Conformal Field Theory and Topology

One of the most remarkable interactions between geometry and physics since 1980 has been an application of quantum field theory to topology and differential geometry. An essential difficulty in quantum field theory comes from infinite-dimensional freedom of a system. Techniques dealing with such infinite-dimensional objects developed in the framework of quantum field theory have been influential in geometry as well. This book focuses on the relationship between two-dimensional quantum field theory and three-dimensional topology which has been studied intensively since the discovery of the Jones polynomial in the middle of the 1980s and Witten's invariant for 3-manifolds which was derived from Chern-Simons gauge theory. This book gives an accessible treatment for a rigorous construction of topological invariants originally defined as partition functions of fields on manifolds. The book is organized as follows: The Introduction starts from classical mechanics and explains basic background materials in quantum field theory and geometry. Chapter 1 presents conformal field theory based on the geometry of loop groups. Chapter 2 deals with the holonomy of conformal field theory. Chapter 3 treats Chern-Simons perturbation theory. The final chapter discusses topological invariants for 3-manifolds derived from Chern-Simons perturbation theory. Readership: Graduate students and research mathematicians interested in topology and algebraic geometry. Cover 1 Conformal Field Theory and Topology 3 Copyright 4 2002 by American Mathematical Society 4 ISBN 082182130x 4 Contents 6 Preface to the English Edition 8 Preface 10 Introduction 12 Lagrangians and Hamiltonians in classical mechanics 12 Complex line bundles and the quantization 14 Sigma models 18 Quantization by Feynman's path integral 19 Wess-Zumino-Witten models and loop groups 20 Jones-Witten theory 21 Chern-Simons perturbation theory 23 Notes 23 CHAPTER 1: Geometric Aspects of Conformal Field Theory 24 1.1. Loop groups and affine Lie algebras 25 1.2. Representations of affine Lie algebras 34 1.3. Wess-Zumino-Witten model 40 1.4. The space of conformal blocks and fusion rules 53 1.5. KZ equation 63 1.6. Vertex operators and OPE 71 CHAPTER 2: Jones-Witten Theory 86 2.1. KZ equation and representations of braid ·groups 86 2.2. Conformal field theory and the Jones polynomial 101 2.3. Witten's invariants for 3-manifolds 114 2.4. Projective representations of mapping class groups 121 2.5. Chern-Simons theory and connections on surfaces 132 CHAPTER 3: Chern-Simons Perturbation Theory 138 3.1. Vassiliev invariants and the Kontsevich integral 138 3.2. Chern-Simons functionals and the Ray-Singer torsion 153 3.3. Chern-Simons perturbative invariants 159 Further Developments and Prospects 168 Developments in conformal field theory 168 Combinatorial aspects of finite type invariants 169 Topological invariants expressed as integrals 170 Relation to the moduli space of surfaces 173 Bibliography 176 Index 180 Back Cover 185 The aim of this book is to provide the reader with an introduction to conformal field theory and its applications to topology. The author starts with a description of geometric aspects of conformal field theory based on loop groups. By means of the holonomy of conformal field theory he defines topological invariants for knots and 3-manifolds. He also gives a brief treatment of Chern-Simons perturbation theory. Toshitake Kohno ; [translated From The Japanese By The Author]. Includes Bibliographical References (p. 165-167) And Index.