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Naber G.L. — Topology, Geometry and Gauge Fields
Naber G.L. — Topology, Geometry and Gauge Fields

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Название: Topology, Geometry and Gauge Fields

Автор: Naber G.L.

Аннотация:

This book covers topology and geometry beginning with an accessible account of the extraordinary and rather mysterious impact of mathematical physics, especially gauge theory, on the study of the geometry and topology of manifolds. Much of the mathematics developed in the book to study the classical field theories of physics (de Rham cohomology, Chern classes, Semi-Riemannian manifolds, Cech cohomology, spinors etc.) is standard, but the treatment always keeps one eye on the physics and unhesitatingly sacrifices generality to clarity. The author brings the reader up to the level needed to conclude with a brief discussion of the Seiberg-Witten invariants. Although this volume can be read independently Naber carries on the program initiated in his earlier volume, Topology, Geometry and Gauge Fields: Foundations, Springer, 1997, and writes in much the same spirit with precisely the same philosophical motivation. A large number of exercises are included to encourage active participation on the part of the reader. This work will be of great interest to researchers and graduate students in the field of mathematical physics.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2000

Количество страниц: 443

Добавлена в каталог: 21.06.2006

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$1^{st}$ Chern class      45 63 352
$1^{st}$ Chern number      68
$1^{st}$ Stiefel — Whitney class      397
$2^{nd}$ Chern class      46 138
$2^{nd}$ Chern number      46 138
$2^{nd}$ Stiefel — Whitney class      401
$C^{\infty}$-manifold      1
$C^{\infty}$-map      2
$C^{\infty}$-related      1
$Spin^c$-structurc      410
$\rho$-tensor product      232
$\rho$-wedge product      232
$\sigma$-algebra      272
1-form      9
1-form, left-invariant      19
1-form, real-valued      9
1-form, restriction      10
2-valued representation      88
Action functional      55
Action functional, Seiberg — Wittcn      415
Action, effective      26
Action, free      26
Action, left      26
Action, right      25
Action, transitive      26
ad(G)-invariant      24
Adjoint bundle      49
Adjoint representation      24
Admissible basis      189
Algebraic homotopy      254 305 320
Almost everywhere      275
Anti-self-dual (ASD)      135
Antiparticle      75
Antipodal map      343
Atiyah — Singer index theorem      207
Atlas      1
Atlas, maximal      1
Atlas, oriented      8
Automorphism of bundles      34
Betti numbers      331
Bianchi identity      56 134 135 268
Bogomolny monopole equations      135
Boosts      192
Borel set      272
BPST potential      42
BPST potential, center      42
BPST potential, instanton      136
BPST potential, scale      42
Bump function      164
Bundle map      33
Bundle splicing      117
Canonical isomorphism      4
Cartan 1-form      19
Cartan structure equation      42
Cech cohomology group      393 397
Cech j-coboundary      393
Cech j-cocycle      393
Characteristic class      64 377
Chart      1
Chart, admissible      1
Chart, consistent with orientation      8
Chern class      377
Chern class, total      387
Chern number      380
Chern — Well homomorphism      377
Chiral representation      98
Classical groups      6
Classification Theorem      34
Clifford algebra      95 99
Clifford multiplication      409
Closed form      63 251
Coboundary      319
Coboundary operator      319 391
Cochain complex      318
Cochain homotopy      320
Cochain map      319
Cocycle      319
Cocycle condition      32
Cohomologous      63 299 319
Cohomology class      299 319
Commutator      15
Complex line bundle      48
Complex scalar field      51
Conjugation representation      104
Connection      35
Connection, existence of      167
Connection, Levi — Civita      186
Connection, linear      186
Connection, Riemannian      186
Coordinate expression      2
Coordinates on $\mathbb{R}^{1,3}$, null      199
Coordinates on $\mathbb{R}^{1,3}$, spherical      198
Cotangent bundle      179
Cotangent space      9
Coulomb field      63
Coupling constant      55
Covariant exterior derivative      51 261
Covariant tensor field of rank two      11
Covariant tensor field of rank two, components      11
Covariant tensor field of rank two, continuous      11
Covariant tensor field of rank two, smooth      11
Covariant tensors      208
Covariant tensors, rank      208
Covariant tensors, rank one      10
Covariant tensors, rank two      10
Covector      9
Critical value      3
Cross-section      32 50
Cross-section, global      34
Curvature      42
de Rham, coboundary      298
de Rham, cocycle      298
de Rham, cohomology group      63 298
de Rham, complex      251
de Sitter spacetime      202 405
Deformed      295
Degree      340
Derivation      7
Derivative      2
Determinant line bundle      410
Diffeomorphic      2
Diffeomorphism      2
Differential form      238 239
Differential form, vector-valued      245
Differentiate manifold      1
Differentiate structure      1
Dimension of a differentiable manifold      1
Dimension of a submanifold      3
Dimension of a topological manifold      1
Dimensional reduction      57 123
Dirac electron, coupled      121
Dirac electron, free      116
Dirac equation      98 411
Dirac magnetic monopole      45 63 150
Dirac matrices      98
Dirac operator      411
Dirac quantization condition      68
Dirac spinor      117
Direct sum of cochains      324
Domain with smooth boundary      289
Domain with smooth boundary, boundary      289
Domain with smooth boundary, exterior      289
Domain with smooth boundary, interior      289
Dominated Convergence Theorem      276
Donaldson theory      407
Einstein cylinder      204 405
Einstein — de Sitter spacetime      200 405
Electric charge      63
electric field      60 61
Electromagnetic theory      57
Elementary symmetric polynomials      360
Elevator experiment      194
Elliptic complex      207
Equation of structure      43
Equations of motion      56
Equivalent bundles      34
Equivariant map      49
Euler — Lagrange equations      56
Euler — Poincare characteristic      339
Events      57 187
Exact form      63 251
Exterior derivative      247 256
Exterior derivative, covariant      51 261
Exterior derivative, of real-valued 1-form      12
Exterior derivative, of real-valued function      9
Exterior differentiation      247
Exterior k-bundle      239
Fiber bundle      47
Finite action      56
Finite type      187
Flat connection      38 43
Forms      127 209
Forms, vector-valued      231
Frame bundle, linear      173
Frame bundle, oriented, orthonormal      184
Frame bundle, oriented, time oriented, orthonormal      197
Frame bundle, orthonormal      182
Frame field      177
Frame field, oriented, orthonormal      182
Frame, linear      172
Frame, moving      177
Frame, oriented, time oriented, orthonormal      197
Frame, orthonormal      181
Fundmental vector field      35
Future directed      196
G-bundle      28
G-bundle, trivial      29
Gauge equivalent      37
Gauge field      43 54
Gauge invariance      79 80
Gauge potential      36 54
Gauge principle      78
Gauge theories      53
Georgi — Glashow potential      126
Ground state      132
Hedgehog solution      144
Higgs fields      51 125
Higgs fields, homotopy classes of      155
Hodge dual      56 60 221
Hodge star operator      127 221 231 224
Holonomy group      38
Homogeneous polynomial      356
Hopf bundle, complex      30
Hopf bundle, quaternionic      31
Hopf invariant      346
Horizontal forms      257
Horizontal lift      264
Horizontal subspace      36
Imbedding      3
Immersion      3
Immersion at p      2
Index of a scmi-Riemannian metric      181
Induced orientation      291 292
Inner product      10
Instanton      41 46 136
Instanton, center      136
Instanton, number      138 387
Instanton, scale      136
Integral curve      7
Internal space      53
Intersection form      339
Isomorphic Lie algebras      15
Isomorphic Lie groups      14
Isotropy subgroup      26
j-cochain      390
j-simplex      390
j-simplex, face      390
j-simplex, support      390
Jacobi identity      15
k-dimensional submanifold      3
k-tensor      208
k-tensor, skew-symmetric      209 231
k-tensor, symmetric      209 232
k-tensor, vector-valued      230
Killing form      24
Klein — Gordon equation      75 94
Laplucian      416 423
Leaning lower of Pisa      193
Lebesguc integral      275
Lebesgue integrable      275
Lebesgue integrable form      278 280 281
Leibnitz Product Rule      2
Lens space      71
Lie algebra      15
Lie algebra, ideal in      24
Lie algebra, of a Lie group      16
Lie bracket      7 15
Lie bracket, trivial      15
Lie group, matrix      14
Lie group, semisimple      24
Light cone      189
Lightlike      189
Local field strength      43
Local gauge      36
Local triviality      28
Local trivialization      29
Locally finite      161
Lorentz group      82 180
Lorentz group, general      180
Lorentz group, proper      180
Lorentz group, proper, orthochronous      82 188
Lorentz invariance      80
Lorentz invariance of the Dirac equation      100
Lorentz metric      195
Lorentz transformation      191
Magnetic charge      46 63 69 387
Magnetic field      60 61
Magnetic monopole      see “Dirac magnetic monopole”
Map induced in cohomology      304
Matrix Lie group      14
Matter field      51 55
Maurer — Cartan equations      19
Maxwell’s equations      62 75
Mayer — Victoris sequence      315 318 324 325
Measurable form      277
Measurable function      274
Measurable set      273 277
Measure zero      272
Metric      11
Metric volume form      127
Metric, bi-invariant      25
Metric, left invariant      25
Metric, Lorentz      195
Metric, Riemannian      11 166
Metric, right invariant      25
Metric, semi-Riemannian      11 181
Minkowski inner product      57 189
Minkowski space      179
Minkowski spacetimc      57 179 405
Monopole number      152 153
Monotone Convergence Theorem      276
Moving frame      177
Multilinear map      208 355
Multilinear map, vector-valued      230
Natural connection, complex Hopf bundle      38 44
Natural connection, quaternionic Hopf bundle      40
Natural diffcrcntiable structures on real vector spaces      4
Nerve      390
Nonstandard diffcrcntiable structure on $\mathbb{R}$      4
Normalized generator      346
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