Giachetta Giovanni, Mangiarotti Luigi, Sardanashvily Gennadi. Advanced Classical Field Theory

World Scientific Publishing, 2009. — 393 p.Contemporary quantum field theory is mainly developed as quantization of classical fields. Classical field theory thus is a necessary step towards quantum field theory. This book provides an exhaust mathematical foundation of Lagrangian classical field theory and its BRST extension for the purpose of quantization.Differential calculus on fibre bundles
Geometry of fibre bundles
Manifold morphisms
Fibred manifolds and fibre bundles
Vector and affine bundles
Vector fields, distributions and foliations
Exterior and tangent-valued forms
Jet manifolds
Connections on fibre bundles
Connections as tangent-valued forms
Connections as jet bundle sections
Curvature and torsion
Linear connections
Affine connections
Flat connections
Second order connections
Composite bundles
Higher order jet manifolds
Differential operators and equations
Infinite order jet formalism
Lagrangian field theory on fibre bundles
Variational bicomplex
Grassmann-graded Lagrangian field theory
Grassmann-graded algebraic calculus
Grassmann-graded differential calculus
Geometry of graded manifolds
Grassmann-graded variational bicomplex
Lagrangian theory of even and odd fields
Cohomology of the Grassmann-graded variational bicomplex
Noether identities. The Koszul–Tate complex
Second Noether theorems in a general setting
BRST operator
BRST extended Lagrangian field theory
Noether identities of differential operators
Geometry of Lie groups
Bundles with structure groups
Principal bundles
Principal connections. Gauge fields
Canonical principal connection
Gauge transformations
Geometry of associated bundles. Matter fields
Yang–Mills gauge theory
Gauge field Lagrangian
Gauge theory on principal bundles
Lagrangian BRST theory
Lagrangian symmetries
Gauge symmetries
First order Lagrangian field theory
Cartan and Hamilton–De Donder equations
Lagrangian conservation laws
Gauge conservation laws. Superpotential
Non-regular quadratic Lagrangians
Reduced second order Lagrangians
Background fields
Variation Euler-Lagrange equation. Jacobi fields
Cohomology of the variational bicomplex
Natural bundles
Linear world connections
Lorentz reduced structure. Gravitational
Space-time structure
Gauge gravitation theory
Energy-momentum conservation law
Affine world connections
Clifford algebras and Dirac spinors
Dirac spinor structure
Universal spinor structure
Dirac fermion fields
Topological characteristics of principal connections
Characteristic classes of principal connections
Flat principal connections
Chern classes of unitary principal connections
Characteristic classes of world connections
Chern–Simons topological field theory
Topological BF theory
Lagrangian theory of submanifolds
Covariant Hamiltonian field theory
Topological field theories
Spinor fields
Conservation laws
BRST extension
Matter field Lagrangian
Yang–Mills supergauge theory
Reduced structure. Higgs fields
Reduction of a structure group
Reduced subbundles
Reducible principal connections
Associated bundles. Matter and Higgs fields
Matter field Lagrangian
Non-linear realization of Lie algebras
Gravitation theory on natural bundles
Polysymplectic Hamiltonian formalism
Associated Hamiltonian and Lagrangian systems
Hamiltonian conservation laws
Quadratic Lagrangian and Hamiltonian systems
Example. Yang–Mills gauge theory
Variation Hamilton equations. Jacobi fields
Commutative algebra
Differential operators on modules
Homology and cohomology of complexes
Cohomology of groups
Cohomology of Lie algebras
Differential calculus over a commutative ring
Sheaf cohomology
Local-ringed spaces
Cohomology of smooth manifolds
Leafwise and fibrewise cohomology