Topics in Physical Mathematics

by Kishore Marathe

Overview

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.

Product Details

ISBN-13:	9781848829381
Publisher:	Springer London
Publication date:	08/18/2010
Edition description:	2010
Pages:	442
Product dimensions:	6.10(w) x 9.25(h) x 0.04(d)

Table of Contents

Preface xi

Acknowledgements xxi

1 Algebra 1

1.1 Introduction 1

1.2 Algebras 2

1.2.1 Graded Algebras 8

1.3 Kac-Moody Algebras 10

1.4 Clifford Algebras 14

1.5 Hopf Algebras 20

1.5.1 Quantum Groups 21

1.6 Monstrous Moonshine 21

1.6.1 Finite Simple Groups 23

1.6.2 Modular Groups and Modular Functions 26

1.6.3 The monster and the Moonshine Conjectures 28

2 Topology 33

2.1 Introduction 33

2.2 Point Set Topology 35

2.3 Homotopy Groups 38

2.3.1 Bott Periodicity 49

2.4 Singular Homology and Cohomology 50

2.5 de Rham Cohomology 58

2.5.1 The Intersection Form 60

2.6 Topological Manifolds 61

2.6.1 Topology of 2-Manifolds 61

2.6.2 Topology of 3-Manifolds 62

2.6.3 Topology of 4-manifolds 64

2.7 The Hopf Invariant 68

2.7.1 Kervaire invariant 70

3 Manifolds 73

3.1 Introduction 73

3.2 Differential Manifolds 74

3.3 Tensors and Differential Forms 82

3.4 Pseudo-Riemannian Manifolds 88

3.5 Symplectic Manifolds 92

3.6 Lie Groups 95

4 Bundles and Connections 107

4.1 Introduction 107

4.2 Principal Bundles 108

4.3 Associated Bundles 116

4.4 Connections and Curvature 119

4.4.1 Universal Connections 125

4.5 Covariant Derivative 127

4.6 Linear Connections 130

4.7 Generalized Connections 135

5 Characteristic Classes 137

5.1 Introduction 137

5.2 Classifying Spaces 138

5.3 Characteristic Classes 139

5.3.1 Secondary Characteristic Classes 153

5.4 K-theory 157

5.5 Index Theorems 164

6 Theory of Fields, I: Classical 169

6.1 Introduction 169

6.2 Physical Background 170

6.3 Gauge Fields 179

6.4 The Space of Gauge Potentials 185

6.5 Gribov Ambiguity 192

6.6 Matter Fields 196

6.7 Gravitational Field Equations 200

6.8 Geometrization Conjecture and Gravity 204

7 Theory of Fields, II: Quantum and Topological 207

7.1 Introduction 207

7.2 Non-perturbative Methods 208

7.3 Semiclassical Approximation 216

7.3.1 Zeta Function Regularization 217

7.3.2 Heat Kernel Regularization 218

7.4 Topological Classical Field Theories (TCFTs) 220

7.4.1 Donaldson Invariants 222

7.4.2 Topological Gravity 223

7.4.3 Chern-Simons (CS) Theory 224

7.5 Topological Quantum Field Theories (TQFTs) 225

7.5.1 Atiyah-Segal Axioms for TQFT 232

8 Yang-Mills-Higgs Fields 235

8.1 Introduction 235

8.2 Electromagnetic Fields 236

8.2.1 Motion in an Electromagnetic Field 239

8.2.2 The Bohm-Aharonov Effect 242

8.3 Yang-Mills Fields 244

8.4 Non-dual Solutions 252

8.5 Yang-Mills-Higgs Fields 255

8.5.1 Monopoles 257

8.6 Spontaneous Symmetry Breaking 259

8.7 Electroweak Theory 263

8.7.1 The Standard Model 268

8.8 Invariant Connections 271

9 4-Manifold Invariants 275

9.1 Introduction 275

9.2 Moduli Spaces of Instantons 276

9.2.1 Atiyah-Jones Conjecture 283

9.3 Topology and Geometry of Moduli Spaces 290

9.3.1 Geometry of Moduli Spaces 294

9.4 Donaldson Polynomials 296

9.4.1 Structure of Polynomial Invariants 301

9.4.2 Relative Invariants and Gluing 303

9.5 Seiberg-Witten Theory 305

9.5.1 Spin Structures and Dirac Operators 306

9.5.2 The Seiberg-Witten (SW) Invariants 307

9.6 Relation between SW and Donaldson Invariants 310

9.6.1 Property P Conjecture 311

10 3-Manifold Invariants 313

10.1 Introduction 313

10.2 Witten Complex and Morse Theory 314

10.3 Chern-Simons Theory 319

10.4 Casson Invariant 325

10.5 Floer Homology 326

10.6 Integer-Graded Instanton Homology 333

10.7 WRT Invariants 338

10.7.1 CFT Approach to WRT Invariants 340

10.7.2 WRT Invariants via Quantum Groups 342

10.8 Chern-Simons and String Theory 345

10.8.1 WRT Invariants and String Amplitudes 347

11 Knot and Link Invariants 351

11.1 Introduction 351

11.2 Invariants of Knots and Links 352

11.3 TQFT Approach to Knot Invariants 362

11.4 Vassiliev Invariants of Singular Knots 367

11.5 Self-linking Invariants of Knots 368

11.6 Categorification of the Jones Polynomial 371

Epilogue 377

Appendix A Correlation of Terminology 379

Appendix B Background Notes 381

Appendix C Categories and Chain Complexes 393

Appendix D Operator Theory 403

References 419

Index 435

Editorial Reviews

From the reviews:

“Topics in physical mathematics is a quite unique account of some of the mathematical background necessary for a beginner entering the field. … the book offers an introduction and overview of some of the remarkable results that have been obtained in the first three decades of research in ‘physical mathematics’. … will be quite useful as a reference and overview of particular subfields within ‘physical mathematics’. I can also … recommending it to a beginning graduate student wishing to quickly fill some gaps of knowledge.” (Johannes Walcher, Mathematical Reviews, Issue 2011 f)

“Reading this book will bring a real pleasure to all mathematically inclined Young physicists having the necessary mathematical luggage in their minds. … it will bring the reader to feel the real correspondence between the real harmony in the physical world and the remarkable mathematical harmony among the various algebraic, geometric and topological structure.” (Stoil Donev, Journal of Geometry and Symmetry in Physics, Vol. 22, 2011)

“The book covers a large subset of today’s mathematics, especially focusing on stuff that has been very active and in vogue for the last half century or so. … The audience of the book … will consist of graduate students and researchers. … I certainly enjoyed it; all in all, reading this book was a fun … .” (Gizem Karaali, The Mathematical Association of America, August, 2012)

From the Publisher