**Translations of Mathematical Monographs
Iwanami Series in Modern Mathematics**

2002; 172 pp; Softcover

MSC: Primary 54; 14; Secondary 46; 20

Print ISBN: 978-0-8218-2130-5

Product Code: MMONO/210

Print ISBN: 978-0-8218-2130-5

Product Code: MMONO/210

List Price: $45.00

AMS Member Price: $36.00

MAA Member Price: $40.50

**Electronic ISBN: 978-1-4704-4635-2
Product Code: MMONO/210.E**

List Price: $45.00

AMS Member Price: $36.00

MAA Member Price: $40.50

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# Conformal Field Theory and Topology

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*Toshitake Kohno*

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.

#### Reviews & Endorsements

All together, this book should be regarded as a highly valuable and welcome contribution … The author has succeeded in presenting a treatise on this ultramodern and mathematically utmost challenging subject … particularly captivating by its conciseness, clarity, systematic representation, mathematical rigour, and endeavour to unify various related approaches … an excellent source of knowledge and inspiration for both physicists and mathematicians, researchers and seasoned graduate students … those readers will undoubtedly acknowledge the mastery of this book.

-- Zentralblatt MATH

#### Table of Contents

# Table of Contents

## Conformal Field Theory and Topology

- Cover Cover11
- Title page iii4
- Contents v6
- Preface to the English edition vii8
- Preface ix10
- Introduction 112
- Geometric aspects of conformal field theory 1324
- Jones-Witten theory 7586
- Chern-Simons perturbation theory 127138
- Further developments and prospects 157168
- Bibliography 165176
- Index 169180
- Copying and reprinting 173184
- Back Cover Back Cover1186