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Geometry, Topology and Physics, Second Edition : Graduate Student Series in Physics - Mikio Nakahara

Geometry, Topology and Physics, Second Edition

Graduate Student Series in Physics


Published: 4th June 2003
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Differential geometry and topology have become essential tools for many theoretical physicists. In particular, they are indispensable in theoretical studies of condensed matter physics, gravity, and particle physics. Geometry, Topology and Physics, Second Edition introduces the ideas and techniques of differential geometry and topology at a level suitable for postgraduate students and researchers in these fields.
The second edition of this popular and established text incorporates a number of changes designed to meet the needs of the reader and reflect the development of the subject. The book features a considerably expanded first chapter, reviewing aspects of path integral quantization and gauge theories. Chapter 2 introduces the mathematical concepts of maps, vector spaces, and topology. The following chapters focus on more elaborate concepts in geometry and topology and discuss the application of these concepts to liquid crystals, superfluid helium, general relativity, and bosonic string theory. Later chapters unify geometry and topology, exploring fiber bundles, characteristic classes, and index theorems. New to this second edition is the proof of the index theorem in terms of supersymmetric quantum mechanics. The final two chapters are devoted to the most fascinating applications of geometry and topology in contemporary physics, namely the study of anomalies in gauge field theories and the analysis of Polakov's bosonic string theory from the geometrical point of view.
Geometry, Topology and Physics, Second Edition is an ideal introduction to differential geometry and topology for postgraduate students and researchers in theoretical and mathematical physics.

"a very impressive book.." -- Australian and New Zealand Physicists" The clarity of the presentation is enhanced by explicit calculations and diagrams; the proof of a theorem is given only when it is instructive and not very technical. There is also a large number of exercises and problems, and last but not least, an index superb layout" -- Zentralblatt fur Mathematick un ihre Grenzgebiete "I believe that the book will not only boost modernization of the traditional courses of theoretical physics but will prompt the specialist in topology and differential geometry to have a closer look at the applications. So I welcome this second edition." --Christopher Gilmour

Preface to the First Editionp. xvii
Preface to the Second Editionp. xix
How to Read this Bookp. xxi
Notation and Conventionsp. xxii
Quantum Physicsp. 1
Analytical mechanicsp. 1
Newtonian mechanicsp. 1
Lagrangian formalismp. 2
Hamiltonian formalismp. 5
Canonical quantizationp. 9
Hilbert space, bras and ketsp. 9
Axioms of canonical quantizationp. 10
Heisenberg equation, Heisenberg picture and Schrodinger picturep. 13
Wavefunctionp. 13
Harmonic oscillatorp. 17
Path integral quantization of a Bose particlep. 19
Path integral quantizationp. 19
Imaginary time and partition functionp. 26
Time-ordered product and generating functionalp. 28
Harmonic oscillatorp. 31
Transition amplitudep. 31
Partition functionp. 35
Path integral quantization of a Fermi particlep. 38
Fermionic harmonic oscillatorp. 39
Calculus of Grassmann numbersp. 40
Differentiationp. 41
Integrationp. 42
Delta-functionp. 43
Gaussian integralp. 44
Functional derivativep. 45
Complex conjugationp. 45
Coherent states and completeness relationp. 46
Partition function of a fermionic oscillatorp. 47
Quantization of a scalar fieldp. 51
Free scalar fieldp. 51
Interacting scalar fieldp. 54
Quantization of a Dirac fieldp. 55
Gauge theoriesp. 56
Abelian gauge theoriesp. 56
Non-Abelian gauge theoriesp. 58
Higgs fieldsp. 60
Magnetic monopolesp. 60
Dirac monopolep. 61
The Wu-Yang monopolep. 62
Charge quantizationp. 62
Instantonsp. 63
Introductionp. 63
The (anti-)self-dual solutionp. 64
Problemsp. 66
Mathematical Preliminariesp. 67
Mapsp. 67
Definitionsp. 67
Equivalence relation and equivalence classp. 70
Vector spacesp. 75
Vectors and vector spacesp. 75
Linear maps, images and kernelsp. 76
Dual vector spacep. 77
Inner product and adjointp. 78
Tensorsp. 80
Topological spacesp. 81
Definitionsp. 81
Continuous mapsp. 82
Neighbourhoods and Hausdorff spacesp. 82
Closed setp. 83
Compactnessp. 83
Connectednessp. 85
Homeomorphisms and topological invariantsp. 85
Homeomorphismsp. 85
Topological invariantsp. 86
Homotopy typep. 88
Euler characteristic: an examplep. 88
Problemsp. 91
Homology Groupsp. 93
Abelian groupsp. 93
Elementary group theoryp. 93
Finitely generated Abelian groups and free Abelian groupsp. 96
Cyclic groupsp. 96
Simplexes and simplicial complexesp. 98
Simplexesp. 98
Simplicial complexes and polyhedrap. 99
Homology groups of simplicial complexesp. 100
Oriented simplexesp. 100
Chain group, cycle group and boundary groupp. 102
Homology groupsp. 106
Computation of H[subscript 0](K)p. 110
More homology computationsp. 111
General properties of homology groupsp. 117
Connectedness and homology groupsp. 117
Structure of homology groupsp. 118
Betti numbers and the Euler-Poincare theoremp. 118
Problemsp. 120
Homotopy Groupsp. 121
Fundamental groupsp. 121
Basic ideasp. 121
Paths and loopsp. 122
Homotopyp. 123
Fundamental groupsp. 125
General properties of fundamental groupsp. 127
Arcwise connectedness and fundamental groupsp. 127
Homotopic invariance of fundamental groupsp. 128
Examples of fundamental groupsp. 131
Fundamental group of torusp. 133
Fundamental groups of polyhedrap. 134
Free groups and relationsp. 134
Calculating fundamental groups of polyhedrap. 136
Relations between H[subscript 1](K) and [pi subscript 1]([vertical bar]K[vertical bar])p. 144
Higher homotopy groupsp. 145
Definitionsp. 146
General properties of higher homotopy groupsp. 148
Abelian nature of higher homotopy groupsp. 148
Arcwise connectedness and higher homotopy groupsp. 148
Homotopy invariance of higher homotopy groupsp. 148
Higher homotopy groups of a product spacep. 148
Universal covering spaces and higher homotopy groupsp. 148
Examples of higher homotopy groupsp. 150
Orders in condensed matter systemsp. 153
Order parameterp. 153
Superfluid [superscript 4]He and superconductorsp. 154
General considerationp. 157
Defects in nematic liquid crystalsp. 159
Order parameter of nematic liquid crystalsp. 159
Line defects in nematic liquid crystalsp. 160
Point defects in nematic liquid crystalsp. 161
Higher dimensional texturep. 162
Textures in superfluid [superscript 3]He-Ap. 163
Superfluid [superscript 3]He-Ap. 163
Line defects and non-singular vortices in [superscript 3]He-Ap. 165
Shankar monopole in [superscript 3]He-Ap. 166
Problemsp. 167
Manifoldsp. 169
Manifoldsp. 169
Heuristic introductionp. 169
Definitionsp. 171
Examplesp. 173
The calculus on manifoldsp. 178
Differentiable mapsp. 179
Vectorsp. 181
One-formsp. 184
Tensorsp. 185
Tensor fieldsp. 185
Induced mapsp. 186
Submanifoldsp. 188
Flows and Lie derivativesp. 188
One-parameter group of transformationsp. 190
Lie derivativesp. 191
Differential formsp. 196
Definitionsp. 196
Exterior derivativesp. 198
Interior product and Lie derivative of formsp. 201
Integration of differential formsp. 204
Orientationp. 204
Integration of formsp. 205
Lie groups and Lie algebrasp. 207
Lie groupsp. 207
Lie algebrasp. 209
The one-parameter subgroupp. 212
Frames and structure equationp. 215
The action of Lie groups on manifoldsp. 216
Definitionsp. 216
Orbits and isotropy groupsp. 219
Induced vector fieldsp. 223
The adjoint representationp. 224
Problemsp. 224
de Rham Cohomology Groupsp. 226
Stokes' theoremp. 226
Preliminary considerationp. 226
Stokes' theoremp. 228
de Rham cohomology groupsp. 230
Definitionsp. 230
Duality of H[subscript r](M) and H[superscript r](M); de Rham's theoremp. 233
Poincare's lemmap. 235
Structure of de Rham cohomology groupsp. 237
Poincare dualityp. 237
Cohomology ringsp. 238
The Kunneth formulap. 238
Pullback of de Rham cohomology groupsp. 240
Homotopy and H[superscript 1](M)p. 240
Riemannian Geometryp. 244
Riemannian manifolds and pseudo-Riemannian manifoldsp. 244
Metric tensorsp. 244
Induced metricp. 246
Parallel transport, connection and covariant derivativep. 247
Heuristic introductionp. 247
Affine connectionsp. 249
Parallel transport and geodesicsp. 250
The covariant derivative of tensor fieldsp. 251
The transformation properties of connection coefficientsp. 252
The metric connectionp. 253
Curvature and torsionp. 254
Definitionsp. 254
Geometrical meaning of the Riemann tensor and the torsion tensorp. 256
The Ricci tensor and the scalar curvaturep. 260
Levi-Civita connectionsp. 261
The fundamental theoremp. 261
The Levi-Civita connection in the classical geometry of surfacesp. 262
Geodesicsp. 263
The normal coordinate systemp. 266
Riemann curvature tensor with Levi-Civita connectionp. 268
Holonomyp. 271
Isometries and conformal transformationsp. 273
Isometriesp. 273
Conformal transformationsp. 274
Killing vector fields and conformal Killing vector fieldsp. 279
Killing vector fieldsp. 279
Conformal Killing vector fieldsp. 282
Non-coordinate basesp. 283
Definitionsp. 283
Cartan's structure equationsp. 284
The local framep. 285
The Levi-Civita connection in a non-coordinate basisp. 287
Differential forms and Hodge theoryp. 289
Invariant volume elementsp. 289
Duality transformations (Hodge star)p. 290
Inner products of r-formsp. 291
Adjoints of exterior derivativesp. 293
The Laplacian, harmonic forms and the Hodge decomposition theoremp. 294
Harmonic forms and de Rham cohomology groupsp. 296
Aspects of general relativityp. 297
Introduction to general relativityp. 297
Einstein-Hilbert actionp. 298
Spinors in curved spacetimep. 300
Bosonic string theoryp. 302
The string actionp. 303
Symmetries of the Polyakov stringsp. 305
Problemsp. 307
Complex Manifoldsp. 308
Complex manifoldsp. 308
Definitionsp. 308
Examplesp. 309
Calculus on complex manifoldsp. 315
Holomorphic mapsp. 315
Complexificationsp. 316
Almost complex structurep. 317
Complex differential formsp. 320
Complexification of real differential formsp. 320
Differential forms on complex manifoldsp. 321
Dolbeault operatorsp. 322
Hermitian manifolds and Hermitian differential geometryp. 324
The Hermitian metricp. 325
Kahler formp. 326
Covariant derivativesp. 327
Torsion and curvaturep. 329
Kahler manifolds and Kahler differential geometryp. 330
Definitionsp. 330
Kahler geometryp. 334
The holonomy group of Kahler manifoldsp. 335
Harmonic forms and [characters not reproducible]-cohomology groupsp. 336
The adjoint operators [characters not reproducible] and [characters not reproducible]p. 337
Laplacians and the Hodge theoremp. 338
Laplacians on a Kahler manifoldp. 339
The Hodge numbers of Kahler manifoldsp. 339
Almost complex manifoldsp. 341
Definitionsp. 342
Orbifoldsp. 344
One-dimensional examplesp. 344
Three-dimensional examplesp. 346
Fibre Bundlesp. 348
Tangent bundlesp. 348
Fibre bundlesp. 350
Definitionsp. 350
Reconstruction of fibre bundlesp. 353
Bundle mapsp. 354
Equivalent bundlesp. 355
Pullback bundlesp. 355
Homotopy axiomp. 357
Vector bundlesp. 357
Definitions and examplesp. 357
Framesp. 359
Cotangent bundles and dual bundlesp. 360
Sections of vector bundlesp. 361
The product bundle and Whitney sum bundlep. 361
Tensor product bundlesp. 363
Principal bundlesp. 363
Definitionsp. 363
Associated bundlesp. 370
Triviality of bundlesp. 372
Problemsp. 372
Connections on Fibre Bundlesp. 374
Connections on principal bundlesp. 374
Definitionsp. 375
The connection one-formp. 376
The local connection form and gauge potentialp. 377
Horizontal lift and parallel transportp. 381
Holonomyp. 384
Definitionsp. 384
Curvaturep. 385
Covariant derivatives in principal bundlesp. 385
Curvaturep. 386
Geometrical meaning of the curvature and the Ambrose-Singer theoremp. 388
Local form of the curvaturep. 389
The Bianchi identityp. 390
The covariant derivative on associated vector bundlesp. 391
The covariant derivative on associated bundlesp. 391
A local expression for the covariant derivativep. 393
Curvature rederivedp. 396
A connection which preserves the inner productp. 396
Holomorphic vector bundles and Hermitian inner productsp. 397
Gauge theoriesp. 399
U(1) gauge theoryp. 399
The Dirac magnetic monopolep. 400
The Aharonov-Bohm effectp. 402
Yang-Mills theoryp. 404
Instantonsp. 405
Berry's phasep. 409
Derivation of Berry's phasep. 410
Berry's phase, Berry's connection and Berry's curvaturep. 411
Problemsp. 418
Characteristic Classesp. 419
Invariant polynomials and the Chern-Weil homomorphismp. 419
Invariant polynomialsp. 420
Chern classesp. 426
Definitionsp. 426
Properties of Chern classesp. 428
Splitting principlep. 429
Universal bundles and classifying spacesp. 430
Chern charactersp. 431
Definitionsp. 431
Properties of the Chern charactersp. 434
Todd classesp. 435
Pontrjagin and Euler classesp. 436
Pontrjagin classesp. 436
Euler classesp. 439
Hirzebruch L-polynomial and A-genusp. 442
Chern-Simons formsp. 443
Definitionp. 443
The Chern-Simons form of the Chern characterp. 444
Cartan's homotopy operator and applicationsp. 445
Stiefel-Whitney classesp. 448
Spin bundlesp. 449
Cech cohomology groupsp. 449
Stiefel-Whitney classesp. 450
Index Theoremsp. 453
Elliptic operators and Fredholm operatorsp. 453
Elliptic operatorsp. 454
Fredholm operatorsp. 456
Elliptic complexesp. 457
The Atiyah-Singer index theoremp. 459
Statement of the theoremp. 459
The de Rham complexp. 460
The Dolbeault complexp. 462
The twisted Dolbeault complex and the Hirzebruch-Riemann-Roch theoremp. 463
The signature complexp. 464
The Hirzebruch signaturep. 464
The signature complex and the Hirzebruch signature theoremp. 465
Spin complexesp. 467
Dirac operatorp. 468
Twisted spin complexesp. 471
The heat kernel and generalized [zeta]-functionsp. 472
The heat kernel and index theoremp. 472
Spectral [zeta]-functionsp. 475
The Atiyah-Patodi-Singer index theoremp. 477
[eta]-invariant and spectral flowp. 477
The Atiyah-Patodi-Singer (APS) index theoremp. 478
Supersymmetric quantum mechanicsp. 481
Clifford algebra and fermionsp. 481
Supersymmetric quantum mechanics in flat spacep. 482
Supersymmetric quantum mechanics in a general manifoldp. 485
Supersymmetric proof of index theoremp. 487
The indexp. 487
Path integral and index theoremp. 490
Problemsp. 500
Anomalies in Gauge Field Theoriesp. 501
Introductionp. 501
Abelian anomaliesp. 503
Fujikawa's methodp. 503
Non-Abelian anomaliesp. 508
The Wess-Zumino consistency conditionsp. 512
The Becchi-Rouet-Stora operator and the Faddeev-Popov ghostp. 512
The BRS operator, FP ghost and moduli spacep. 513
The Wess-Zumino conditionsp. 515
Descent equations and solutions of WZ conditionsp. 515
Abelian anomalies versus non-Abelian anomaliesp. 518
m dimensions versus m + 2 dimensionsp. 520
The parity anomaly in odd-dimensional spacesp. 523
The parity anomalyp. 524
The dimensional ladder: 4-3-2p. 525
Bosonic String Theoryp. 528
Differential geometry on Riemann surfacesp. 528
Metric and complex structurep. 528
Vectors, forms and tensorsp. 529
Covariant derivativesp. 531
The Riemann-Roch theoremp. 533
Quantum theory of bosonic stringsp. 535
Vacuum amplitude of Polyakov stringsp. 535
Measures of integrationp. 538
Complex tensor calculus and string measurep. 550
Moduli spaces of Riemann surfacesp. 554
One-loop amplitudesp. 555
Moduli spaces, CKV, Beltrami and quadratic differentialsp. 555
The evaluation of determinantsp. 557
Referencesp. 560
Indexp. 565
Table of Contents provided by Rittenhouse. All Rights Reserved.

ISBN: 9780750306065
ISBN-10: 0750306068
Series: Graduate Student Series in Physics
Audience: Professional
Format: Paperback
Language: English
Number Of Pages: 596
Published: 4th June 2003
Publisher: Taylor & Francis Ltd
Country of Publication: GB
Dimensions (cm): 23.4 x 15.9  x 3.2
Weight (kg): 1.03
Edition Number: 2
Edition Type: New edition