Table of Contents
Preface v
Introduction 1
1 Differential calculus on fibre bundles 5
1.1 Geometry of fibre bundles 5
1.1.1 Manifold morphisms 6
1.1.2 Fibred manifolds and fibre bundles 7
1.1.3 Vector and affine bundles 12
1.1.4 Vector fields, distributions and foliations 18
1.1.5 Exterior and tangent-valued forms 21
1.2 Jet manifolds 26
1.3 Connections on fibre bundles 29
1.3.1 Connections as tangent-valued forms 30
1.3.2 Connections as jet bundle sections 32
1.3.3 Curvature and torsion 34
1.3.4 Linear connections 36
1.3.5 Affine connections 38
1.3.6 Flat connections 39
1.3.7 Second order connections 41
1.4 Composite bundles 42
1.5 Higher order jet manifolds 46
1.6 Differential operators and equations 51
1.7 Infinite order jet formalism 54
2 Lagrangian field theory on fibre bundles 61
2.1 Variational bicomplex 61
2.2 Lagrangian symmetries 66
2.3 Gauge Symmetries 70
2.4 First order Lagrangian field theory 73
2.4.1 Cartan and Hamilton-De Donder equations 75
2.4.2 Lagrangian conservation laws 78
2.4.3 Gauge conservation laws. Superpotential 80
2.4.4 Non-regular quadratic Lagrangians 83
2.4.5 Reduced second order Lagrangians 87
2.4.6 Background fields 88
2.4.7 Variation Euler-Lagrange equation. Jacobi fields 90
2.5 Appendix. Cohomology of the variational bicomplex 92
3 Grassmann-graded Lagrangian field theory 99
3.1 Grassmann-graded algebraic calculus 99
3.2 Grassmann-graded differential calculus 104
3.3 Geometry of graded manifolds 107
3.4 Grassmann-graded variational bicomplex 115
3.5 Lagrangian theory of even and odd fields 120
3.6 Appendix. Cohomology of the Grassmann-graded variational bicomplex 125
4 Lagrangian BRSTtheory 129
4.1 Noether identities. The Koszul-Tate complex 130
4.2 Second Noether theorems in a general setting 140
4.3 BRST operator 147
4.4 BRST extended Lagrangian field theory 150
4.5 Appendix. Noether identities of differential operators 154
5 Gauge theory on principal bundles 165
5.1 Geometry of Lie groups 165
5.2 Bundles with structure groups 169
5.3 Principal bundles 171
5.4 Principal connections. Gauge fields 175
5.5 Canonical principal connection 179
5.6 Gauge transformations 181
5.7 Geometry of associated bundles. Matter fields 184
5.8 Yang-Mills gauge theory 188
5.8.1 Gauge field Lagrangian 188
5.8.2 Conservation laws 190
5.8.3 BRST extension 192
5.8.4 Matter field Lagrangian 194
5.9 Yang-Mills supergauge theory 196
5.10 Reduced structure. Higgs fields 198
5.10.1 Reduction of a structure group 198
5.10.2 Reduced subbundles 200
5.10.3 Reducible principal connections 202
5.10.4 Associated bundles. Matter and Higgs fields 203
5.10.5 Matter field Lagrangian 207
5.11 Appendix. Non-linear realization of Lie algebras 211
6 Gravitation theory on natural bundles 215
6.1 Natural bundles 215
6.2 Linear world connections 219
6.3 Lorentz reduced structure. Gravitational fields 223
6.4 Space-time structure 228
6.5 Gauge gravitation theory 232
6.6 Energy-momentum conservation law 236
6.7 Appendix. Affine world connections 238
7 Spinor fields 243
7.1 Clifford algebras and Dirac spinors 243
7.2 Dirac spinor structure 246
7.3 Universal spinor structure 252
7.4 Dirac fermion fields 258
8 Topological field theories 263
8.1 Topological characteristics of principal connections 263
8.1.1 Characteristic classes of principal connections 264
8.1.2 Flat principal connections 266
8.1.3 Chern classes of unitary principal connections 270
8.1.4 Characteristic classes of world connections 274
8.2 Chern-Simons topological field theory 278
8.3 Topological BF theory 283
8.4 Lagrangian theory of submanifolds 286
9 Covariant Hamiltonian field theory 293
9.1 Polysymplectic Hamiltonian formalism 293
9.2 Associated Hamiltonian and Lagrangian systems 298
9.3 Hamiltonian conservation laws 304
9.4 Quadratic Lagrangian and Hamiltonian systems 306
9.5 Example. Yang-Mills gauge theory 313
9.6 Variation Hamilton equations. Jacobi fields 316
10 Appendixes 319
10.1 Commutative algebra 319
10.2 Differential operators on modules 324
10.3 Homology and cohomology of complexes 327
10.4 Cohomology of groups 330
10.5 Cohomology of Lie algebras 333
10.6 Differential calculus over a commutative ring 334
10.7 Sheaf cohomology 337
10.8 Local-ringed spaces 346
10.9 Cohomology of smooth manifolds 348
10.10 Leafwise and fibrewise cohomology 354
Bibliography 359
Index 369
