyakov measure, using elements of algebraic geometry. Its existence was first proven by Belavin and Kniznik [17]. The formulation of string theory in terms of algebraic geometry allows one to discuss in detail the relation between this theory and conformal field theory. For ex- ample, it follows that the amplitudes of a consistent string theory can be expressed in terms of correlation functions of a conformal field theory. Some conformal field theories describe the propagation of a string on compactified space-time. Others do not have such an obvious space-time interpretation. However, as we will ex- plain later, the property of conformal invariance allows one to interpret arbitrary combinations of conformal field theories as a space-time, so that one concludes that a conformally invariant two-dimensional quantum field theory describes the prop- agation of a string in a static background. The Ward identities of the conformal field theory guarantee the unitarity of the string scattering amplitudes. We thus 9 conclude that it is relevant for both string theory and statistical systems to study the classification of two-dimensional conformal field theories. In this thesis we will study the classification of two-dimensional conformal field theories in terms of algebraic principles. The first attempts to formulate a classifica- tion program for conformal field theories are described in [18,55,52]. As we will sec below the properties of a conformal field theory are conveniently described by means of the representation theory of its symmetry algebra. The most familiar example of such an algebra is the Virasoro Lie algebra. The Hilbert space of a conformal field theory with the Virasoro algebra as its symmetry algebra, can be decomposed into highest weight representations of this algebra. An important property of the symmetry algebra is the existence of so-called degenerate representations which are highest weight representations that contain invariant subspaces, spanned by null- vectors, which vectors othogonal to every other vector inducing themselves. In the physical Hilbert space these subspaces have to be excluded. This amounts to con- sidering the 'irreducible quotient', by simply putting these vectors to zero, which results ,n a smaller representation of the symmetry algebra. (This is similar to the situation in quantum electrodynamics where one uses the Gupta-Bleuler method to construct the physical Hilbcrt space.) All these representations are specified by two parameters.
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