N on-Semisimple Topological Quantum Field Theories for 3-Manifolds with Corners

Thomas Kerler, Volodymyr V. Lyubashenko9783540424161, 3-540-42416-4

This book presents the (to date) most general approach to combinatorial constructions of topological quantum field theories (TQFTs) in three dimensions. The authors describe extended TQFTs as double functors between two naturally defined double categories: one of topological nature, made of 3-manifolds with corners, the other of algebraic nature, made of linear categories, functors, vector spaces and maps. Atiyah’s conventional notion of TQFTs as well as the notion of modular functor from axiomatic conformal field theory are unified in this concept. A large class of such extended modular catergory is constructed, assigning a double functor to every abelian modular category, which does not have to be semisimple.

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Table of contents :
Lecture Notes in Mathematics……Page 1
Contents……Page 5
0.1 Atiyah’s TQFT Axioms via Categories……Page 7
0.2 Double Categories……Page 9
0.3 Extended TQFT’s……Page 12
0.4.1 Specializations and Generalizations……Page 14
0.4.2 Strategy of Construction and Summary of Content……Page 16
Summary of Content……Page 22
1.1 The 0-1-Arrow Category of Surfaces with Boundaries……Page 24
1.2 2-Arrows from Cobordisms with Corners……Page 29
1.3 Basic Consequences and Generalizations of the Double Category Picture……Page 33
1.4.1 Mapping Class Groups in Cob……Page 41
1.4.2 Framing Extension of Braid Groups, and the Ribbon Element……Page 47
1.4.3 Framed Braid Groups in Cob……Page 53
1.5.1 Handles, Surgery, Isotopies and Cancellation……Page 57
1.5.2 4-dim Handles, 3-dim Surgery, and “Bridged Links”……Page 62
1.5.3 3-dim Handles and Surgery……Page 67
1.6 The Central Extension 0mega_4 $rrow$ tilde{Cob} $rrow$ Cob……Page 74
1.6.1 2-Framings, and Closure of 3-Cobordisms……Page 75
1.6.2 Bounding 4-manifolds and 2-Arrows of tilde{Cob}……Page 79
1.6.3 Compositions in tilde{COB} and tilde{Cob}……Page 82
1.6.4 Double Category Properties of tilde{Cob}……Page 95
Summary of Content……Page 104
2.1.1 Horizontal I-Arrows and Intervals on R {+-1}……Page 105
2.1.3 Local Pictures of Elementary Slices……Page 108
2.2.1 Vertical 1-Arrows, and Squares……Page 110
2.2.2 Types of Strands……Page 111
2.2.3 Conditions for Admissibility……Page 114
2.3 Equivalence Moves of Tangles, and the 2-Arrows in $tau$gl……Page 115
2.3.1 Local Moves relating Isotopies and Projections……Page 116
2.3.2 Local Moves for Coupons and Auxiliary Ribbons……Page 118
2.3.3 Local Moves for Surgery in Interior and at Boundaries……Page 119
2.3.4 Definition of 2-Arrow sets of $tau$gl……Page 121
2.4 Tangles in Three-Space……Page 122
2.4.1 Tangles over R^2……Page 123
2.4.2 Tangles over S^2……Page 141
2.4.3 Removing Auxiliary Tangles……Page 144
2.5 Alternative Calculi and Further Equivalences……Page 149
2.5.1 From Coupons to Bridged Links……Page 150
2.5.2 Kirby and Fenn Rourke Moves……Page 156
2.6.1 Vertical Compositions……Page 159
2.6.2 Horizontal Compositions……Page 162
2.6.3 Double Category Structure of $tau$gl……Page 168
2.7.1 Isolated Strands Category for $beta$(1) = 1……Page 172
2.7.2 IXB-Decomposition and Specializations of Compositions……Page 176
Summary of Content……Page 180
3.1 Trading and Eliminating Handles……Page 181
3.1.1 Handle Decompositions, Connectivity, and Elimination of top Handles……Page 182
3.1.2 Handle Trading and Elimination of 3-Handles……Page 183
3.1.3 4-dim Surgery and $beta$-Moves……Page 187
3.2 Stratified Function Spaces and External Strands on W……Page 193
3.2.1 Natural Stratification of Functions onW……Page 194
3.2.2 Stratification in Presence of External Strands……Page 199
3.2.3 Construction of Cobordisms and Surgery Calculi……Page 202
3.3.1 From Tangles over S^2 to Bridged Links on H……Page 205
3.3.2 The Boundary Move and Factorization into Classes……Page 208
3.3.3 Bijectivity of 2-Arrows Sets……Page 211
3.4 Verification of Compositions……Page 213
4.1 Ribbon monoidal categories……Page 223
4.1.1 Rigid monoidal categories……Page 224
4.1.2 Braided categories……Page 228
4.1.3 Ribbon categories……Page 229
4.2.1 Algebra in a monoidal category……Page 232
4.2.2 Dual Hopf algebras……Page 233
4.2.3 Integrals for Hopf algebras……Page 234
4.2.4 Self-dual Hopf algebras……Page 243
4.3 Abelian categories form a monoidal 2-category……Page 248
4.3.1 Deligne’s tensor product of abelian categories……Page 249
4.3.2 Semistrict monoidal 2-categories……Page 250
4.3.3 A semistrict version of the 2-category of categories of modules……Page 252
4.3.4 Construction of the 2-monoidal structure……Page 255
4.3.5 Braided monoidal 2-structures……Page 263
5.1 The coend……Page 267
5.1.1 General coends……Page 268
5.1.2 A particular coend……Page 269
5.1.3 Coends for bounded categories……Page 272
5.2.1 General properties……Page 276
5.2.2 Braided functions as a Hopf algebra……Page 277
5.2.3 Modular categories……Page 282
5.2.4 Coupon transformation……Page 286
6.1 Main result……Page 289
6.2 Colorations, Natural Transformations, and Liftings……Page 290
6.3 Topological Invariance……Page 296
6.4 Compositions over Colored Surfaces……Page 298
6.5 Lifting V(M) to Color-Independent Natural Transformation……Page 300
6.6 Horizontal Compositions……Page 305
6.7 Topological moves imply the properties of integrals and modularity……Page 310
7.1 Enhanced cobordism categories……Page 319
7.1.2 Enhanced cobordisms……Page 320
7.3 Sketch of the construction of enhanced TQFT……Page 321
7.4.1 Semisimple Abelian Modular Categories……Page 326
7.4.2 Examples of Semisimple Modular Categories……Page 333
7.4.3 Further Related Constructions……Page 334
7.4.5 Ribbon Hopf algebras……Page 335
7.4.6 The braided Hopf algebra F and the coend F……Page 336
7.4.7 The Hennings invariant……Page 338
7.4.8 Quantum Invariants via Cell Decompositions……Page 339
A.1 Witten-Chern-Simons Theory and Conformal Field Theory……Page 341
A.2 Developing the Axiomatics for Extended TQFT’s……Page 344
A.3 Generalized TQFT’s in Gauge Theory……Page 347
B.1 Double Categories……Page 349
B.2 Double pseudofunctors……Page 351
B.2.1 Compositions of double pseudofunctors……Page 353
B.2.2 Functors from double categories to 2-categories……Page 354
B.2.3 Double transformations……Page 357
CA Monoidal bicategory of thick tangles……Page 359
C.2 Representation of thick tangles by abelian categories……Page 371
Bibliography……Page 375
Index……Page 382