Topics in Physical Mathematics (eBook)

Kishore Marathe (Autor)

eBook Download: PDF

2010 | 2010
XXII, 442 Seiten
Springer London (Verlag)
978-1-84882-939-8 (ISBN)

Infos

As many readers will know, the 20th century was a time when the fields of mathematics and the sciences were seen as two separate entities. Caused by the rapid growth of the physical sciences and an increasing abstraction in mathematical research, each party, physicists and mathematicians alike, suffered a misconception; not only of the opposition's theoretical underpinning, but of how the two subjects could be intertwined and effectively utilized. One sub-discipline that played a part in the union of the two subjects is Theoretical Physics. Breaking it down further came the fundamental theories, Relativity and Quantum theory, and later on Yang-Mills theory. Other areas to emerge in this area are those derived from the works of Donaldson, Chern-Simons, Floer-Fukaya, and Seiberg-Witten. Aimed at a wide audience, Physical Topics in Mathematics demonstrates how various physical theories have played a crucial role in the developments of Mathematics and in particular, Geometric Topology. Issues are studied in great detail, and the book steadfastly covers the background of both Mathematics and Theoretical Physics in an effort to bring the reader to a deeper understanding of their interaction. Whilst the world of Theoretical Physics and Mathematics is boundless; it is not the intention of this book to cover its enormity. Instead, it seeks to lead the reader through the world of Physical Mathematics; leaving them with a choice of which realm they wish to visit next.

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This title adopts the view that physics is the primary driving force behind a number of developments in mathematics. Previously, science and mathematics were part of natural philosophy and many mathematical theories arose as a result of trying to understand natural phenomena. This situation changed at the beginning of last century as science and mathematics diverged. These two fields are collaborating once again; 'Topics in Mathematical Physics' takes the reader through this journey.The author discusses topics where the interaction of physical and mathematical theories has led to new points of view and new results in mathematics. The area where this is most evident is that of geometric topology of low dimensional manifolds. These include the theories of Donaldson, Chern-Simons, Floer-Fukaya, Seiberg-Witten, and Topological (Quantum) Field Theory.The author also discusses the interaction of CFT, Supersymmetry, String Theory and Gravity with diverse areas of mathematics. Several of these ideas have led to new insights into old mathematical structures and some have led to surprising new results The term "e;Physical Mathematics'' has been coined to describe collectively these new and fast growing areas of research, and regards the work of Donaldson and Witten as belonging to this new area of physical mathematics. Study of this work forms an important part of this book.

Inhalt

Contents 8
Preface 12
Acknowledgements 22
Chapter 1:Algebra 24
1.1 Introduction 24
1.2 Algebras 25
1.2.1 Graded Algebras 31
1.3 Kac--Moody Algebras 33
1.4 Clifford Algebras 37
1.5 Hopf Algebras 43
1.5.1 Quantum Groups 44
1.6 Monstrous Moonshine 44
1.6.1 Finite Simple Groups 46
1.6.2 Modular Groups and Modular Functions 49
1.6.3 The monster and the Moonshine Conjectures 51
Chapter 2:Topology 56
2.1 Introduction 56
2.2 Point Set Topology 58
2.3 Homotopy Groups 61
2.3.1 Bott Periodicity 72
2.4 Singular Homology and Cohomology 73
2.5 de Rham Cohomology 81
2.5.1 The Intersection Form 83
2.6 Topological Manifolds 84
2.6.1 Topology of 2-Manifolds 84
2.6.2 Topology of 3-Manifolds 85
2.6.3 Topology of 4-manifolds 87
2.7 The Hopf Invariant 91
2.7.1 Kervaire invariant 93
Chapter 3:Manifolds 95
3.1 Introduction 95
3.2 Differential Manifolds 96
3.3 Tensors and Differential Forms 104
3.4 Pseudo-Riemannian Manifolds 110
3.5 Symplectic Manifolds 114
3.6 Lie Groups 117
Chapter 4:Bundles and Connections 128
4.1 Introduction 128
4.2 Principal Bundles 129
4.3 Associated Bundles 137
4.4 Connections and Curvature 140
4.4.1 Universal Connections 146
4.5 Covariant Derivative 148
4.6 Linear Connections 151
4.7 Generalized Connections 156
Chapter 5:Characteristic Classes 158
5.1 Introduction 158
5.2 Classifying Spaces 159
5.3 Characteristic Classes 160
5.3.1 Secondary Characteristic Classes 174
5.4 K-theory 178
5.5 Index Theorems 185
Chapter 6:Theory of Fields, I: Classical 189
6.1 Introduction 189
6.2 Physical Background 190
6.3 Gauge Fields 199
6.4 The Space of Gauge Potentials 205
6.5 Gribov Ambiguity 212
6.6 Matter Fields 216
6.7 Gravitational Field Equations 220
6.8 Geometrization Conjecture and Gravity 224
Chapter 7:Theory of Fields, II: Quantum and Topological 227
7.1 Introduction 227
7.2 Non-perturbative Methods 228
7.3 Semiclassical Approximation 236
7.3.1 Zeta Function Regularization 237
7.3.2 Heat Kernel Regularization 238
7.4 Topological Classical Field Theories (TCFTs) 240
7.4.1 Donaldson Invariants 242
7.4.2 Topological Gravity 243
7.4.3 Chern--Simons (CS) Theory 244
7.5 Topological Quantum Field Theories (TQFTs) 245
7.5.1 Atiyah--Segal Axioms for TQFT 252
Chapter 8:Yang--Mills--Higgs Fields 255
8.1 Introduction 255
8.2 Electromagnetic Fields 256
8.2.1 Motion in an Electromagnetic Field 259
8.2.2 The Bohm--Aharonov Effect 262
8.3 Yang--Mills Fields 264
8.4 Non-dual Solutions 272
8.5 Yang--Mills--Higgs Fields 275
8.5.1 Monopoles 277
8.6 Spontaneous Symmetry Breaking 279
8.7 Electroweak Theory 283
8.7.1 The Standard Model 288
8.8 Invariant Connections 291
Chapter 9:4-Manifold Invariants 295
9.1 Introduction 295
9.2 Moduli Spaces of Instantons 296
9.2.1 Atiyah--Jones Conjecture 303
9.3 Topology and Geometry of Moduli Spaces 310
9.3.1 Geometry of Moduli Spaces 314
9.4 Donaldson Polynomials 316
9.4.1 Structure of Polynomial Invariants 321
9.4.2 Relative Invariants and Gluing 323
9.5 Seiberg--Witten Theory 325
9.5.1 Spin Structures and Dirac Operators 326
9.5.2 The Seiberg--Witten (SW) Invariants 327
9.6 Relation between SW and Donaldson Invariants 330
9.6.1 Property P Conjecture 331
Chapter 10:3-Manifold Invariants 333
10.1 Introduction 333
10.2 Witten Complex and Morse Theory 334
10.3 Chern--Simons Theory 339
10.4 Casson Invariant 345
10.5 Floer Homology 346
10.6 Integer-Graded Instanton Homology 353
10.7 WRT Invariants 358
10.7.1 CFT Approach to WRT Invariants 360
10.7.2 WRT Invariants via Quantum Groups 362
10.8 Chern--Simons and String Theory 365
10.8.1 WRT Invariants and String Amplitudes 367
Chapter 11:Knot and Link Invariants 371
11.1 Introduction 371
11.2 Invariants of Knots and Links 372
11.3 TQFT Approach to Knot Invariants 382
11.4 Vassiliev Invariants of Singular Knots 387
11.5 Self-linking Invariants of Knots 388
11.6 Categorification of the Jones Polynomial 391
Epilogue 396
Appendix A:Correlation of Terminology 398
Appendix B:Background Notes 400
Appendix C:Categories and Chain Complexes 411
Appendix D:Operator Theory 420
References 435
Index 451

Produkt-Details

• Erscheint lt. Verlag: 9.8.2010
• Zusatzinfo: XXII, 442 p.
• Verlagsort: London
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
• Themenwelt: Technik
• Schlagworte: classical and quantum field theories • differential geometry and topology • invariants of 3- and 4-dimensional manifolds • manifold • Mathematical Physics • physical mathematics • Ring Theory • theoretical physics
• ISBN-10: 1-84882-939-6 / 1848829396
• ISBN-13: 978-1-84882-939-8 / 9781848829398