Noncommutative geometry is a novel approach which is opening up new possibilities for geometry from a mathematical viewpoint. It is also providing new tools for the investigation of quantum space-time in physics. Recent developments in string theory have supported the idea of quantum spaces, and have strongly stimulated the research in this field. This self-contained volume contains survey lectures and research articles which address these issues and related topics. The book is accessible to both researchers and graduate students beginning to study this subject.
Author(s): Ursula Carow-watamura, Satoshi Watamura, Yoshiaki Maeda, Hitoshi Moriyoshi, Zhangju Liu
Publisher: World Scientific Publishing Company
Year: 2007
Language: English
Pages: 333
Contents......Page 8
Preface......Page 6
I DEFORMATIONS AND NONCOMMUTATIVITY......Page 11
1 Introduction......Page 12
2.1 Fundamental product formulas and intertwiners......Page 13
2.2 The star exponential function et(z+s 1ih uk)*......Page 15
3.1 The star exponential function et(z+1ih uv)*......Page 16
3.2 Several estimates......Page 18
4 Inverses and their analytic continuation......Page 22
4.1 Analytic continuation of inverses......Page 24
5 An in nite product formula......Page 27
5.1 The product with (z+ 1ih uv)-1+* and with (1 + 1m (z + 1ih uv))-1+*......Page 28
6 Star gamma functions......Page 30
6.1 Analytic continuation of *(z + uv hi)......Page 31
6.2 An infinite product formula......Page 32
7 Products with sin (z + 1ih uv)......Page 35
7.1 Additional support for the discrete interpretation......Page 36
7.2 The residue of etv(z + 1ih uv)......Page 37
References......Page 39
2 Quasi-Hamiltonian Quotients as Disjoint Unions of Symplectic Manifolds Florent Scha hauser......Page 40
1.1 Definition......Page 41
1.2 Examples......Page 42
1.3 Properties of quasi-hamiltonian spaces......Page 43
2.1 Symplectic reduction in the usual hamiltonian setting......Page 44
2.2 The smooth case......Page 46
2.3 The stratified case......Page 48
3 Application to representation spaces of surface groups......Page 59
References......Page 63
1 Introduction......Page 64
2.1 Definition and property......Page 66
2.2 Cup product and integration......Page 67
3 A field theory......Page 68
3.1 Action functional......Page 69
3.2 Equation of motion......Page 70
3.3 Analogy of the Polyakov-Wiegmann formula......Page 71
3.5 Toward the quantum theory......Page 72
4.1 Representations of smooth Deligne cohomology......Page 73
4.2 Relationship to Henningson's work......Page 75
4.3 An analogy of the space of conformal blocks......Page 76
References......Page 77
1 Introduction......Page 80
2 Review of elements of Poisson geometry......Page 81
3.1 Lie algebroids......Page 83
3.2 Dirac structures......Page 84
4 Leibniz algebra and Leibniz algebroid......Page 86
4.1 Abelian extension of Leibniz algebra......Page 87
4.2 Super Leibniz algebra and its cochain complex......Page 91
References......Page 105
1 Introduction......Page 106
2 Groupoids......Page 107
3 Spin groupoids and spinor torsors......Page 110
4 Analytic properties......Page 114
5 Concluding remarks......Page 115
References......Page 116
1 Introduction......Page 118
2.1 Lie algebroids and its representations......Page 119
2.2 The localization of 1-cohomology of transitive Lie algebroids......Page 121
3 Kernel of the localization map 1......Page 122
4 Application of the localization theories for principal bundles and their associated bundles......Page 128
5 Transitive Lie Bialgebroids......Page 133
References......Page 136
1 Introduction......Page 138
2 System of second-order PDE's......Page 140
3 Compactification of the at system......Page 143
4 Orbit of diffeomorphisms......Page 144
5 Construction of Diffeomorphisms......Page 147
6 System of third-order PDE's......Page 148
7 Compactification of the at system for 3OPDE......Page 150
8 Orbit of contact diffeomorphisms and Contact Schwarzian Derivatives......Page 151
9 Construction of Contact Diffeomorphisms......Page 153
References......Page 157
1 Introduction......Page 160
2 AKSZ formulation of Batalin-Vilkovisky formalism on Graded Bundles......Page 161
3.1 BF case......Page 164
3.2 Chern-Simons with BF case......Page 166
4 Deformation......Page 167
5.1 n = 2......Page 169
5.2 n = 3......Page 171
6 Quantum Version of Deformation......Page 175
7 Summary and Outlook......Page 177
References......Page 178
II DEFORMED FIELD THEORY AND SOLUTIONS......Page 182
1 Introduction......Page 184
2.1 Solitons in d = 1+2 scalar eld theory......Page 185
2.3 Moyal star product......Page 186
3.1 U*(1) sigma model in d = 0+2s......Page 187
3.2 Grassmannian subsectors......Page 188
3.3 BPS con gurations......Page 189
4.2 Commutative Ward solitons......Page 190
4.3 Co-moving coordinates......Page 191
4.4 Time-space versus space-space deformation......Page 192
4.6 Ward multi-solitons......Page 193
4.8 U*(n) Ward solitons......Page 194
5.1 Manton's paradigm......Page 195
5.2 Ward model metric......Page 196
5.3 Adiabatic two-soliton scattering......Page 197
6.1 Fluctuation Hessian......Page 199
6.2 Diagonal U*(1) soliton: uctuation spectrum......Page 200
6.4 Instability in unitary sigma model......Page 202
7.1 Reduction to d = (1+1) : instantons......Page 203
7.2 d = 1+1 sigma model metric......Page 204
7.4 Reduction to d = (1+1) : solitons......Page 205
7.6 Noncommutative sine-Gordon kinks......Page 206
7.8 Tree-level scattering of elementary quanta......Page 207
References......Page 208
1 Introduction......Page 210
2 Non-linear sigma-models......Page 211
3 Supersymmetric NLSM......Page 212
4 NLSM in extended superspace......Page 215
5 Non-anticommutative deformation of four-dimensional supersymmetric NLSM......Page 217
6 Example: NAC-deformed CP(1) model......Page 224
References......Page 225
2 N.C. Cohomological field theory......Page 228
3 Universality of Partition Functions......Page 230
4 Z of N=4 Super U(1) N.C.Theory......Page 231
5 @N = 2 SUSY Gauge Theory on N.C.R4......Page 232
7 Deformed BRS Operators......Page 236
8 Solutions......Page 237
9 Localization Theorem......Page 243
10 Conclusion......Page 244
References......Page 245
1 Introduction......Page 248
2 Instanton Counting......Page 249
3 Superstring Perspective and Discrete Matrix Model......Page 251
4 Large N Limit......Page 253
5 Topological M-theory and Non-critical M-theory......Page 257
References......Page 260
1 Introduction......Page 262
2 N = 1=2 SYM theory......Page 263
3 Differential algebra in the deformed superspace......Page 264
4 Review of the N = 1 super ADHM construction......Page 265
5 Deformed super ADHM construction......Page 267
References......Page 268
1 Introduction......Page 270
2 Hopf algebra......Page 272
3 Twisted Poincar e symmetry......Page 273
4 Super-Poincar e algebra and Non(anti)commutative Superspace......Page 276
5 Conclusion......Page 279
References......Page 280
1 Introduction......Page 282
2 Notations......Page 284
3 Differential Operators realization of Lie superalgebra gl(2|2)......Page 287
4 Free Field realization of Lie superalgebra gl(2|2)......Page 289
5 Primary Field of the Superalgebra......Page 292
References......Page 296
1 Introduction......Page 298
2 The Beltrami de-Sitter spacetime......Page 299
3 The connections and Yang-Mills equation on BdS spacetime......Page 300
References......Page 303
1 Introduction......Page 306
2 Description of the exact solution......Page 307
References......Page 309
18 Difference Discrete Geometry on Lattice Ke Wu, Wei-Zhong Zhao, Han-Ying Guo......Page 9
1 Introduction......Page 310
2 Difference Discrete Mechanics......Page 313
3 Difference and Differential Form on Lattice......Page 315
4.1 Connection, Gauge Transformations and Holonomy......Page 317
4.2 Difference Discrete Curvature, Bianchi Identity and Abelian Chern Class......Page 324
5.1 Discrete Connection over Randam Lattice......Page 326
5.2 Topological Number in Two Dimension......Page 327
6.1 Lattice Gauge Theory and Difference Discrete Connection......Page 329
6.2 Geometric Meaning of Discrete Lax Pair......Page 330
7 Remarks and Discussions......Page 331
References......Page 332
