Geometry, topology, and physics

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Title:

Personal Author:

Nakahara, Mikio

Publication Information:

Bristol : Adam Hilger, 1990

ISBN:

9780852740941

Subject Term:

Geometry, Differential

Topology

Mathematical physics

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Library	Item Barcode	Call Number	Material Type	Item Category 1	Status
Searching... PSZ JB	30000001532898	QA641.N34 1990	Open Access Book	Book	Searching... Unknown

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Preface to the First Edition	p. xvii
Preface to the Second Edition	p. xix
How to Read this Book	p. xxi
Notation and Conventions	p. xxii
1 Quantum Physics	p. 1
1.1 Analytical mechanics	p. 1
1.1.1 Newtonian mechanics	p. 1
1.1.2 Lagrangian formalism	p. 2
1.1.3 Hamiltonian formalism	p. 5
1.2 Canonical quantization	p. 9
1.2.1 Hilbert space, bras and kets	p. 9
1.2.2 Axioms of canonical quantization	p. 10
1.2.3 Heisenberg equation, Heisenberg picture and Schrodinger picture	p. 13
1.2.4 Wavefunction	p. 13
1.2.5 Harmonic oscillator	p. 17
1.3 Path integral quantization of a Bose particle	p. 19
1.3.1 Path integral quantization	p. 19
1.3.2 Imaginary time and partition function	p. 26
1.3.3 Time-ordered product and generating functional	p. 28
1.4 Harmonic oscillator	p. 31
1.4.1 Transition amplitude	p. 31
1.4.2 Partition function	p. 35
1.5 Path integral quantization of a Fermi particle	p. 38
1.5.1 Fermionic harmonic oscillator	p. 39
1.5.2 Calculus of Grassmann numbers	p. 40
1.5.3 Differentiation	p. 41
1.5.4 Integration	p. 42
1.5.5 Delta-function	p. 43
1.5.6 Gaussian integral	p. 44
1.5.7 Functional derivative	p. 45
1.5.8 Complex conjugation	p. 45
1.5.9 Coherent states and completeness relation	p. 46
1.5.10 Partition function of a fermionic oscillator	p. 47
1.6 Quantization of a scalar field	p. 51
1.6.1 Free scalar field	p. 51
1.6.2 Interacting scalar field	p. 54
1.7 Quantization of a Dirac field	p. 55
1.8 Gauge theories	p. 56
1.8.1 Abelian gauge theories	p. 56
1.8.2 Non-Abelian gauge theories	p. 58
1.8.3 Higgs fields	p. 60
1.9 Magnetic monopoles	p. 60
1.9.1 Dirac monopole	p. 61
1.9.2 The Wu-Yang monopole	p. 62
1.9.3 Charge quantization	p. 62
1.10 Instantons	p. 63
1.10.1 Introduction	p. 63
1.10.2 The (anti-)self-dual solution	p. 64
Problems	p. 66
2 Mathematical Preliminaries	p. 67
2.1 Maps	p. 67
2.1.1 Definitions	p. 67
2.1.2 Equivalence relation and equivalence class	p. 70
2.2 Vector spaces	p. 75
2.2.1 Vectors and vector spaces	p. 75
2.2.2 Linear maps, images and kernels	p. 76
2.2.3 Dual vector space	p. 77
2.2.4 Inner product and adjoint	p. 78
2.2.5 Tensors	p. 80
2.3 Topological spaces	p. 81
2.3.1 Definitions	p. 81
2.3.2 Continuous maps	p. 82
2.3.3 Neighbourhoods and Hausdorff spaces	p. 82
2.3.4 Closed set	p. 83
2.3.5 Compactness	p. 83
2.3.6 Connectedness	p. 85
2.4 Homeomorphisms and topological invariants	p. 85
2.4.1 Homeomorphisms	p. 85
2.4.2 Topological invariants	p. 86
2.4.3 Homotopy type	p. 88
2.4.4 Euler characteristic: an example	p. 88
Problems	p. 91
3 Homology Groups	p. 93
3.1 Abelian groups	p. 93
3.1.1 Elementary group theory	p. 93
3.1.2 Finitely generated Abelian groups and free Abelian groups	p. 96
3.1.3 Cyclic groups	p. 96
3.2 Simplexes and simplicial complexes	p. 98
3.2.1 Simplexes	p. 98
3.2.2 Simplicial complexes and polyhedra	p. 99
3.3 Homology groups of simplicial complexes	p. 100
3.3.1 Oriented simplexes	p. 100
3.3.2 Chain group, cycle group and boundary group	p. 102
3.3.3 Homology groups	p. 106
3.3.4 Computation of H[subscript 0](K)	p. 110
3.3.5 More homology computations	p. 111
3.4 General properties of homology groups	p. 117
3.4.1 Connectedness and homology groups	p. 117
3.4.2 Structure of homology groups	p. 118
3.4.3 Betti numbers and the Euler-Poincare theorem	p. 118
Problems	p. 120
4 Homotopy Groups	p. 121
4.1 Fundamental groups	p. 121
4.1.1 Basic ideas	p. 121
4.1.2 Paths and loops	p. 122
4.1.3 Homotopy	p. 123
4.1.4 Fundamental groups	p. 125
4.2 General properties of fundamental groups	p. 127
4.2.1 Arcwise connectedness and fundamental groups	p. 127
4.2.2 Homotopic invariance of fundamental groups	p. 128
4.3 Examples of fundamental groups	p. 131
4.3.1 Fundamental group of torus	p. 133
4.4 Fundamental groups of polyhedra	p. 134
4.4.1 Free groups and relations	p. 134
4.4.2 Calculating fundamental groups of polyhedra	p. 136
4.4.3 Relations between H[subscript 1](K) and [pi subscript 1]([vertical bar]K[vertical bar])	p. 144
4.5 Higher homotopy groups	p. 145
4.5.1 Definitions	p. 146
4.6 General properties of higher homotopy groups	p. 148
4.6.1 Abelian nature of higher homotopy groups	p. 148
4.6.2 Arcwise connectedness and higher homotopy groups	p. 148
4.6.3 Homotopy invariance of higher homotopy groups	p. 148
4.6.4 Higher homotopy groups of a product space	p. 148
4.6.5 Universal covering spaces and higher homotopy groups	p. 148
4.7 Examples of higher homotopy groups	p. 150
4.8 Orders in condensed matter systems	p. 153
4.8.1 Order parameter	p. 153
4.8.2 Superfluid [superscript 4]He and superconductors	p. 154
4.8.3 General consideration	p. 157
4.9 Defects in nematic liquid crystals	p. 159
4.9.1 Order parameter of nematic liquid crystals	p. 159
4.9.2 Line defects in nematic liquid crystals	p. 160
4.9.3 Point defects in nematic liquid crystals	p. 161
4.9.4 Higher dimensional texture	p. 162
4.10 Textures in superfluid [superscript 3]He-A	p. 163
4.10.1 Superfluid [superscript 3]He-A	p. 163
4.10.2 Line defects and non-singular vortices in [superscript 3]He-A	p. 165
4.10.3 Shankar monopole in [superscript 3]He-A	p. 166
Problems	p. 167
5 Manifolds	p. 169
5.1 Manifolds	p. 169
5.1.1 Heuristic introduction	p. 169
5.1.2 Definitions	p. 171
5.1.3 Examples	p. 173
5.2 The calculus on manifolds	p. 178
5.2.1 Differentiable maps	p. 179
5.2.2 Vectors	p. 181
5.2.3 One-forms	p. 184
5.2.4 Tensors	p. 185
5.2.5 Tensor fields	p. 185
5.2.6 Induced maps	p. 186
5.2.7 Submanifolds	p. 188
5.3 Flows and Lie derivatives	p. 188
5.3.1 One-parameter group of transformations	p. 190
5.3.2 Lie derivatives	p. 191
5.4 Differential forms	p. 196
5.4.1 Definitions	p. 196
5.4.2 Exterior derivatives	p. 198
5.4.3 Interior product and Lie derivative of forms	p. 201
5.5 Integration of differential forms	p. 204
5.5.1 Orientation	p. 204
5.5.2 Integration of forms	p. 205
5.6 Lie groups and Lie algebras	p. 207
5.6.1 Lie groups	p. 207
5.6.2 Lie algebras	p. 209
5.6.3 The one-parameter subgroup	p. 212
5.6.4 Frames and structure equation	p. 215
5.7 The action of Lie groups on manifolds	p. 216
5.7.1 Definitions	p. 216
5.7.2 Orbits and isotropy groups	p. 219
5.7.3 Induced vector fields	p. 223
5.7.4 The adjoint representation	p. 224
Problems	p. 224
6 de Rham Cohomology Groups	p. 226
6.1 Stokes' theorem	p. 226
6.1.1 Preliminary consideration	p. 226
6.1.2 Stokes' theorem	p. 228
6.2 de Rham cohomology groups	p. 230
6.2.1 Definitions	p. 230
6.2.2 Duality of H[subscript r](M) and H[superscript r](M); de Rham's theorem	p. 233
6.3 Poincare's lemma	p. 235
6.4 Structure of de Rham cohomology groups	p. 237
6.4.1 Poincare duality	p. 237
6.4.2 Cohomology rings	p. 238
6.4.3 The Kunneth formula	p. 238
6.4.4 Pullback of de Rham cohomology groups	p. 240
6.4.5 Homotopy and H[superscript 1](M)	p. 240
7 Riemannian Geometry	p. 244
7.1 Riemannian manifolds and pseudo-Riemannian manifolds	p. 244
7.1.1 Metric tensors	p. 244
7.1.2 Induced metric	p. 246
7.2 Parallel transport, connection and covariant derivative	p. 247
7.2.1 Heuristic introduction	p. 247
7.2.2 Affine connections	p. 249
7.2.3 Parallel transport and geodesics	p. 250
7.2.4 The covariant derivative of tensor fields	p. 251
7.2.5 The transformation properties of connection coefficients	p. 252
7.2.6 The metric connection	p. 253
7.3 Curvature and torsion	p. 254
7.3.1 Definitions	p. 254
7.3.2 Geometrical meaning of the Riemann tensor and the torsion tensor	p. 256
7.3.3 The Ricci tensor and the scalar curvature	p. 260
7.4 Levi-Civita connections	p. 261
7.4.1 The fundamental theorem	p. 261
7.4.2 The Levi-Civita connection in the classical geometry of surfaces	p. 262
7.4.3 Geodesics	p. 263
7.4.4 The normal coordinate system	p. 266
7.4.5 Riemann curvature tensor with Levi-Civita connection	p. 268
7.5 Holonomy	p. 271
7.6 Isometries and conformal transformations	p. 273
7.6.1 Isometries	p. 273
7.6.2 Conformal transformations	p. 274
7.7 Killing vector fields and conformal Killing vector fields	p. 279
7.7.1 Killing vector fields	p. 279
7.7.2 Conformal Killing vector fields	p. 282
7.8 Non-coordinate bases	p. 283
7.8.1 Definitions	p. 283
7.8.2 Cartan's structure equations	p. 284
7.8.3 The local frame	p. 285
7.8.4 The Levi-Civita connection in a non-coordinate basis	p. 287
7.9 Differential forms and Hodge theory	p. 289
7.9.1 Invariant volume elements	p. 289
7.9.2 Duality transformations (Hodge star)	p. 290
7.9.3 Inner products of r-forms	p. 291
7.9.4 Adjoints of exterior derivatives	p. 293
7.9.5 The Laplacian, harmonic forms and the Hodge decomposition theorem	p. 294
7.9.6 Harmonic forms and de Rham cohomology groups	p. 296
7.10 Aspects of general relativity	p. 297
7.10.1 Introduction to general relativity	p. 297
7.10.2 Einstein-Hilbert action	p. 298
7.10.3 Spinors in curved spacetime	p. 300
7.11 Bosonic string theory	p. 302
7.11.1 The string action	p. 303
7.11.2 Symmetries of the Polyakov strings	p. 305
Problems	p. 307
8 Complex Manifolds	p. 308
8.1 Complex manifolds	p. 308
8.1.1 Definitions	p. 308
8.1.2 Examples	p. 309
8.2 Calculus on complex manifolds	p. 315
8.2.1 Holomorphic maps	p. 315
8.2.2 Complexifications	p. 316
8.2.3 Almost complex structure	p. 317
8.3 Complex differential forms	p. 320
8.3.1 Complexification of real differential forms	p. 320
8.3.2 Differential forms on complex manifolds	p. 321
8.3.3 Dolbeault operators	p. 322
8.4 Hermitian manifolds and Hermitian differential geometry	p. 324
8.4.1 The Hermitian metric	p. 325
8.4.2 Kahler form	p. 326
8.4.3 Covariant derivatives	p. 327
8.4.4 Torsion and curvature	p. 329
8.5 Kahler manifolds and Kahler differential geometry	p. 330
8.5.1 Definitions	p. 330
8.5.2 Kahler geometry	p. 334
8.5.3 The holonomy group of Kahler manifolds	p. 335
8.6 Harmonic forms and [characters not reproducible]-cohomology groups	p. 336
8.6.1 The adjoint operators [characters not reproducible] and [characters not reproducible]	p. 337
8.6.2 Laplacians and the Hodge theorem	p. 338
8.6.3 Laplacians on a Kahler manifold	p. 339
8.6.4 The Hodge numbers of Kahler manifolds	p. 339
8.7 Almost complex manifolds	p. 341
8.7.1 Definitions	p. 342
8.8 Orbifolds	p. 344
8.8.1 One-dimensional examples	p. 344
8.8.2 Three-dimensional examples	p. 346
9 Fibre Bundles	p. 348
9.1 Tangent bundles	p. 348
9.2 Fibre bundles	p. 350
9.2.1 Definitions	p. 350
9.2.2 Reconstruction of fibre bundles	p. 353
9.2.3 Bundle maps	p. 354
9.2.4 Equivalent bundles	p. 355
9.2.5 Pullback bundles	p. 355
9.2.6 Homotopy axiom	p. 357
9.3 Vector bundles	p. 357
9.3.1 Definitions and examples	p. 357
9.3.2 Frames	p. 359
9.3.3 Cotangent bundles and dual bundles	p. 360
9.3.4 Sections of vector bundles	p. 361
9.3.5 The product bundle and Whitney sum bundle	p. 361
9.3.6 Tensor product bundles	p. 363
9.4 Principal bundles	p. 363
9.4.1 Definitions	p. 363
9.4.2 Associated bundles	p. 370
9.4.3 Triviality of bundles	p. 372
Problems	p. 372
10 Connections on Fibre Bundles	p. 374
10.1 Connections on principal bundles	p. 374
10.1.1 Definitions	p. 375
10.1.2 The connection one-form	p. 376
10.1.3 The local connection form and gauge potential	p. 377
10.1.4 Horizontal lift and parallel transport	p. 381
10.2 Holonomy	p. 384
10.2.1 Definitions	p. 384
10.3 Curvature	p. 385
10.3.1 Covariant derivatives in principal bundles	p. 385
10.3.2 Curvature	p. 386
10.3.3 Geometrical meaning of the curvature and the Ambrose-Singer theorem	p. 388
10.3.4 Local form of the curvature	p. 389
10.3.5 The Bianchi identity	p. 390
10.4 The covariant derivative on associated vector bundles	p. 391
10.4.1 The covariant derivative on associated bundles	p. 391
10.4.2 A local expression for the covariant derivative	p. 393
10.4.3 Curvature rederived	p. 396
10.4.4 A connection which preserves the inner product	p. 396
10.4.5 Holomorphic vector bundles and Hermitian inner products	p. 397
10.5 Gauge theories	p. 399
10.5.1 U(1) gauge theory	p. 399
10.5.2 The Dirac magnetic monopole	p. 400
10.5.3 The Aharonov-Bohm effect	p. 402
10.5.4 Yang-Mills theory	p. 404
10.5.5 Instantons	p. 405
10.6 Berry's phase	p. 409
10.6.1 Derivation of Berry's phase	p. 410
10.6.2 Berry's phase, Berry's connection and Berry's curvature	p. 411
Problems	p. 418
11 Characteristic Classes	p. 419
11.1 Invariant polynomials and the Chern-Weil homomorphism	p. 419
11.1.1 Invariant polynomials	p. 420
11.2 Chern classes	p. 426
11.2.1 Definitions	p. 426
11.2.2 Properties of Chern classes	p. 428
11.2.3 Splitting principle	p. 429
11.2.4 Universal bundles and classifying spaces	p. 430
11.3 Chern characters	p. 431
11.3.1 Definitions	p. 431
11.3.2 Properties of the Chern characters	p. 434
11.3.3 Todd classes	p. 435
11.4 Pontrjagin and Euler classes	p. 436
11.4.1 Pontrjagin classes	p. 436
11.4.2 Euler classes	p. 439
11.4.3 Hirzebruch L-polynomial and A-genus	p. 442
11.5 Chern-Simons forms	p. 443
11.5.1 Definition	p. 443
11.5.2 The Chern-Simons form of the Chern character	p. 444
11.5.3 Cartan's homotopy operator and applications	p. 445
11.6 Stiefel-Whitney classes	p. 448
11.6.1 Spin bundles	p. 449
11.6.2 Cech cohomology groups	p. 449
11.6.3 Stiefel-Whitney classes	p. 450
12 Index Theorems	p. 453
12.1 Elliptic operators and Fredholm operators	p. 453
12.1.1 Elliptic operators	p. 454
12.1.2 Fredholm operators	p. 456
12.1.3 Elliptic complexes	p. 457
12.2 The Atiyah-Singer index theorem	p. 459
12.2.1 Statement of the theorem	p. 459
12.3 The de Rham complex	p. 460
12.4 The Dolbeault complex	p. 462
12.4.1 The twisted Dolbeault complex and the Hirzebruch-Riemann-Roch theorem	p. 463
12.5 The signature complex	p. 464
12.5.1 The Hirzebruch signature	p. 464
12.5.2 The signature complex and the Hirzebruch signature theorem	p. 465
12.6 Spin complexes	p. 467
12.6.1 Dirac operator	p. 468
12.6.2 Twisted spin complexes	p. 471
12.7 The heat kernel and generalized [zeta]-functions	p. 472
12.7.1 The heat kernel and index theorem	p. 472
12.7.2 Spectral [zeta]-functions	p. 475
12.8 The Atiyah-Patodi-Singer index theorem	p. 477
12.8.1 [eta]-invariant and spectral flow	p. 477
12.8.2 The Atiyah-Patodi-Singer (APS) index theorem	p. 478
12.9 Supersymmetric quantum mechanics	p. 481
12.9.1 Clifford algebra and fermions	p. 481
12.9.2 Supersymmetric quantum mechanics in flat space	p. 482
12.9.3 Supersymmetric quantum mechanics in a general manifold	p. 485
12.10 Supersymmetric proof of index theorem	p. 487
12.10.1 The index	p. 487
12.10.2 Path integral and index theorem	p. 490
Problems	p. 500
13 Anomalies in Gauge Field Theories	p. 501
13.1 Introduction	p. 501
13.2 Abelian anomalies	p. 503
13.2.1 Fujikawa's method	p. 503
13.3 Non-Abelian anomalies	p. 508
13.4 The Wess-Zumino consistency conditions	p. 512
13.4.1 The Becchi-Rouet-Stora operator and the Faddeev-Popov ghost	p. 512
13.4.2 The BRS operator, FP ghost and moduli space	p. 513
13.4.3 The Wess-Zumino conditions	p. 515
13.4.4 Descent equations and solutions of WZ conditions	p. 515
13.5 Abelian anomalies versus non-Abelian anomalies	p. 518
13.5.1 m dimensions versus m + 2 dimensions	p. 520
13.6 The parity anomaly in odd-dimensional spaces	p. 523
13.6.1 The parity anomaly	p. 524
13.6.2 The dimensional ladder: 4-3-2	p. 525
14 Bosonic String Theory	p. 528
14.1 Differential geometry on Riemann surfaces	p. 528
14.1.1 Metric and complex structure	p. 528
14.1.2 Vectors, forms and tensors	p. 529
14.1.3 Covariant derivatives	p. 531
14.1.4 The Riemann-Roch theorem	p. 533
14.2 Quantum theory of bosonic strings	p. 535
14.2.1 Vacuum amplitude of Polyakov strings	p. 535
14.2.2 Measures of integration	p. 538
14.2.3 Complex tensor calculus and string measure	p. 550
14.2.4 Moduli spaces of Riemann surfaces	p. 554
14.3 One-loop amplitudes	p. 555
14.3.1 Moduli spaces, CKV, Beltrami and quadratic differentials	p. 555
14.3.2 The evaluation of determinants	p. 557
References	p. 560
Index	p. 565

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