Preface to the first edition |
Preface to the second edition |
Analysis over supernumbers / 1: |
Supernumbers and superanalytic functions / 1.1: |
Grassmann algebras |
Supernumbers |
c-numbers and a-numbers |
Superanalytic functions of supernumbers |
Integration of superanalytic functions of supernumbers |
Real supernumbers. Differentiable functions of real c-numbers and their integrals / 1.2: |
Complex conjugation |
Functions, distributions and integrals over R[subscript c] |
Fourier transforms over R[subscript c] |
Functions and integrals over R[subscript a] / 1.3: |
Basic definitions |
Fourier transforms over R[subscript a] |
Integrals over R[superscript n][subscript a] |
Supervector spaces / 1.4: |
Definition |
Bases |
Pure bases |
Pure real bases |
Standard bases |
Linear transformations, supertranspositions and dual supervector spaces / 1.5: |
Change of basis |
Shifting indices. The supertranspose |
Extensions of the supertransposition rules |
Dual supervector spaces |
Dual bases |
Further index-shifting conventions |
The supertrace and the superdeterminant / 1.6: |
The supertrace |
The superdeterminant |
The superdeterminant in special cases |
The superdeterminant in the general case |
Integration over R[superscript m][subscript c] [times] R[superscript n][subscript a] / 1.7: |
Notation |
Integration |
Homogeneous linear transformations of the a-number coordinates |
Homogeneous linear transformations of all the coordinates |
Nonlinear transformations |
Gaussian integrals over R[superscript m][subscript c] [times] R[superscript n][subscript a] |
Exercises |
Comments on chapter 1 |
Supermanifolds / 2: |
Definition and structure of supermanifolds / 2.1: |
Topology of R[superscript m][subscript c] [times] R[superscript n][subscript a]. Differentiable mappings |
Supermanifolds, charts and atlases |
Scalar fields and supercurves |
Diffeomorphisms and embeddings |
Ordinary manifolds. Skeleton and body of a supermanifold |
Projectively Hausdorff, compact, paracompact and orientable supermanifolds. Realizations of the body |
Supervector structures on supermanifolds / 2.2: |
Scalar fields as supervectors |
Contravariant vector fields |
Alternative presentation of contravariant vector fields |
Components |
Tangent spaces |
Tangents to supercurves |
Super Lie brackets, local frames and covariant vector fields / 2.3: |
Supercommutators and antisupercommutators |
A matter of notation |
The super Lie bracket |
Local frames |
Super Lie brackets of local frame fields |
Covariant vector fields |
Differentials |
Tensor fields / 2.4: |
Tensors at a point |
The supervector space T[superscript r] [subscript s](p) |
Tensor products |
Tensor and multitensor fields |
Index-shifting conventions. Contractions |
The unit tensor field |
The Lie derivative / 2.5: |
Explicit forms |
Lie derivations as supervectors |
The derivative mapping |
Integral supercurves. Congruences |
Dragging of tensor fields |
Forms / 2.6: |
The exterior product |
Bases for forms |
Derivations of forms |
The exterior derivative |
The inner product |
Connections / 2.7: |
The connection components |
Multiple covariant derivatives. The torsion |
The Riemann tensor field |
The super Bianchi identity |
Parallel transport. Supergeodesics |
Distant parallelism |
Riemannian supermanifolds / 2.8: |
The metric tensor field |
Canonical form of the metric tensor at a point |
Canonical or orthosymplectic bases |
Riemannian connections |
The curvature tensor field |
The Ricci tensor field |
Flat Riemannian supermanifolds |
Conformally related Riemannian supermanifolds. The Weyl tensor field |
Conformally flat Riemannian supermanifolds |
Killing vector fields |
Conformal Killing vector fields |
The global conformal group |
Integration over supermanifolds / 2.9: |
Integration over R[superscript m][subscript c] [times] R[superscript n][subscript a]. Measure functions |
Locally finite atlases and partitions of unity |
Integration over paracompact orientable supermanifolds |
Integration over Riemannian supermanifolds |
Integrals of total divergences |
The compact case |
An example |
Comments on chapter 2 |
Super Lie groups. General theory / 3: |
Definition and structure of super Lie groups / 3.1: |
Canonical diffeomorphisms |
Left- and right-invariant vector fields |
Left- and right-invariant local frame fields |
Left- and right-invariant congruences |
One-parameter Abelian subgroups |
The exponential mapping. Canonical coordinates |
The super Lie algebra |
The structure constants |
The right and left auxiliary functions |
Identities satisfied by the auxiliary functions |
Construction of a super Lie group from its super Lie algebra |
Realizations of super Lie groups / 3.2: |
Orbits |
Transitive realizations |
Isotropy subgroups |
Coset spaces |
Killing flows |
Properties of the coordinate components of the Q[subscript a] |
A special canonical coordinate system |
Coordinates for the coset spaces |
Classification of transitive realizations |
Matrix representations of super Lie groups |
Contragredient representations |
Inner automorphisms. The adjoint representation |
Matrix representations of the super Lie algebra |
Geometry of coset spaces / 3.3: |
Invariant tensor fields |
Differential equations for geometrical structures |
Integrability of the differential equations |
A special coordinate system |
Condition for the existence of a group-invariant measure function |
Condition for the existence of a group-invariant metric tensor field |
Condition for the existence of a group-invariant connection |
Solutions of the differential equations |
Geometry of the group supermanifold |
Identity of the left- and right-invariant connections |
Parallelism at a distance in the group supermanifold |
Integration over the group |
A special class of super Lie groups |
Comments on chapter 3 |
Super Lie groups. Examples / 4: |
Construction of super Lie algebras and super Lie groups / 4.1: |
Properties of the structure constants |
Conventional super Lie groups, Z[subscript 2]-graded algebras |
Unconventional super Lie groups |
Structure of conventional super Lie Groups. The extending representation |
Construction of a class of super Lie algebras |
The classical super Lie groups / 4.2: |
The group GL (m, n) |
The group SL (m, n) |
The group SL (m, m)/GL (1, 0) |
The orthosymplectic group OSp (m, n) |
The Kac notation |
The group P(m) |
The group Q(m) |
The exceptional simple super Lie groups / 4.3: |
The groups D(2, 1, [alpha]) |
The group F(4) |
The structure of F(4) |
Pseudorepresentation of F(4) |
The group G(3) |
The structure of G[subscript 2] |
The structure of G(3) |
Pseudorepresentation of G(3) |
Super Lie groups of basic importance in physics / 4.4: |
The super de Sitter group |
The super Poincare group |
The coset space: super Poincare group/SO(1, 3) |
Killing flows and invariant connections |
Riemannian geometry of the coset space |
The super Lorentz group |
The Cartan super Lie groups / 4.5: |
The diffeomorphism group Diff(M) |
The group SDiff(M, [mu]) |
The canonical transformation group Can(M, [omega]) |
The group of contact transformations |
The case m = 0 |
The group W(n) |
The groups S(n) and S(n) |
The groups H(n) and H(n) |
Comments on chapter 4 |
Selected applications of supermanifold theory / 5: |
Superclassical dynamical systems / 5.1: |
Configuration spaces |
Supermanifolds as configuration spaces |
Space of histories |
The action functional and the dynamical equations |
Infinitesimal disturbances and Green's functions |
Reciprocity relations |
The Peierls bracket |
Peierls bracket identities |
Super Hilbert spaces / 5.2: |
Linear operators |
Physical observables |
Quantum systems / 5.3: |
Transition to the quantum theory |
The Schwinger variational principle |
External sources |
Chronologically ordered form of the operator dynamical equations |
The Feynman functional integral |
A simple Fermi system / 5.4: |
Action functional and Green's functions |
Eigenvectors of x |
The energy |
A pure basis |
An alternative representation |
The functional integral representation of [x", t"|x', t'] |
Evaluation of the functional integral |
The average superclassical trajectory |
Propagator for x[subscript av](t) |
The Fermi oscillator / 5.5: |
Mode functions and Hamiltonian |
Basic supervectors |
Eigenvectors of x[subscript 1] and x[subscript 2]. Choice of pure basis |
Coherent states |
The functional integral representation of [a"*, t" |
Direct evaluation of the functional integral |
The importance of endpoint contributions |
The stationary trajectory as a matrix element |
The Feynman propagator |
The Bose oscillator / 5.6: |
Energy eigenvectors |
Hamilton-Jacobi theory |
The amplitude [x', t'|x', t'] and its functional integral representation |
The functional-integral representation of [a"*, t" |
The stationary path between coherent states |
Energy eigenfunctions |
Bose-Fermi supersymmetry / 5.7: |
The simplest model |
New conserved quantities |
The Bose-Fermi supersymmetry group |
Eigenvectors of Q[subscript 1] and Q[subscript 2] |
The supersymmetry group as a transformation group |
Auxiliary variable |
Nonlinear Bose-Fermi supersymmetry |
The supersymmetry group |
The energy spectrum |
Spontaneously broken supersymmetry |
Comments on chapter 5 |
Applications involving topology / 6: |
Nontrivial configuration spaces / 6.1: |
Standard canonical systems |
Green's functions |
Equivalence of Peierls and Poisson brackets |
Quantization / 6.2: |
Problems with the naive quantization rule |
Operator-valued forms. The projection m-form |
The position operator |
Vector operators |
The momentum operator |
Restriction to a local chart |
Lack of uniqueness of the momentum operator |
Overlapping charts. Transformation of coordinates |
The position representation |
The momentum operator in the position representation |
The Schrodinger equation |
The position representation of the projection m-form |
Curved configuration spaces / 6.3: |
A special class of systems |
Covariant variation |
Covariant differentiation with respect to t |
The dynamical equations |
Covariant functional differentiation |
The Feynman functional integral and its meaning / 6.4: |
Formal computation of det G[superscript +][x] |
The functional integral |
Normalization |
Ambiguity in the functional integral |
Homotopy |
Homotopy mesh |
The total amplitude |
Change of homotopy mesh |
The role of homology |
The universal covering space |
The total amplitude revisited |
The Hamiltonian operator: a nonlattice derivation / 6.5: |
Integration over phase space |
Evaluation of the chronologically ordered Hamiltonian |
Approximate evaluation of the path integral / 6.6: |
Brief review of Hamilton-Jacobi theory |
The Van Vleck-Morette determinant |
Jacobi fields and the Green's function for the trajectory x[subscript c] |
Determinantal relations |
The loop expansion |
The WKB approximation |
The heat kernel expansion |
Role of the two-loop term in the independent verification of (6.5.25) |
New variables |
Computation of the two-loop term |
Supersymmetry and the Euler-Poincare characteristic / 6.7: |
Inclusion of a-type dynamical variables |
Green's functions and Peierls brackets |
Energy and supersymmetry group |
Basis supervectors |
Differential representation of operators |
The Euler-Poincare characteristic |
Functional integral for the coherent-state transition amplitude |
The Chern-Gauss-Bonnet formula |
Comments on chapter 6 |
References |
Index |