Contents -- Preface -- Introduction -- 1. Differential calculus on fibre bundles -- 1.1 Geometry of fibre bundles -- 1.1.1 Manifold morphisms -- 1.1.2 Fibred manifolds and fibre bundles -- 1.1.3 Vector and affine bundles -- 1.1.4 Vector fields, distributions and foliations -- 1.1.5 Exterior and tangent-valued forms -- 1.2 Jet manifolds -- 1.3 Connections on fibre bundles -- 1.3.1 Connections as tangent-valued forms -- 1.3.2 Connections as jet bundle sections -- 1.3.3 Curvature and torsion -- 1.3.4 Linear connections -- 1.3.5 Affine connections -- 1.3.6 Flat connections -- 1.3.7 Second order connections -- 1.4 Composite bundles -- 1.5 Higher order jet manifolds -- 1.6 Differential operators and equations -- 1.7 Infinite order jet formalism -- 2. Lagrangian field theory on fibre bundles -- 2.1 Variational bicomplex -- 2.2 Lagrangian symmetries -- 2.3 Gauge symmetries -- 2.4 First order Lagrangian field theory -- 2.4.1 Cartan and Hamilton-De Donder equations -- 2.4.2 Lagrangian conservation laws -- 2.4.3 Gauge conservation laws. Superpotential -- 2.4.4 Non-regular quadratic Lagrangians -- 2.4.5 Reduced second order Lagrangians -- 2.4.6 Background fields -- 2.4.7 Variation Euler-Lagrange equation. Jacobi fields -- 2.5 Appendix. Cohomology of the variational bicomplex -- 3. Grassmann-graded Lagrangian field theory -- 3.1 Grassmann-graded algebraic calculus -- 3.2 Grassmann-graded differential calculus -- 3.3 Geometry of graded manifolds -- 3.4 Grassmann-graded variational bicomplex -- 3.5 Lagrangian theory of even and odd fields -- 3.6 Appendix. Cohomology of the Grassmann-graded variational bicomplex -- 4. Lagrangian BRST theory -- 4.1 Noether identities. The Koszul-Tate complex -- 4.2 Second Noether theorems in a general setting -- 4.3 BRST operator -- 4.4 BRST extended Lagrangian field theory.

4.5 Appendix. Noether identities of di erential operators -- 5. Gauge theory on principal bundles -- 5.1 Geometry of Lie groups -- 5.2 Bundles with structure groups -- 5.3 Principal bundles -- 5.4 Principal connections. Gauge fields -- 5.5 Canonical principal connection -- 5.6 Gauge transformations -- 5.7 Geometry of associated bundles. Matter fields -- 5.8 Yang-Mills gauge theory -- 5.8.1 Gauge field Lagrangian -- 5.8.2 Conservation laws -- 5.8.3 BRST extension -- 5.8.4 Matter field Lagrangian -- 5.9 Yang-Mills supergauge theory -- 5.10 Reduced structure. Higgs fields -- 5.10.1 Reduction of a structure group -- 5.10.2 Reduced subbundles -- 5.10.3 Reducible principal connections -- 5.10.4 Associated bundles. Matter and Higgs fields -- 5.10.5 Matter field Lagrangian -- 5.11 Appendix. Non-linear realization of Lie algebras -- 6. Gravitation theory on natural bundles -- 6.1 Natural bundles -- 6.2 Linear world connections -- 6.3 Lorentz reduced structure. Gravitational field -- 6.4 Space-time structure -- 6.5 Gauge gravitation theory -- 6.6 Energy-momentum conservation law -- 6.7 Appendix. A ne world connections -- 7. Spinor fields -- 7.1 Clifford algebras and Dirac spinors -- 7.2 Dirac spinor structure -- 7.3 Universal spinor structure -- 7.4 Dirac fermion fields -- 8. Topological field theories -- 8.1 Topological characteristics of principal connections -- 8.1.1 Characteristic classes of principal connections -- 8.1.2 Flat principal connections -- 8.1.3 Chern classes of unitary principal connections -- 8.1.4 Characteristic classes of world connections -- 8.2 Chern-Simons topological field theory -- 8.3 Topological BF theory -- 8.4 Lagrangian theory of submanifolds -- 9. Covariant Hamiltonian field theory -- 9.1 Polysymplectic Hamiltonian formalism -- 9.2 Associated Hamiltonian and Lagrangian systems -- 9.3 Hamiltonian conservation laws.

9.4 Quadratic Lagrangian and Hamiltonian systems -- 9.5 Example. Yang-Mills gauge theory -- 9.6 Variation Hamilton equations. Jacobi fields -- 10. Appendixes -- 10.1 Commutative algebra -- 10.2 Differential operators on modules -- 10.3 Homology and cohomology of complexes -- 10.4 Cohomology of groups -- 10.5 Cohomology of Lie algebras -- 10.6 Differential calculus over a commutative ring -- 10.7 Sheaf cohomology -- 10.8 Local-ringed spaces -- 10.9 Cohomology of smooth manifolds -- 10.10 Leafwise and fibrewise cohomology -- Bibliography -- Index.