**Fields Institute Monographs**

Volume: 24;
2008;
168 pp;
Hardcover

MSC: Primary 81; 14; 17;

Print ISBN: 978-0-8218-4088-7

Product Code: FIM/24

List Price: $65.00

Individual Member Price: $52.00

**Electronic ISBN: 978-1-4704-3150-1
Product Code: FIM/24.E**

List Price: $65.00

Individual Member Price: $52.00

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#### Supplemental Materials

# Conformal Field Theory with Gauge Symmetry

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*Kenji Ueno*

A co-publication of the AMS and Fields Institute

This book presents a systematic approach to conformal field theory
with gauge symmetry from the point of view of complex algebraic
geometry. After presenting the basic facts of the theory of compact
Riemann surfaces and the representation theory of affine Lie algebras
in Chapters 1 and 2, conformal blocks for pointed Riemann surfaces
with coordinates are constructed in Chapter 3. In Chapter 4 the sheaf
of conformal blocks associated to a family of pointed Riemann surfaces
with coordinates is constructed, and in Chapter 5 it is shown that
this sheaf supports a projective flat connection—one of the most
important facts of conformal field theory. Chapter 6 is devoted to the
study of the detailed structure of the conformal field theory over
\(\mathbb{P}^1\).

Recently it was shown that modular functors can be constructed from
conformal field theory, giving an interesting relationship between algebraic
geometry and topological quantum field theory. This book provides a timely
introduction to an intensively studied topic of conformal field theory with
gauge symmetry by a leading algebraic geometer, and includes all the
necessary techniques and results that are used to construct the modular
functor.

Titles in this series are co-published with The Fields Institute for Research in Mathematical Sciences (Toronto, Ontario, Canada).

#### Table of Contents

# Table of Contents

## Conformal Field Theory with Gauge Symmetry

- Cover Cover11
- Title page i2
- Contents iii4
- Preface v6
- Notation vii8
- Riemann surfaces and stable curves 110
- Affine Lie algebras and integrable highest weight representations 3746
- Conformal blocks and correlation functions 4756
- Sheaf of conformal blocks 7382
- Projectively flat connections 91100
- Vertex operators and KZ equations 131140
- Appendix 161170
- Bibliography 163172
- Index 167176
- Back Cover Back Cover1178

#### Readership

Graduate students and research mathematicians interested in algebraic/arithmetic geometry, theoretical physics (high energy) string theory.