Marathe K. Topics in Physical Mathematics
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Topics in Physical Mathematics
Kishore Marathe
Topics in Physical
Mathematics
ABC
Prof. Kishore Marathe
City University of New York
Brooklyn College
Bedford Avenue 2900
11210-2889 Brooklyn
New York
USA
KMarathe@brooklyn.cuny.edu
Whilst we have made considerable efforts to contact all holders of copyright material
contained in this book, we have failed to locate some of them. Should holders wish to contact
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ISBN 978-1-84882-938-1 e-ISBN 978-1-84882-939-8
DOI 10.1007/978-1-84882-939-8
Springer London Dordrecht Heidelberg New York
British Library Cataloguing in Publication Data
A catalogue record for this book is available from the British Library
Library of Congress Control Number: 2010933531
Mathematics Subject Classification (2010): 53C25, 57M25, 57N10, 57R57, 58J60, 81T13, 81T30
c
° Springer-Verlag London Limited 2010
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Dedicated to the memory of my mother,
Indumati (1920 – 2005),
whopassedontomeherloveoflearning.
==========================
Memories
Your voice is silent now.
But the sound of your soft
Music will always resonate
In my heart.
The wisp of morning incense
Floats in the air no more.
It now resides only in my
Childhood memories.
Contents
Preface ....................................................... xi
Acknowledgements ........................................... xxi
1Algebra................................................... 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
vii
viii Contents
3Manifolds................................................. 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
Contents ix
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