Conformal Field Theory and Topology

✍ Scribed by Toshitake Kohno

• Publisher: American Mathematical Society
• Year: 2002
• Tongue: English
• Leaves: 185
• Series: Translations of Mathematical Monographs 210
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

Cover
Conformal Field Theory and Topology
Copyright
2002 by American Mathematical Society
ISBN 082182130x
Contents
Preface to the English Edition
Preface
Introduction
Lagrangians and Hamiltonians in classical mechanics
Complex line bundles and the quantization
Sigma models
Quantization by Feynman's path integral
Wess-Zumino-Witten models and loop groups
Jones-Witten theory
Chern-Simons perturbation theory
Notes
CHAPTER 1: Geometric Aspects of Conformal Field Theory
1.1. Loop groups and affine Lie algebras
1.2. Representations of affine Lie algebras
1.3. Wess-Zumino-Witten model
1.4. The space of conformal blocks and fusion rules
1.5. KZ equation
1.6. Vertex operators and OPE
CHAPTER 2: Jones-Witten Theory
2.1. KZ equation and representations of braid ·groups
2.2. Conformal field theory and the Jones polynomial
2.3. Witten's invariants for 3-manifolds
2.4. Projective representations of mapping class groups
2.5. Chern-Simons theory and connections on surfaces
CHAPTER 3: Chern-Simons Perturbation Theory
3.1. Vassiliev invariants and the Kontsevich integral
3.2. Chern-Simons functionals and the Ray-Singer torsion
3.3. Chern-Simons perturbative invariants
Further Developments and Prospects
Developments in conformal field theory
Combinatorial aspects of finite type invariants
Topological invariants expressed as integrals
Relation to the moduli space of surfaces
Bibliography
Index
Back Cover