Homotopy quantum field theory

✍ Scribed by Turaev V.

• Publisher: EMS
• Year: 2010
• Tongue: English
• Leaves: 290
• Series: EMS Tracts in Mathematics
• Category: Library

✦ Synopsis

Homotopy Quantum Field Theory (HQFT) is a branch of Topological Quantum Field Theory founded by E. Witten and M. Atiyah. It applies ideas from theoretical physics to study principal bundles over manifolds and, more generally, homotopy classes of maps from manifolds to a fixed target space. This book is the first systematic exposition of Homotopy Quantum Field Theory. It starts with a formal definition of an HQFT and provides examples of HQFTs in all dimensions. The main body of the text is focused on $2$-dimensional and $3$-dimensional HQFTs. A study of these HQFTs leads to new algebraic objects: crossed Frobenius group-algebras, crossed ribbon group-categories, and Hopf group-coalgebras. These notions and their connections with HQFTs are discussed in detail. The text ends with several appendices including an outline of recent developments and a list of open problems. Three appendices by M. Müger and A. Virelizier summarize their work in this area. The book is addressed to mathematicians, theoretical physicists, and graduate students interested in topological aspects of quantum field theory. The exposition is self-contained and well suited for a one-semester graduate course. Prerequisites include only basics of algebra and topology. A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.

✦ Table of Contents

Contents......Page 7
Introduction......Page 11
I.1 Basic definitions......Page 15
I.2 Cohomological HQFTs and transfer......Page 22
I.3 Aspherical targets......Page 25
I.4 Hermitian and unitary HQFTs......Page 30
I.5 Proof of Lemmas 1.3.1–1.3.3......Page 31
II.1 G-algebras......Page 37
II.2 Inner products and Frobenius G-algebras......Page 38
II.3 Crossed Frobenius G-algebras......Page 39
II.4 Transfer......Page 42
II.5 Semisimple crossed G-algebras......Page 45
II.6 Semisimple crossed Frobenius G-algebras......Page 50
II.7 Hermitian G-algebras......Page 52
III.1 The underlying G-algebra......Page 54
III.2 Computation for cohomological HQFTs......Page 63
III.3 Equivalence of categories......Page 66
III.4 Proof of Theorem 3.1......Page 68
III.5 Hermitian two-dimensional HQFTs......Page 77
IV.1 Frobenius G-algebras re-examined......Page 80
IV.2 Biangular G-algebras......Page 84
IV.3 Lattice HQFTs......Page 88
IV.4 Skeletons of surfaces......Page 95
IV.5 Hermitian biangular G-algebras......Page 99
V.1 Enumeration problem for homomorphisms......Page 102
V.2 Linear representations of Gamma and cohomology......Page 106
V.3 Projective representations of Gamma......Page 110
V.4 Properties of kappa_rho and zeta_rho......Page 115
V.5 Equivalence of two approaches......Page 121
V.6 A generalization and a proof of Theorem 1.2.1......Page 124
V.7 A homological obstruction to lifting......Page 129
V.8 Applications of Theorem 1.2.1......Page 135
V.9 Further applications of Theorem 1.2.1......Page 140
VI.1 G-categories......Page 144
VI.2 Crossed, braided, and ribbon G-categories......Page 147
VI.3 Colored G-tangles and their invariants......Page 154
VI.4 Colored G-graphs and their invariants......Page 163
VI.5 Trace, dimension, and algebra of colors......Page 168
VII.1 Modular crossed G-categories......Page 172
VII.2 Invariants of 3-dimensional G-manifolds......Page 177
VII.3 Homotopy modular functor......Page 184
VII.4 Two-dimensional HQFT......Page 190
VII.5 Three-dimensional HQFT......Page 193
VIII.1 Hopf G-coalgebras......Page 200
VIII.2 Canonical extensions......Page 207
VIII.3 Transfer of categories......Page 210
VIII.4 Quasi-abelian cohomology of groups......Page 213
VIII.5 Remarks on group-algebras......Page 214
Appendix 1. Relative HQFTs......Page 219
Appendix 2. State sum invariants of 3-dimensional G-manifolds......Page 227
Appendix 3. Recent work on HQFTs......Page 231
Appendix 4. Open problems......Page 233
5.1 Braided crossed G-categories......Page 235
5.2 The G-fixed category of a braided crossed G-category......Page 237
5.3 From braided categories containing Rep G to braided G-crossed categories......Page 239
5.4 Classification and coherence for braided crossed G-categories......Page 243
5.5 Braided crossed G-categories as crossed products......Page 244
5.6 Remarks on applications in conformal field theory......Page 247
6.1 Hopf G-coalgebras......Page 250
6.2 Quasitriangular Hopf G-coalgebras......Page 255
6.3 The twisted double construction......Page 259
7.1 Kuperberg-type invariants......Page 267
7.2 Hennings–Kauffman–Radford-type invariants......Page 272
Bibliography......Page 277
Index......Page 287