**University Lecture Series**

Volume: 21;
2001;
221 pp;
Softcover

MSC: Primary 18; 81; 57;
Secondary 17

Print ISBN: 978-0-8218-2686-7

Product Code: ULECT/21

List Price: $36.00

AMS Member Price: $28.80

MAA Member Price: $32.40

**Electronic ISBN: 978-1-4704-2168-7
Product Code: ULECT/21.E**

List Price: $36.00

AMS Member Price: $28.80

MAA Member Price: $32.40

# Lectures on Tensor Categories and Modular Functors

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*Bojko Bakalov; Alexander Kirillov, Jr.*

This book gives an exposition of the relations among the following
three topics: monoidal tensor categories (such as a category of representations
of a quantum group), 3-dimensional topological quantum field theory, and
2-dimensional modular functors (which naturally arise in 2-dimensional
conformal field theory). The following examples are discussed in detail: the
category of representations of a quantum group at a root of unity and the
Wess-Zumino-Witten modular functor.

The idea that these topics are related first appeared in the physics literature in
the study of quantum field theory. Pioneering works of Witten and Moore-Seiberg
triggered an avalanche of papers, both physical and mathematical, exploring
various aspects of these relations. Upon preparing to lecture on the topic at
MIT, however, the authors discovered that the existing literature was difficult
and that there were gaps to fill.

The text is wholly expository and finely succinct. It gathers results, fills
existing gaps, and simplifies some proofs. The book makes an important addition
to the existing literature on the topic. It would be suitable as a course text
at the advanced-graduate level.

#### Readership

Graduate students and research mathematicians interested in representation theory and mathematical physics

#### Table of Contents

# Table of Contents

## Lectures on Tensor Categories and Modular Functors

- Cover Cover11 free
- Title v6 free
- Copyright vi7 free
- Contents viii9 free
- Introduction 111 free
- CHAPTER 1 Braided Tensor Categories 919 free
- CHAPTER 2 Ribbon Categories 2939
- CHAPTER 3 Modular Tensor Categories 4757
- CHAPTER 4 3-Dimensional Topological Quantum Field Theory 7181
- CHAPTER 5 Modular Functors 93103
- 5.1. Modular functors 93103
- 5.2. The Lego game 99109
- 5.3. Ribbon categories via the Horn spaces 108118
- 5.4. Modular functors in genus zero and tensor categories 116126
- 5.5. Modular categories and modular functors for zero central charge 119129
- 5.6. Towers of groupoids 124134
- 5.7. Central extensions of modular functors 130140
- 5.8. From 2D MF to 3D TQFT 135145

- CHAPTER 6 Moduli Spaces and Complex Modular Functors 141151
- 6.1. Moduli spaces and complex Teichmiiller tower 141151
- 6.2. Compactification of the moduli space and gluing 147157
- 6.3. Connections with regular singularities 152162
- 6.4. Complex-analytic modular functors 158168
- 6.5. Example: Drinfeld's category 161171
- 6.6. Twisted D-modules 164174
- 6.7. Complex modular functors with central charge 167177

- CHAPTER 7 Wess-Zumino—Witten Model 173183
- 7.1. Preliminaries on affine Lie algebras 174184
- 7.2. Reminders from algebraic geometry 175185
- 7.3. Conformal blocks: definition 177187
- 7.4. Flat connection in the bundle of conformal blocks 181191
- 7.5. From local parameters to tangent vectors 188198
- 7.6. Families of curves over a formal base 191201
- 7.7. Coinvariants for singular curves 194204
- 7.8. The gluing axiom for the bundle of coinvariants 195205
- 7.9. WZW modular functor 202212
- 7.10. Calculation of the central charge 203213

- Bibliography 213223
- Index 217227
- Index of Notation 219229
- Back Cover Back Cover1232