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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.

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Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2000

Количество страниц: 459

Добавлена в каталог: 17.11.2013

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Предметный указатель

$(U_{S},\varphi_{S})$ , $(U_{N},\varphi_{N})$ , stereographic projection charts      4
$1^{st}$ 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
$A(\omega, \phi)$ , action functional      55
$A\otimes_{\rho} B$ , $\rho$ -tensor product      232
$A^{#}$ , fundamental vector field      35
$B^{j}(\mathcal{U},\mathds{Z}_{2})$ , Cech j-coboundaries of $\mathcal{U}$       393
$C^{j}(\mathcal{U},\mathds{Z}_{2})$ , Cech j-cochain group      390 391
$C^{\infty}$ -manifold      1
$C^{\infty}$ -map      2
$C^{\infty}$ -related      1
$C^{\infty}(X)$ , real-valued $C^{\infty}$ -functions on X      1
$c_{1}(P)$ , $1^{st}$ Chern class of P      352
$c_{k}(P)$ , $k^{th}$ Chern class of P      377
$df_{p} = df(p)$       9
$d\omega$ , exterior derivative of $\omega$       247 256
$d^{k}$ , exterior differentiation operator      247
$D^{n}$ , unit disc in $\mathds{R}^{n}$       289
$d^{\omega}$ , covariant exterior derivative      51
$d^{\omega}\varphi$ , covariant exterior derivative of $\varphi$       261
$E(A, \psi)$ , Seiberg — Witten action      415
$f^{#}$ , map induced in cohomology      303
$f_{*p}$ , derivative of f at p      2
$g \cdot p$ , left action of g on p      26
$G \hookrightarrow P \frac{\mathcal{P}}{\rightarrow} X$ , principal G-bundle over X      29
$GL(n, \mathds{R}) \hookrightarrow L(X) \frac{\mathcal{P}_{L}}{\rightarrow} X$ , linear frame bundle      173
$Hor_{p}(P)$ , horizontal vectors at p in P      36
$H^{j}(X,\mathds{Z}_{2})$ , $j^{th}$ Cech cohomology group of X      397
$H^{j}(\mathcal{U},\mathds{Z}_{2})$ , $j^{th}$ Cech cohomology group of $\mathcal{U}$       393
$H^{k}(C^{*})$ , $k^{th}$ cohomology group of $C^{*}$       319
$H^{k}_{de\ R}(X)$ , $k^{th}$ de Rham cohomology group of X      63 298
$I^{k}(G)=S^{k}_{ad}(\mathcal{G})$       358
$N(\mathcal{A},\phi)$ , monopole number      152
$N(\mathcal{U})$ , nerve of $\mathcal{U}$       390
$O(k,n-k) \hookrightarrow F(X) \frac{\mathcal{P}_{F}}{\rightarrow} X$ , orthonormal frame bundle of X      182
$p \cdot q$ , right action of g on p      26
$P(\mathcal{V})$ , direct sum of the $P^{k}(\mathcal{V})$       356
$P\times_{g}F$ , associated bundle      47
$P^{k}(\mathcal{V})$ , homogeneous polynomials of degree k on $\mathcal{V}$       356
$P^{k}_{\rho}(\mathcal{V})$ , $\rho$ -invariant subspace of $P^{k}(\mathcal{V})$       358
$P_{\rho}(\mathcal{V})$ , direct sum of the $P^{k}(\mathcal{V})$       358
$Q_{X}$ , intersection form on X      339
$S(\mathcal{V})$ , direct sum of the $S^{k}(\mathcal{V})$       355
$SO(k,n-k) \hookrightarrow F_{+}(X) \frac{\mathcal{P}_{F}}{\rightarrow} X$ , oriented, orthonormal frame bundle of X      184
$Spine^{c} (4)$       408
$spin^{c}$ -structure      410
$Spin^{c}(4) \hookrightarrow S^{c}(X)\frac{\mathcal{P}_{c}}{\rightarrow} X$ , $spin^{c}$ -structure      410
$S^{k}(\mathcal{V})$ , complex-valued symmetric k-multilinear maps on $\mathcal{V}$       355
$S^{k}_{\rho}(\mathcal{V})$ , $\rho$ -invariant subspace of $S^{k}(\mathcal{V})$       358
$S_{0},S_{1}...S_{n}$ , elementary symmetric polynomials      360
$S_{\rho}(\mathcal{V})$ , direct sum of the $S^{k}(\mathcal{V})$       358
$T^{*}$ , pullback      208 230 240
$T^{*}(X)$ , cotangent bundle of X      179
$T_{p}^{*}(X)$ , cotangent space at p      9
$Vert_{p}(P)$ , vertical vectors at p in P      35
$V^{H}$ , horizontal part of V      36
$V^{V}$ , vertical part of V      36
$Z^{j}(\mathcal{U},\mathds{Z}_{2})$ , Cech j-cocycles of $\mathcal{U}$       393
$[e_{1},...,e_{n}]$ , orientation class of ${e_{1},...,e_{n}}$       8
$[\alpha,\beta]$       234
$\alpha\dot{\wedge}\beta$       263
$\alpha\wedge_{\rho}\beta$ , $\rho$ -wedge product      232
$\alpha^{'}(t_{0})$ , velocity vector to $\alpha$ at $t_{0}$       2
$\chi_{M}$ , characteristic function of M      275
$\delta^{j}$ , coboundary operator      391
$\eta$ , Minkowski matrix      58
$\eta_{\alpha\beta}$ , Minkowski metric components      57
$\gamma^{0}$ , $\gamma^{1}$ , $\gamma^{2}$ , $\gamma^{3}$ , Dirac matrices      96
$\Lambda^{*}(X)$ , cochain of forms on X      320
$\Lambda^{k}(E)$ , k-forms on E      209
$\Lambda^{k}(E,\mathcal{V})$ , k-forms with values in $\mathcal{V}$       231
$\Lambda^{k}(X)$ , real-valued k-forms on X      127 238
$\Lambda^{k}(X,\mathcal{V})$ , $\mathcal{V}$ -valued differential k-forms on X      257
$\Lambda^{k}_{\rho}(P,\mathcal{V})$ , tensorial forms of type $\rho$ on P      257
$\langle\ ,\ \rangle_{k}$ , inner product on $\mathds{R}^{k, n-k}$       179
$\langle\alpha,\beta\rangle$ , inner product on forms      127
$\mathcal{A}$ , gauge potential      36 54
$\mathcal{A}_{\lambda, n}$ , BPST potential      42 163
$\mathcal{C}$ , Einstein cylinder      204
$\mathcal{D}$ , deSitter spacetime      202
$\mathcal{E}$ , Einstein — deSitter spacetime      200
$\mathcal{F}$ , gauge field strength      43 54 163
$\mathcal{GL}(n, \mathds{F})$       6 17
$\mathcal{G}$ , Lie algebra of the Lie group G      16
$\mathcal{J}^{k}(E)$ , multilinear maps on E      208
$\mathcal{L}$ , Lorentz group      82 180
$\mathcal{L}^{\uparrow}_{+}$ , proper, orthochronous Lorentz group      82 188
$\mathcal{L}_{+}^{\uparrow} \hookrightarrow \mathcal{L}(X) \frac{\mathcal{P}_{\mathcal{L}}}{\rightarrow} X$ , oriented, time oriented, orthonormal frame bundle of X      197
$\mathcal{X}(X)$ , smooth vector fields on X      7
$\mathcal{X}^{*}(X)$ , smooth 1-forms on X      9
$\mathcal{YM}$ , Yang — Mills action      46 135
$\mathds{D}$ , Dirac operator      94
$\mathds{FP}^{n-1}$ , projective spaces      5
$\mathds{R}^{1,3}$ , Minkowski spacetime      57 179
$\mathds{R}^{k, n-k}$       179
$\mathds{R}^{n}_{+}$       289
$\nabla_{A}$ , covariant derivative      411
$\not{D}_{A}$ , Dirac operator      411
$\omega_{1}(X)$ , $1^{st}$ Stiefel — Whitney class of X      397
$\omega_{2}(X)$ , $2^{nd}$ Stiefel — Whitney class of X      401
$\omega_{\lambda, n}$ , BPST connection      42 136
$\otimes$ , tensor product      10 209
$\partial$ D, boundary of D      289
$\rho$ , left action      26
$\rho$ -tensor product      232
$\rho$ -wedge product      232
$\rho_{g}$ , left action by g      26
$\sigma$ , right action      25 28
$\sigma$ -algebra      272
$\sigma_{1}$ , $\sigma_{2}$ , $\sigma_{3}$ , Pauli spin matrices      18
$\sigma_{g}$ , right action by g      26
$\sigma_{p}$ , left action on p      35
$\Theta(V)=\Theta V$       9
$\varepsilon_{ijk}$ , $\varepsilon_{\alpha\beta\gamma\delta}$ , Levi-Civita symbols      59
$\varepsilon_{j_{1}...j_{n}}$ , Levi-Civita symbol      227
$\varphi\times\psi$       6
$\wedge$ , wedge product      11 210
$\wedge_{\rho}$ , $\rho$ -wedge product      13 232
$^{\sigma}P$       360
$|\sigma|$ , support of the simplex $\sigma$       390
1-form      9
1-form, left-invariant      19
1-form, real-valued      9
1-form, restriction      10
2-valued representation      88
< , >, inner product on $\mathds{F}^{n}$       5
a.e., almost everywhere      275
Action functional      55
Action functional, Seiberg — Witten      415
Action, effective      26
Action, free      26
Action, left      26
Action, right      25
Action, transitive      26
ad P, adjoint bundle      49
ad(G)-invariant      24
Adjoint bundle      49
Adjoint representation      24
Admissible basis      189
Algebraic homotopy      254 305 320
Almost everywhere      275
Alt(A)      211
Anti-self-dual (ASD)      135
Antiparticle      75
Antipodal map      343
ASD, anti-self-dual      135

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
C*, cochain complex      318
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 — Weil homomorphism      377
Chiral representation      98
Cl(4), Clifford algebra of $\mathds{R}^{4}$       408
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 $\mathds{R}^{1,3}$ , null      199
Coordinates on $\mathds{R}^{1,3}$ , 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
deg(f), degree of f      340
Degree      340
Derivation      7
Derivative      2
Determinant line bundle      410
df, exterior derivative of f      9
Diffeomorphic      2
Diffeomorphism      2
Differentiable structure      1
Differential form      238 239
Differential form, vector-valued      245
Differentiate manifold      1
dim X, dimension of the manifold X      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
dm, Lebesgue measure on $\mathds{R}^{n}$       275
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
exp(A) = $e^{A}$       18
Exterior derivative      247 256
Exterior derivative of real-valued 1-form      12
Exterior derivative of real-valued function      9
Exterior derivative, covariant      51 261
Exterior differentiation      247
Exterior k-bundle      239
F(X), orthonormal frames on X      181
F*, pullback by F      9
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

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