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Topics in Physical Mathematics 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 [email protected] 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 the Publisher, we will make every effort to come to some arrangement with them. 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 Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licenses issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publishers. The use of registered names, trademarks, etc., in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant laws and regulations and therefore free for general use. The publisher makes no representation, express or implied, with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that may be made. Cover design: deblik Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) 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 x Contents 11 Knot and Link Invariants ................................. 351 11.1 Introduction ........................................... 351 11.2 Invariants of
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