Supermanifolds

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Paperback | 27 July 1992 | Edition Number 2

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428 Pages

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Revised

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22.86 x 15.24 x 2.41

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This is an updated and expanded second edition of a successful and well-reviewed text presenting a detailed exposition of the modern theory of supermanifolds, including a rigorous account of the super-analogs of all the basic structures of ordinary manifold theory. The exposition opens with the theory of analysis over supernumbers (Grassman variables), Berezin integration, supervector spaces and the superdeterminant. This basic material is then applied to the theory of supermanifolds, with an account of super-analogs of Lie derivatives, connections, metric, curvature, geodesics, Killing flows, conformal groups, etc. The book goes on to discuss the theory of super Lie groups, super Lie algebras, and invariant geometrical structures on coset spaces. Complete descriptions are given of all the simple super Lie groups. The book then turns to applications. Chapter 5 contains an account of the Peierals bracket for superclassical dynamical systems, super Hilbert spaces, path integration for fermionic quantum systems, and simple models of Boseâ"Fermi supersymmetry. The sixth and final chapter, which is new in this revised edition, examines dynamical systems for which the topology of the configuration supermanifold is important. A concise but complete account is given of the pathintegral derivation of the Chernâ"Gaussâ"Bonnet formula for the Eulerâ"Poincar© characteristic of an ordinary manifold, which is based on a simple extension of a point particle moving freely in this manifold to a supersymmetric dynamical system moving in an associated supermanifold. Many exercises are included to complement the text.

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'Supermanifolds is destined to become the standard work for all serious study of super-symmetric theories of physics.' Nature

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Non-Fiction Science Physics Mathematical Physics Mathematics Geometry Differential & Riemannian Geometry Topology Algebraic Topology

Preface to the first edition	p. xvii
Preface to the second edition	p. xviii
Analysis over supernumbers	p. 1
Supernumbers and superanalytic functions	p. 1
Grassmann algebras	p. 1
Supernumbers	p. 1
c-numbers and a-numbers	p. 2
Superanalytic functions of supernumbers	p. 3
Integration of superanalytic functions of supernumbers	p. 5
Real supernumbers. Differentiable functions of real c-numbers and their integrals	p. 5
Complex conjugation	p. 5
Functions, distributions and integrals over R[subscript c]	p. 6
Fourier transforms over R[subscript c]	p. 8
Functions and integrals over R[subscript a]	p. 8
Basic definitions	p. 8
Fourier transforms over R[subscript a]	p. 10
Integrals over R[superscript n][subscript a]	p. 12
Supervector spaces	p. 14
Definition	p. 14
Bases	p. 16
Pure bases	p. 17
Pure real bases	p. 20
Standard bases	p. 24
Linear transformations, supertranspositions and dual supervector spaces	p. 24
Change of basis	p. 24
Shifting indices. The supertranspose	p. 25
Extensions of the supertransposition rules	p. 27
Dual supervector spaces	p. 28
Dual bases	p. 30
Further index-shifting conventions	p. 31
The supertrace and the superdeterminant	p. 33
The supertrace	p. 33
The superdeterminant	p. 34
The superdeterminant in special cases	p. 35
The superdeterminant in the general case	p. 36
Integration over R[superscript m][subscript c] [times] R[superscript n][subscript a]	p. 37
Notation	p. 37
Integration	p. 38
Homogeneous linear transformations of the a-number coordinates	p. 39
Homogeneous linear transformations of all the coordinates	p. 40
Nonlinear transformations	p. 41
Gaussian integrals over R[superscript m][subscript c] [times] R[superscript n][subscript a]	p. 44
Exercises	p. 46
Comments on chapter 1	p. 49
Supermanifolds	p. 50
Definition and structure of supermanifolds	p. 50
Topology of R[superscript m][subscript c] [times] R[superscript n][subscript a]. Differentiable mappings	p. 50
Supermanifolds, charts and atlases	p. 52
Scalar fields and supercurves	p. 52
Diffeomorphisms and embeddings	p. 53
Ordinary manifolds. Skeleton and body of a supermanifold	p. 54
Projectively Hausdorff, compact, paracompact and orientable supermanifolds. Realizations of the body	p. 56
Supervector structures on supermanifolds	p. 57
Scalar fields as supervectors	p. 57
Contravariant vector fields	p. 58
Alternative presentation of contravariant vector fields	p. 60
Components	p. 61
Tangent spaces	p. 62
Tangents to supercurves	p. 63
Super Lie brackets, local frames and covariant vector fields	p. 64
Supercommutators and antisupercommutators	p. 64
A matter of notation	p. 65
The super Lie bracket	p. 66
Local frames	p. 67
Super Lie brackets of local frame fields	p. 69
Covariant vector fields	p. 69
Differentials	p. 70
Tensor fields	p. 72
Tensors at a point	p. 72
The supervector space T[superscript r] [subscript s](p)	p. 73
Tensor products	p. 74
Tensor and multitensor fields	p. 73
Index-shifting conventions. Contractions	p. 76
The unit tensor field	p. 77
The Lie derivative	p. 78
Definition	p. 78
Explicit forms	p. 78
Lie derivations as supervectors	p. 79
The derivative mapping	p. 81
Integral supercurves. Congruences	p. 81
Dragging of tensor fields	p. 82
Forms	p. 83
Definition	p. 83
The exterior product	p. 84
Bases for forms	p. 85
Derivations of forms	p. 85
The exterior derivative	p. 86
The inner product	p. 87
Connections	p. 88
Definition	p. 88
The connection components	p. 89
Explicit forms	p. 90
Multiple covariant derivatives. The torsion	p. 90
The Riemann tensor field	p. 91
The super Bianchi identity	p. 93
Parallel transport. Supergeodesics	p. 93
Distant parallelism	p. 96
Riemannian supermanifolds	p. 97
The metric tensor field	p. 97
Canonical form of the metric tensor at a point	p. 98
Canonical or orthosymplectic bases	p. 99
Riemannian connections	p. 100
The curvature tensor field	p. 101
The Ricci tensor field	p. 102
Flat Riemannian supermanifolds	p. 103
Conformally related Riemannian supermanifolds. The Weyl tensor field	p. 103
Conformally flat Riemannian supermanifolds	p. 104
Killing vector fields	p. 106
Conformal Killing vector fields	p. 106
The global conformal group	p. 107
Integration over supermanifolds	p. 110
Integration over R[superscript m][subscript c] [times] R[superscript n][subscript a]. Measure functions	p. 110
Locally finite atlases and partitions of unity	p. 110
Integration over paracompact orientable supermanifolds	p. 112
Integration over Riemannian supermanifolds	p. 112
Integrals of total divergences	p. 113
The compact case	p. 113
An example	p. 115
Exercises	p. 116
Comments on chapter 2	p. 121
Super Lie groups. General theory	p. 123
Definition and structure of super Lie groups	p. 123
Definition	p. 123
Canonical diffeomorphisms	p. 124
Left- and right-invariant vector fields	p. 125
Left- and right-invariant local frame fields	p. 125
Left- and right-invariant congruences	p. 126
One-parameter Abelian subgroups	p. 127
The exponential mapping. Canonical coordinates	p. 129
The super Lie algebra	p. 130
The structure constants	p. 131
The right and left auxiliary functions	p. 132
Identities satisfied by the auxiliary functions	p. 134
Construction of a super Lie group from its super Lie algebra	p. 135
Realizations of super Lie groups	p. 137
Definition	p. 137
Orbits	p. 137
Transitive realizations	p. 138
Isotropy subgroups	p. 139
Coset spaces	p. 140
Killing flows	p. 141
Properties of the coordinate components of the Q[subscript a]	p. 143
A special canonical coordinate system	p. 144
Coordinates for the coset spaces	p. 146
Classification of transitive realizations	p. 147
Matrix representations of super Lie groups	p. 149
Contragredient representations	p. 149
Inner automorphisms. The adjoint representation	p. 150
Matrix representations of the super Lie algebra	p. 151
Geometry of coset spaces	p. 152
Invariant tensor fields	p. 152
Differential equations for geometrical structures	p. 154
Integrability of the differential equations	p. 154
A special coordinate system	p. 156
Condition for the existence of a group-invariant measure function	p. 158
Condition for the existence of a group-invariant metric tensor field	p. 158
Condition for the existence of a group-invariant connection	p. 159
Solutions of the differential equations	p. 161
Geometry of the group supermanifold	p. 163
Identity of the left- and right-invariant connections	p. 165
Parallelism at a distance in the group supermanifold	p. 165
Integration over the group	p. 166
A special class of super Lie groups	p. 168
Exercises	p. 170
Comments on chapter 3	p. 171
Super Lie groups. Examples	p. 173
Construction of super Lie algebras and super Lie groups	p. 173
Properties of the structure constants	p. 173
Conventional super Lie groups, Z[subscript 2]-graded algebras	p. 173
Unconventional super Lie groups	p. 174
Structure of conventional super Lie Groups. The extending representation	p. 176
Construction of a class of super Lie algebras	p. 176
Notation	p. 179
The classical super Lie groups	p. 180
The group GL (m, n)	p. 180
The group SL (m, n)	p. 183
The group SL (m, m)/GL (1, 0)	p. 185
The orthosymplectic group OSp (m, n)	p. 187
The Kac notation	p. 190
The group P(m)	p. 191
The group Q(m)	p. 193
The group Q(m)	p. 193
The exceptional simple super Lie groups	p. 194
The groups D(2, 1, [alpha])	p. 194
The group F(4)	p. 197
The structure of F(4)	p. 199
Pseudorepresentation of F(4)	p. 200
The group G(3)	p. 201
The structure of G[subscript 2]	p. 203
The structure of G(3)	p. 204
Pseudorepresentation of G(3)	p. 206
Super Lie groups of basic importance in physics	p. 207
The super de Sitter group	p. 207
The super Poincare group	p. 209
The coset space: super Poincare group/SO(1, 3)	p. 210
Killing flows and invariant connections	p. 211
Riemannian geometry of the coset space	p. 212
The super Lorentz group	p. 213
The Cartan super Lie groups	p. 216
The diffeomorphism group Diff(M)	p. 216
The group SDiff(M, [mu])	p. 216
The canonical transformation group Can(M, [omega])	p. 217
The group of contact transformations	p. 219
The case m = 0	p. 220
The group W(n)	p. 221
The groups S(n) and S(n)	p. 221
The groups H(n) and H(n)	p. 222
Exercises	p. 223
Comments on chapter 4	p. 225
Selected applications of supermanifold theory	p. 227
Superclassical dynamical systems	p. 227
Configuration spaces	p. 227
Supermanifolds as configuration spaces	p. 229
Space of histories	p. 230
The action functional and the dynamical equations	p. 232
Infinitesimal disturbances and Green's functions	p. 232
Reciprocity relations	p. 234
The Peierls bracket	p. 235
Peierls bracket identities	p. 235
Super Hilbert spaces	p. 237
Definition	p. 237
Linear operators	p. 238
Physical observables	p. 239
Quantum systems	p. 240
Transition to the quantum theory	p. 240
The Schwinger variational principle	p. 242
External sources	p. 245
Chronologically ordered form of the operator dynamical equations	p. 246
The Feynman functional integral	p. 248
A simple Fermi system	p. 250
Action functional and Green's functions	p. 250
Eigenvectors of x	p. 250
The energy	p. 251
A pure basis	p. 252
An alternative representation	p. 253
The functional integral representation of [x", t" x', t']	p. 254
Evaluation of the functional integral	p. 256
The average superclassical trajectory	p. 259
Propagator for x[subscript av](t)	p. 260
The Fermi oscillator	p. 261
Action functional and Green's functions	p. 261
Mode functions and Hamiltonian	p. 262
Basic supervectors	p. 263
Eigenvectors of x[subscript 1] and x[subscript 2]. Choice of pure basis	p. 265
Coherent states	p. 266
The functional integral representation of [a"*, t"
Direct evaluation of the functional integral	p. 269
The importance of endpoint contributions	p. 271
The stationary trajectory as a matrix element	p. 272
The Feynman propagator	p. 272
The Bose oscillator	p. 273
Action functional and Green's functions	p. 273
Mode functions and Hamiltonian	p. 275
Energy eigenvectors	p. 276
Coherent states	p. 277
Hamilton-Jacobi theory	p. 279
The amplitude [x', t' x', t'] and its functional integral representation	p. 281
The functional-integral representation of [a"*, t"
The stationary path between coherent states	p. 283
The Feynman propagator	p. 283
Energy eigenfunctions	p. 284
Bose-Fermi supersymmetry	p. 285
The simplest model	p. 285
New conserved quantities	p. 287
The Bose-Fermi supersymmetry group	p. 289
Eigenvectors of Q[subscript 1] and Q[subscript 2]	p. 290
The supersymmetry group as a transformation group	p. 291
Auxiliary variable	p. 292
Nonlinear Bose-Fermi supersymmetry	p. 292
The supersymmetry group	p. 295
A pure basis	p. 296
The energy spectrum	p. 297
Spontaneously broken supersymmetry	p. 299
Exercises	p. 303
Comments on chapter 5	p. 306
Applications involving topology	p. 307
Nontrivial configuration spaces	p. 307
Standard canonical systems	p. 307
Green's functions	p. 309
Equivalence of Peierls and Poisson brackets	p. 310
Quantization	p. 311
Problems with the naive quantization rule	p. 311
Operator-valued forms. The projection m-form	p. 312
The position operator	p. 313
Vector operators	p. 314
The momentum operator	p. 316
Restriction to a local chart	p. 316
Lack of uniqueness of the momentum operator	p. 318
Overlapping charts. Transformation of coordinates	p. 319
The position representation	p. 321
The momentum operator in the position representation	p. 322
The Schrodinger equation	p. 325
The position representation of the projection m-form	p. 327
Curved configuration spaces	p. 327
A special class of systems	p. 327
Covariant variation	p. 328
Covariant differentiation with respect to t	p. 329
The dynamical equations	p. 330
Covariant functional differentiation	p. 331
The Feynman functional integral and its meaning	p. 332
Formal computation of det G[superscript +][x]	p. 332
The functional integral	p. 334
Normalization	p. 336
Ambiguity in the functional integral	p. 336
Homotopy	p. 337
Homotopy mesh	p. 339
The total amplitude	p. 340
Change of homotopy mesh	p. 341
The role of homology	p. 342
The universal covering space	p. 343
The total amplitude revisited	p. 345
The Hamiltonian operator: a nonlattice derivation	p. 347
Integration over phase space	p. 347
The Schrodinger equation	p. 348
Evaluation of the chronologically ordered Hamiltonian	p. 349
Approximate evaluation of the path integral	p. 352
Brief review of Hamilton-Jacobi theory	p. 352
The Van Vleck-Morette determinant	p. 354
Jacobi fields and the Green's function for the trajectory x[subscript c]	p. 356
Determinantal relations	p. 358
The loop expansion	p. 360
The WKB approximation	p. 361
The heat kernel expansion	p. 363
Role of the two-loop term in the independent verification of (6.5.25)	p. 365
New variables	p. 366
Computation of the two-loop term	p. 368
Supersymmetry and the Euler-Poincare characteristic	p. 370
Inclusion of a-type dynamical variables	p. 370
Green's functions and Peierls brackets	p. 371
Energy and supersymmetry group	p. 373
Quantization	p. 375
Basis supervectors	p. 377
Differential representation of operators	p. 379
Coherent states	p. 381
Energy eigenfunctions	p. 383
The Euler-Poincare characteristic	p. 385
Functional integral for the coherent-state transition amplitude	p. 386
The Chern-Gauss-Bonnet formula	p. 388
Exercises	p. 389
Comments on chapter 6	p. 396
References	p. 397
Index	p. 401
Table of Contents provided by Syndetics. All Rights Reserved.

Supermanifolds

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