9780821826867

Lectures on Tensor Categories and Modular Functors

by ;
  • ISBN13:

    9780821826867

  • ISBN10:

    0821826867

  • Format: Paperback
  • Copyright: 2001-06-01
  • Publisher: Amer Mathematical Society

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Summary

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.

Table of Contents

Introduction 1(8)
Braided Tensor Categories
9(20)
Monoidal tensor categories
9(6)
Braided tensor categories
15(4)
Quantum groups
19(2)
Drinfeld's category
21(8)
Ribbon Categories
29(18)
Rigid monoidal categories
29(4)
Ribbon categories
33(2)
Graphical calculus for morphisms
35(9)
Semisimple categories
44(3)
Modular Tensor Categories
47(24)
Modular tensor categories
47(12)
Example: Quantum double of a finite group
59(3)
Quantum groups at roots of unity
62(9)
3-Dimensional Topological Quantum Field Theory
71(22)
Invariants of 3-manifolds
71(5)
Topological quantum field theory
76(2)
1+1 dimensional TQFT
78(4)
3D TQFT from MTC
82(5)
Examples
87(6)
Modular Functors
93(48)
Modular functors
93(6)
The Lego game
99(9)
Ribbon categories via the Hom spaces
108(8)
Modular functors in genus zero and tensor categories
116(3)
Modular categories and modular functors for zero central charge
119(5)
Towers of groupoids
124(6)
Central extensions of modular functors
130(5)
From 2D MF to 3D TQFT
135(6)
Moduli Spaces and Complex Modular Functors
141(32)
Moduli spaces and complex Teichmuller tower
141(6)
Compactification of the moduli space and gluing
147(5)
Connections with regular singularities
152(6)
Complex-analytic modular functors
158(3)
Example: Drinfeld's category
161(3)
Twisted D-modules
164(3)
Complex modular functors with central charge
167(6)
Wess-Zumino-Witten Model
173(40)
Preliminaries on affine Lie algebras
174(1)
Reminders from algebraic geometry
175(2)
Conformal blocks: definition
177(4)
Flat connection in the bundle of conformal blocks
181(7)
From local parameters to tangent vectors
188(3)
Families of curves over a formal base
191(3)
Coinvariants for singular curves
194(1)
The gluing axiom for the bundle of coinvariants
195(7)
WZW modular functor
202(1)
Calculation of the central charge
203(10)
Bibliography 213(4)
Index 217(2)
Index of Notation 219

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