Preface to the second edition
Preface
Contents
Conventions
I. REVIEW OF FUNDAMENTAL NOTIONS OF ANALYSIS
1.Graded algebras
2.Berezinian
3.Tensor product of algebras
4.Clifford algebras
5.Clifford algebra as a coset of the tensor algebra
6.Fierz identity
7.Pin and Spin groups
8.Weyl spinors, helicity operator; Majorana pinors, charge
conjugation
9.Representations of Spin(n, m), n+m odd
10.Dirac adjoint
11.Lie algebra of Pin(n, m) and Spin(n, m)
12.Compact spaces
13.Compactness in weak sr oology
14.Homotopy groups, general properties
15.Homotopy of topological groups
16.Spectrum of closed and self-adjoint linear operators
I. IFFERENTIAL CALCULUS ON BANACH SPACES
1.Supersmooth mappings
2.Berezin integration; Gaussian integrals
3.Noethers theorems I
4.Noethers theorems II
5.Invarianceof the equations of motion
6.String action
7.Stress-energy tensor; energy with respect to a timelike vector field
II. IFFERENTIABLE MANIFOLDS
1.Sheaves
2.Differentiable submanifolds
3.Subgroups of Lie groups. When are they Lie subgroups?
4.Cartan-Killing form on the Lie algebra g of a Lie group G
5.Direct and semidirect products of Lie groups and their Lie algebra
6.Homomorphisms and antihomomorphisms of a Lie algebra into
spaces of vector fields
7.Homogeneous spaces; symmetric spaces
8.Examples of homogeneous spaces, Stiefel and Grassmann manifolds
9.Abelian representations of nonabelian groups
10.Irreducibility and reducibility
11.Characters
12.Solvable Lie groups
13.Lie algebras of linear groups
14.Graded bundles
IV. INTEGRATION ON MANIFOLDS
1.Cohomology. Definitions and exercises
2.Obstruction to the construction of Spin and Pin bundles;
Stiefel-Whitney classes
3.Inequivalent spin structures
4.Cohomology of groups
5.Lifting a group action
6.Short exact sequence; Weyl Heisenberg group
……