-1- TOPICS IN TWO DIMENSIONAL CONFORMAL FIELD THEORY by Gavin Waterson A thesis presented for the Degree of Doctor of Philosophy of the University of London and the Diploma of Membership of Imperial College. Department of Physics Blackett Laboratory Imperial College London SW7 2BZ. August 1987 -2- To my Parents ABSTRACT Kac-Moody and Virasoro algebras provide the mathematical tools for understanding the structure of two dimensional con­ formal field theory. These algebras are intimately related in that it is possible to construct Virasoro generators from Kac- Moody ones by means of the Sugawara construction. Both alge­ bras also possess various supersymmetric extensions. It is possible to associate vertex operators with a variety of integral lattices. For example, it is known that Kac-Moody generators can be realised from vertex operators corresponding to lattices of points of length squared 2, while fermionic fields can be realised from lattices of points of squared length 1. Here we find another type of vertex operator associated with a lattice of points with square length 3, which provides a realisation of the supercharges for an N=2 super-Virasoro algebra. The second part of this thesis is concerned with the realisation of Kac-Moody and Virasoro algebras in terms of symplectic bosons rather than the well known fermionic con­ structions. Again there exists a Sugawara construction for the Virasoro generators and the conditions for this to equal a Virasoro algebra constructed from free symplectic bosons are provided by a ’superalgebra theorem’. Both fermionic and bosonic constructions can be combined to provide realisations of certain Kac-Moody superalgebras and the corresponding ’ super-Sugawara construction’ can be performed. The condition for this to equal the sum of free fermion and^ free symplectic boson Virasoro generators is provided by a ’supersymmetric space theorem’. -4- These results are then applied to the specific case of the ghost system of fields arising in covariant string theory, It is found that extra derivatives of u(l) Kac-Moody generat­ ors have to be added to the various Sugawara constructions in order to reproduce the string theory results. PREFACE The work presented in this thesis was carried out in the Department of Physics, Imperial College, London between Oct­ ober 1984 and July 1987, under the supervision of Professor D.I. Olive. Unless otherwise stated, the work is original and has not been submitted before for a degree of this or any other University. Chapters 1 to 3 are mainly introductory while Chapter 4 is based on a paper appearing in Phys. Lett. B171 (1986) 77. Chapters 5 and 6 are the result of work done in collaboration with P. Goddard and D. Olive (Imperial preprint TP/86-87/15 to appear in Communications in Mathematical Physics). In Chapters 7 and 8, our formalism is applied to the BRST string ghosts though no new results are obtained. I would like to thank David Olive for providing ever helpful advice and guidance throughout. I am grateful to Peter Goddard for discussions and the Science and Engineering Research Council for financial support. Finally, I would like to thank all my friends and colleagues in the Theory Group at Imperial College. -6- CONTENTS ABSTRACT.............................................. 3 PREFACE............................................... 5 CONTENTS.............................................. 6 CHAPTER 1 - INTRODUCTION 1.1 Motivation.................................. 10 1.2 Classical Conformal Symmetry..................13 1.3 Radial Ordering and Quantum Conformal Symmetry..16 1.4 Conformal Fields...... 18 1.5 The Bosonic String........................... 20 1.6 Layout of Thesis................. 24 CHAPTER 2 - KAC-MOODY AND VIRASORO ALGEBRAS 2.1 Virasoro Algebras............................ 27 2.2 Kac-Moody Algebras........................... 34 2.3 The Sugawara Construction.................... 37 2.4 The Quark Model and Quantum Equivalences.......41 2.5 The N=1 Superconformal Algebra.................46 2.6 Virasoro Algebras with 'u(l) current' terms ....49 CHAPTER 3 - VERTEX OPERATORS 3.1 Basic Construction..................... ......52 3.2 Vertex Operators and Lattices.................55 3.3 Lattices with points of square length 1........59 3.4 Lattices with points of square length 2........61 3.5 Vertex Operators for Virasoro algebras with 1 'u(l) current' terms 65 -7- CHAPTER 4 - VERTEX OPERATORS AND SUPERCONFORMAL GENERATORS 4.1 The N=2 Superconf ormal Algebra................ 69 4.2 Vertex Operators for points of square length 3...71 4.3 Further Realisations of N=2 Superconformal Algebras....... ............................. 77 4.4 Vertex Operators for points of square length 4?..84 CHAPTER 5 - SYMPLECTIC BOSON CONSTRUCTIONS 5.1 Realisations of Kac-Moody Algebras.............88 5.2 Two Virasoro Algebras and a Superalgebra Theorem......................................92 5.3 Examples of the Superalgebra Theorem........... 99 5.4 Critical Representations..................... 102 5.5 Symplectic Fermions..........................104 CHAPTER 6 - SUPERAFFINE ALGEBRAS AND SUPERSYMMETRIC SPACES 6.1 Affine Superalgebras.........................107 6.2 Realisations of Affine Superalgebras.......... 109 6.3 The Super-Sugawara Construction.............. 114 6.4 A Super-Symmetric Space Theorem.............. 119 CD LO • Examples of the Super-Symmetric Space Theorem...124 CHAPTER 7 - BRST METHODS 7.1 Motivation and Applications in String Theory.... 128 7.2 BRST Approach to Quantisation.......... ...... 130 7.3 Application to Virasoro Algebras..............132 7.4 Superconformal Ghosts and Fermionic Strings... 134 7.5 Applications to Affine Algebras...............137 - 8 - CHAPTER 8 - GHOST FIELDS IN COVARIANT STRING THEORY 8.1 Conformal Ghost Field Constructions........... 141 8.2 Superconformal Ghost Field Constructions...... 145 8.3 Algebraic Structure of Combined Ghost System.... 148 CHAPTER 9 - CONCLUSIONS AND OUTLOOK.................... 153 REFERENCES 156 -SI- CHAPTER 1 INTRODUCTION -10- 1.1 Motivation Two dimensional conformal field theory has recently attracted much attention due mainly to the resurgence of int­ erest in string theories, which at present are the most prom­ ising candidates for providing a unified theory of all fund­ amental forces, including gravity (see Scherk,1975 and Schwarz 1982 for reviews). The basic object occurring in these theories is a one dimensional string, which sweeps out a two dimensional worldsheet as it propagates through a higher dimensional spacetime. Conformal properties are guaranteed, as will be seen in the next section, by the existence of a symmetric, traceless energy-momentum tensor, 0a^, (for string theory Qa^ = 0 on shell). The conformal group in more than two dimensions is finite dimensional but, as we shall see, in two dimensions it is infinite dimensional and corresponds to analytic (and anti- analytic) transformations of the complex plane. Thus the powerful and elegant mathematical techniques of complex anal­ ysis become applicable, enabling a very rich mathematical structure to be developed for such two dimensional conformal theories. This leads to a better understanding of string theories as well as simplifying some of the calculations. Since string theory contains quantum gravity, the strings should, in principle, determine the geometry of the background spacetime through which they propagate. This unsolved problem will not be addressed here as we shall assume a Minkowski 1 background, though the techniques we shall be using could be useful in the more general case. The ultimate goal is to -11- elucidate whatever fundamental symmetry underlies string theory. Of course, the formalism we shall be using can be applied to any conformally invariant two dimensional theory and there are several other physically interesting applicat­ ions as well as strings. One notable example is in the case of various two dimensional statistical models, which exhibit conformal invariance at second order phase transitions (see Cardy, 1985 for a review). Conformal invariance determines the various correlation functions of these models more or less uniquely and provides an explanation for the observed critical exponents. Other applications include current algebras and non-linear sigma models with Wess-Zumino terms. The formalism is also of interest to pure mathematicians as, for example, it provides connections between modular forms and the Fischer-Griess Monster Group (see e.g. Frenkel, Lepow- sky and Meurman, 1984 and references therein). It seems remarkable that such apparently disparate areas of mathematics and physics should, in some sense, be related. One common feature of all these applications is the widespread use of vertex operators, which were originally introduced as emission vertices in string theory. It was later realised that these operators are intimately related to the theory of lattices, providing further mathematical structure and insight. Having motivated a study of the algebraic properties of two dimensional conformal theories, we shall discuss their fundamental properties and give an account of the two dimen­ sional conformal algebra in the rest of this chapter. In section (1.5) a brief account of the bosonic string is given, 12- partly as an example of a two dimensional conformal model and partly to establish notation. The layout of the rest of the material is given in section (1.6). -13- 1.2 Classical Conformal Symmetry Conformal field theories possess energy-momentum tensors 0^v , which can be taken to be