Non-Semisimple Topological Quantum Field Theories for 3-Manifolds with Corners

Cover......Page 1
Non-Semisimple Topological Quantum Field Theories for 3-Manifolds with Corners......Page 4
Contents......Page 6
0.1 Atiyah's TQFT Axioms via Categories......Page 8
0.2 Double Categories......Page 10
0.3 Extended TQFT's......Page 13
0.4.1 Specializations and Generalizations......Page 15
0.4.2 Strategy of Construction and Summary of Content......Page 17
1. The Double Category of Framed, Relative 3-Cobordisms......Page 22
Summary of Content......Page 23
1.1 The 0-1-Arrow Category of Surfaces with Boundaries......Page 25
1.2 2-Arrows from Cobordisms with Corners......Page 30
1.3 Basic Consequences and Generalizations of the Double Category Picture......Page 34
1.4.1 Mapping Class Groups in Cob......Page 42
1.4.2 Framing Extension of Braid Groups, and the Ribbon Element......Page 48
1.4.3 Framed Braid Groups in Cob......Page 54
1.5.1 Handles, Surgery, Isotopies and Cancellation......Page 58
1.5.2 4-dim Handles, 3-dim Surgery, and "Bridged Links"......Page 63
1.5.3 3-dim Handles and Surgery......Page 68
1.6 The Central Extension 0mega_4 $\arrow$ \tilde{Cob} $\arrow$ Cob......Page 75
1.6.1 2-Framings, and Closure of 3-Cobordisms......Page 76
1.6.2 Bounding 4-manifolds and 2-Arrows of \tilde{Cob}......Page 80
1.6.3 Compositions in \tilde{COB} and \tilde{Cob}......Page 83
1.6.4 Double Category Properties of \tilde{Cob}......Page 96
2. Tangle-Categories and Presentation of Cobordisms......Page 104
Summary of Content......Page 105
2.1.1 Horizontal I-Arrows and Intervals on R {+-1}......Page 106
2.1.3 Local Pictures of Elementary Slices......Page 109
2.2.1 Vertical 1-Arrows, and Squares......Page 111
2.2.2 Types of Strands......Page 112
2.2.3 Conditions for Admissibility......Page 115
2.3 Equivalence Moves of Tangles, and the 2-Arrows in $\tau$gl......Page 116
2.3.1 Local Moves relating Isotopies and Projections......Page 117
2.3.2 Local Moves for Coupons and Auxiliary Ribbons......Page 119
2.3.3 Local Moves for Surgery in Interior and at Boundaries......Page 120
2.3.4 Definition of 2-Arrow sets of $\tau$gl......Page 122
2.4 Tangles in Three-Space......Page 123
2.4.1 Tangles over R^2......Page 124
2.4.2 Tangles over S^2......Page 142
2.4.3 Removing Auxiliary Tangles......Page 145
2.5 Alternative Calculi and Further Equivalences......Page 150
2.5.1 From Coupons to Bridged Links......Page 151
2.5.2 Kirby and Fenn Rourke Moves......Page 157
2.6.1 Vertical Compositions......Page 160
2.6.2 Horizontal Compositions......Page 163
2.6.3 Double Category Structure of $\tau$gl......Page 169
2.7.1 Isolated Strands Category for $\beta$(1) = 1......Page 173
2.7.2 IXB-Decomposition and Specializations of Compositions......Page 177
3. Isomorphism between Tangle and Cobordism Double Categories......Page 180
Summary of Content......Page 181
3.1 Trading and Eliminating Handles......Page 182
3.1.1 Handle Decompositions, Connectivity, and Elimination of top Handles......Page 183
3.1.2 Handle Trading and Elimination of 3-Handles......Page 184
3.1.3 4-dim Surgery and $\beta$-Moves......Page 188
3.2 Stratified Function Spaces and External Strands on W......Page 194
3.2.1 Natural Stratification of Functions onW......Page 195
3.2.2 Stratification in Presence of External Strands......Page 200
3.2.3 Construction of Cobordisms and Surgery Calculi......Page 203
3.3.1 From Tangles over S^2 to Bridged Links on H......Page 206
3.3.2 The Boundary Move and Factorization into Classes......Page 209
3.3.3 Bijectivity of 2-Arrows Sets......Page 212
3.4 Verification of Compositions......Page 214
4.1 Ribbon monoidal categories......Page 224
4.1.1 Rigid monoidal categories......Page 225
4.1.2 Braided categories......Page 229
4.1.3 Ribbon categories......Page 230
4.2.1 Algebra in a monoidal category......Page 233
4.2.2 Dual Hopf algebras......Page 234
4.2.3 Integrals for Hopf algebras......Page 235
4.2.4 Self-dual Hopf algebras......Page 244
4.3 Abelian categories form a monoidal 2-category......Page 249
4.3.1 Deligne's tensor product of abelian categories......Page 250
4.3.2 Semistrict monoidal 2-categories......Page 251
4.3.3 A semistrict version of the 2-category of categories of modules......Page 253
4.3.4 Construction of the 2-monoidal structure......Page 256
4.3.5 Braided monoidal 2-structures......Page 264
5.1 The coend......Page 268
5.1.1 General coends......Page 269
5.1.2 A particular coend......Page 270
5.1.3 Coends for bounded categories......Page 273
5.2.1 General properties......Page 277
5.2.2 Braided functions as a Hopf algebra......Page 278
5.2.3 Modular categories......Page 283
5.2.4 Coupon transformation......Page 287
6.1 Main result......Page 290
6.2 Colorations, Natural Transformations, and Liftings......Page 291
6.3 Topological Invariance......Page 297
6.4 Compositions over Colored Surfaces......Page 299
6.5 Lifting V(M) to Color-Independent Natural Transformation......Page 301
6.6 Horizontal Compositions......Page 306
6.7 Topological moves imply the properties of integrals and modularity......Page 311
7.1 Enhanced cobordism categories......Page 320
7.1.2 Enhanced cobordisms......Page 321
7.3 Sketch of the construction of enhanced TQFT......Page 322
7.4.1 Semisimple Abelian Modular Categories......Page 327
7.4.2 Examples of Semisimple Modular Categories......Page 334
7.4.3 Further Related Constructions......Page 335
7.4.5 Ribbon Hopf algebras......Page 336
7.4.6 The braided Hopf algebra F and the coend F......Page 337
7.4.7 The Hennings invariant......Page 339
7.4.8 Quantum Invariants via Cell Decompositions......Page 340
A.1 Witten-Chern-Simons Theory and Conformal Field Theory......Page 342
A.2 Developing the Axiomatics for Extended TQFT's......Page 345
A.3 Generalized TQFT's in Gauge Theory......Page 348
B.1 Double Categories......Page 350
B.2 Double pseudofunctors......Page 352
B.2.1 Compositions of double pseudofunctors......Page 354
B.2.2 Functors from double categories to 2-categories......Page 355
B.2.3 Double transformations......Page 358
C.1 Monoidal bicategory of thick tangles......Page 360
C.2 Representation of thick tangles by abelian categories......Page 372
Bibliography......Page 376
Index......Page 383