Ideas and Methods of Supersymmetry and Supergravity: a Walk Through Superspace

Ideas and Methods of Supersymmetry and Supergravity or A Walk Through Superspace......Page 1
Contents......Page 3
Preface to the First Edition......Page 13
Preface to the Revised Edition......Page 16
1.1.1. Definitions......Page 18
1.1.2. Useful decomposition in SO(3,1)T......Page 20
1.1.3. Universal covering group of the Lorentz group......Page 21
1.1.4. Universal covering group of the Poincare group......Page 23
1.2.1. Connection between representations of SO(3,1)t and SL(2,C)......Page 24
1.2.2. Construction of SL(2,C) irreducible representations......Page 25
1.2.3. Invariant Lorentz tensors......Page 28
1.3. The Lorentz algebra......Page 30
1.4. Two-component and four-component spinors......Page 34
1.4.1. Two-component spinors......Page 35
1.4.2. Dirac spinors......Page 36
1.4.4. Majorana spinors......Page 38
1.4.5. The reduction rule and the Fierz identity......Page 39
1.5.1. The Poincare algebra......Page 40
1.5.3. Unitary representations......Page 43
1.5.4. Stability subgroup......Page 44
1.5.5. Massive irreducible representations......Page 46
1.5.6. Massless irreducible representations......Page 47
1.6.1. Lorentz manifolds......Page 49
1.6.2. Covariant differentiation of world tensors......Page 52
1.6.3 Covariant differentiation of the Lorentz tensor......Page 53
1.6.4. Frame deformations......Page 55
1.6.5. The Weyl tensor......Page 56
1.6.6. Four-dimensional topological invariants......Page 57
1.6.7. Einstein gravity and conformal gravity......Page 58
1.6.8. Energy-momentum tensor......Page 60
1.6.9. The covariant derivatives algebra in spinor notation......Page 61
1.7.1. Conformal Killing vectors......Page 62
1.7.2. Conformal Killing vectors in Minkowski space......Page 63
1.7.3. The conformal algebra......Page 64
1.7.4. Conformal transformations......Page 65
1.7.5. Matrix realization of the conformal group......Page 67
1.7.6. Conformal invariance......Page 68
1.7.7. Examples of conformally invariant theories......Page 70
1.7.8. Example of a non-conformal massless theory......Page 72
1.8.1. Massive field representations of the Poincare group......Page 73
1.8.2. Real massive field representations......Page 76
1.8.3. Massless field representations of the Poincare group......Page 78
1.8.4. Examples of massless fields......Page 79
1.8.5. Massless field representations of the conformal group......Page 83
1.9. Elements of algebra with supernumbers......Page 85
1.9.1. Grassman algebrus AN and A x......Page 87
1.9.2. Supervector spaces......Page 89
1.9.3. Finite-dimensional supervector spaces......Page 91
1.9.4. Linear operators and supermatrices......Page 94
1.9.5. Dual supervector spaces, supertransposition......Page 102
1.9.6. Bi-linear forms......Page 105
1.10.1. Superfunctions......Page 108
1.10.2. Integration over RPIQ......Page 113
1.10.3. Linear replacements of variables on RPIq......Page 118
1.10.4. C-type supermatrices revisited......Page 120
1.1 1. The supergroup of general coordinate transformations on Rp14......Page 123
1.11.1 The exponential form for general coordinate transformations......Page 124
1.11.2. The operators K and k......Page 126
1.11.3. Theorem......Page 127
1.11.4. The transformation law for the volume element on RpIq......Page 129
1.11.5 Basic properties of integration theory over W12q......Page 130
2.0. Introduction: from IwpI4 to supersymmetry......Page 134
2.1. Superalgebras, Grassmann shells and super Lie groups......Page 138
2.1.1 Superalgebras......Page 139
2.1.2. Examples of superalgebras......Page 141
2.1.3. The Grassmann shell of a superalgebra......Page 142
2.1.4. Examples of Berezin superalgebras and super Lie algebras......Page 145
2.1.5. Representations of (Berezin) superalgebras and super Lie algebras......Page 149
2.1.6. Super Lie groups......Page 152
2.1.7. Unitary representations of real superalgebras......Page 154
2.2.1. Uniqueness of the N = 1 Poincare superalgebra......Page 155
2.2.2. Extended Poincure superalgebras......Page 158
2.2.3. Matrix realization of the Poincare superalgebra......Page 160
2.2.4. Grassman shell of the Poincare superalgebru......Page 161
2.2.5. The super Poincare group......Page 162
2.3.1. Positivity of energy......Page 163
2.3.2. Casimir operators of the Poincare superalgebra......Page 164
2.3.3. Massive irreducible representations......Page 166
2.3.4. Massless irreducible representations......Page 169
2.3.5. Superhelicity......Page 170
2.3.6. Equality of bosonic and fermionic degrees of freedom......Page 171
2.4.1. Minkowski space as the coset space l l j S......Page 172
2.4.2. Real superspace R4I4......Page 174
2.4.4. Superfields......Page 177
2.4.5. Superfield representations of the super Poincare group......Page 179
2.5. Complex superspace C4I2, chiral superfields and covariant derivatives......Page 182
2.5.1. complex superspace 4I2......Page 183
2.5.2. Holomorphic superfields......Page 184
2.5.3. W4 as a surface in C41\'......Page 185
2.5.4. Chiral superfields......Page 186
2.5.5. Covariant derivatives......Page 187
2.5.6. Properties of covariant derivatives......Page 188
2.6. The on-shell massive superfield representations......Page 189
2.6.1. On-shell massive superfields......Page 190
2.6.2. Extended super-Poincare algebra......Page 191
2.6.3. The superspin operator......Page 192
2.6.4. Decomposition of ~Ffi),~) into irreducible representations......Page 193
2.6.6. Real representations......Page 196
2.7.1. Consistency conditions......Page 198
2.7.2. On-shell massless superfields......Page 199
2.7.3. Superhelicity......Page 202
2.8.1. Chiral scalar superfield......Page 203
2.8.2. Chiral tensor superfield of Lorentz type ( n / 2 , 0......Page 205
2.8.3. Real scalar superfield......Page 207
2.9. The superconformal group......Page 208
2.9.1. Superconformal transformations......Page 209
2.9.2. The supersymmetric interval and superconformal transformations......Page 211
2.9.3. The superconformal algebra......Page 212
3.1.1. Quick review of field theory......Page 215
3.1.2. The space of superfield histories; the action superfunctional......Page 218
3.1.3. Integration over R4I4 and superfunctional derivatives......Page 219
3.1.4. Local supersymmetric field theories......Page 225
3.1.6. Chiral representation......Page 228
3.2.1. Massive chiral scalar superfield model......Page 230
3.2.3. Wess-Zumino model......Page 232
3.2.4. Wess-Zumino model in component form......Page 233
3.2.5. Auxiliary fields......Page 234
3.2.6. Wess-Zumino model after auxiliary field elimination......Page 235
3.2.7. Generalization of the model......Page 237
3.3.1. Four-dimensional o-models......Page 238
3.3.2. Supersymmetric a-models......Page 239
3.3.3. Kahler manifolds......Page 241
3.3.4. Kahler geometry and supersymmetric o-models......Page 243
3.4.1. Massive vector multiplet model......Page 245
3.4.3. Wess-Zumino gauge......Page 247
3.4.4. Supersymmetry transformations......Page 249
3.4.6. Massive vector multiplet model revisited......Page 250
3.5.1. Supersymmetric scalar electrodynamics......Page 253
3.5.2. Supersymmetric spinor electrodynamics......Page 256
3.5.3. Non-Abelian gauge superfeld......Page 257
3.5.4. Infinitesimal gauge transformations......Page 258
3.5.5. Super Yang-Mills action......Page 260
3.5.6. Super Yang-Mills models......Page 262
3.5.7. Red representations......Page 263
3.6.1. Complex and c-number shells of compact Lie groups......Page 264
3.6.2. K-supergroup and A-supergroup......Page 267
3.6.3. Gauge superfield......Page 268
3.6.4. Gauge covariant derivatives......Page 271
3.6.5. Matter equations of motion......Page 274
3.6.6. Gauge superfield dynamical equations......Page 275
3.7. Classically equivalent theories......Page 276
3.7.1 Massive chiral spinor superfield model......Page 277
3.7.2. Massless chiral spinor superfield model......Page 278
3.7.3. Superfield redefinitions......Page 280
3.8. Non-minimal scalar multiplet......Page 281
3.8.1. Complex linear scalar superfield......Page 282
3.8.2. Free non-minimal scalar multiplet......Page 283
3.8.3. Mass generation I......Page 285
3.8.4. Mass generation II......Page 286
3.8.5. Supersymmetric electrodynamics......Page 287
3.8.7. Nonlinear sigma models......Page 290
4.1. Picture-change operators......Page 292
4.1.1. Functional supermatrices......Page 293
4.1.2. Superfunctional supermatrices......Page 295
4.1.3. (Super) functional derivatives......Page 300
4.1.4. Picture-change operators......Page 302
4.2.1. (Super)field Green\'s functions......Page 306
4.2.2. Generating functional......Page 309
4.2.3. Generating superfunctional......Page 311
4.2.4. Coincidence of Z[J] and Z[J]......Page 313
4.3. Effective action (super)functional......Page 315
4.3.1. Effective action......Page 316
4.3.2. Super Poincare invariance of W[J] and r[C]......Page 319
4.3.3. Short excursion into renormalization theory......Page 322
4.3.4. Finite pathalogical supersymmetric theories......Page 324
4.4.1. Preliminary discussion......Page 326
4.4.2. Feynman superpropagator......Page 328
4.4.3. Generating superfunctional......Page 330
4.4.4. Standard Feynman rules......Page 333
4.4.5. Improved Feynman rules......Page 336
4.4.6. Example of supergraph calculations......Page 341
4.4.7 Supersymmetric analytic regularization......Page 342
4.4.8. Non-renormalization theorem......Page 343
4.5. Note about gauge theories......Page 345
4.5.1. Gauge theories......Page 346
4.5.2. Feynman rules for irreducible gauge theories with closed algebras......Page 349
4.5.3. Supersymmetric gauge theories......Page 354
4.6.1. Quantization of the pure super Yany-Mills model......Page 357
4.6.2. Propagators and vertices......Page 360
4.6.3. Feynman rules for general super Yang-Mills models......Page 363
4.7.1. Superficial degree of divergence......Page 367
4.7.2. Structure of counterterms......Page 370
4.7.3. Questions of regularization......Page 374
4.8.1. One-loop counterterms of matter in an external super Yang-Mills field......Page 378
4.8.2. One-loop counterterms of the general Wess-Zumino model......Page 383
4.9.1. Effective potential in quantum field theory (brief survey)......Page 388
4.9.2. Superfield effective potential......Page 391
4.9.3. Superfield effective potential in the Wess-Zumino model......Page 393
4.9.4. Calculation of the one-loop kahlerian effective potential......Page 396
4.9.5. Calculation of the two-loop effective chiral superpotential......Page 399
5.1.1. Curved superspace......Page 403
5.1.2. Conformal supergravity......Page 407
5.1.3. Einstein supergravity......Page 413
5.1.4. Einstein supergravity (second formulation)......Page 414
5.1.5. Einstein supergravity multiplet......Page 415
5.1.6. Flat superspace (final definition) and conformally flat superspace......Page 418
5.2. Superspace differential geometry......Page 419
5.2.1. Superfield representations of the general coordinate transformation supergroup......Page 420
5.2.2. The general coordinate transformation supergroup in exponential form......Page 422
5.2.3. Tangent and cotangent supervector spaces......Page 423
5.2.4. Supervierbein......Page 424
5.2.5. Superlocal Lorentz group......Page 425
5.2.6. Superconnection and covariant derivatives......Page 427
5.2.7. Bianchi identities and the Dragon theorem......Page 429
5.2.8. Integration by parts......Page 430
5.2.9. Flat superspace geometry......Page 431
5.3.1. Conformal supergravity constraints......Page 433
5.3.2. The Bianchi identities......Page 437
5.3.3. Solution to the dim = 1 Bianchi identities......Page 440
5.3.4. Solution to the dim = 4 Bianchi identities......Page 441
5.3.6. Algebra of covariant derivatives......Page 443
5.3.8. Generalized super Weyl transformations......Page 444
5.4. Prepotentials......Page 445
5.4.1. Solution to constraints (5.3.15a)......Page 446
5.4.2. Useful gauges on the superlocal Lorentz group......Page 447
5.4.3. The /I-supergroup......Page 449
5.4.4. Expressions for E, T, and R......Page 450
5.4.5. Gauge fixing for the K- and A-supergroups......Page 452
5.4.6. Chiral representation......Page 453
5.4.7. Gravitational superfield......Page 455
5.4.8. Gauge fixing on the generalized super Weyl group......Page 458
5.5.2. Chiral compensator......Page 459
5.5.3. MinimaI algebra of covariant derivatives......Page 460
5.5.4. Super Weyl transformations......Page 462
5.5.6. Chiral integration rule......Page 463
5.5.7. Matter dynamical systems in a supergravity background......Page 465
5.6.1. Modified parametrization of prepotentials......Page 467
5.6.2. Background-quantum splitting......Page 471
5.6.3. Background-quantum splitting in Einstein supergravity......Page 477
5.6.4. First-order expressions......Page 478
5.6.5. Topological invariants......Page 480
5.7.1. Basic construction......Page 482
5.7.2. The relation with ordinary currents......Page 485
5.7.3. The supercurrent and the supertrace in flat superspace......Page 486
5.7.4. Super Weyl invariant models......Page 488
5.7.5 Example......Page 489
5.8. Supergravity in components......Page 490
5.8.1. Space projections of covariant derivatives......Page 491
5.8.2. Space projections of R, R and G,......Page 495
5.8.3. Basic construction......Page 496
5.8.4. Algebraic structure of the curvature with torsion......Page 500
5.8.5. Space projections of a\'$ ace9cyGa,b, and 9(a WBra)......Page 502
5.8.6. Component fields and local supersymmetry transformation laws......Page 503
5.8.7. From superfield action to component action......Page 505
6.1. Pure supergravity dynamics......Page 508
6.1.1. Einstein supergravity action superfunctional......Page 509
6.1.2. Supergravity dynamical equations......Page 510
6.1.3. Einstein supergravity action functional......Page 511
6.1.4. Supergravity with a cosmological term......Page 513
6.1.6. Renormalizable supergravity models......Page 515
6.1.7. Pathological supergravity model......Page 516
6.2.1. Linearized Einstein supergravity action......Page 517
6.2.2. Linearized superfield strengths and dynamical equations......Page 520
6.2.3. Linearized conformal supergravity......Page 521
6.3. Supergravity-matter dynamical systems......Page 522
6.3.1. Chiral scalar models......Page 523
6.3.2. Vector multiplet models......Page 526
6.3.3. Super Yang-Mills models......Page 528
6.3.4. Chiral spinor model......Page 532
6.4. (Conformal) Killing supervectors. Superconformal models......Page 534
6.4.1. (Conformal) Killing supervector fields......Page 535
6.4.2. The gravitational superfield and conformal Killing supervectors......Page 539
6.4.3. (Conformal) Killing supervectors in flat global superspace......Page 540
6.4.4. Superconformal models......Page 542
6.4.5. On-shell massless conformal superfields......Page 545
6.5.1 Flat superspace......Page 548
6.5.2. Conformally flat superspace......Page 550
6.5.3. Physical sense of conformal flatness......Page 552
6.5.4. Anti-de Sitter superspace......Page 553
6.5.5. Killing supervectors of anti-de Sitter superspace......Page 554
6.6.1. Preliminary discussion......Page 556
6.6.2. Complex linear compensator......Page 558
6.6.3. Non-minimal supergeometry......Page 560
6.6.4. Dynamics in non-minimal supergravity......Page 562
6.6.5. Prepotentials and field content in non-minimal supergravity......Page 563
6.6.6. Geometrical approach to non-minimal supergravity......Page 564
6.6.7. Linearized non-minimal supergravity......Page 567
6.7.1. Real linear compensator......Page 568
6.7.2. Dynamics in new minimal supergravity......Page 571
6.7.3. Gauge fixing and field content in new minimal supergravity......Page 573
6.7.4. Linearized new minimal supergravity......Page 576
6.8.1. Non-minimal chiral compensator......Page 577
6.8.2. Matter dynamical systems in a non-minimal supergravity background......Page 580
6.8.3. New minimal supergravity and supersymmetric a-models......Page 581
6.9. Free massless higher superspin theories......Page 582
6.9.1. Free massless theories of higher integer spins......Page 583
6.9.2. Free massless theories of higher half-integer spins......Page 587
6.9.3. Free massless theories of higher half-integer superspins......Page 590
6.9.4. Free massless theories of higher integer superspins......Page 594
6.9.5. Massless gravitino multiplet......Page 597
7.1.1. When the proper-time technique can be applied......Page 601
7.1.2. Schwinger\'s kernel......Page 603
7.1.3. One-loop divergences of effective action......Page 607
7.1.4. Conformal anomaly......Page 610
7.1.5. The coefficients a,(x, x) and a,(x, x)......Page 614
7.2. Proper-time representation for covariantly chiral scalar superpropagator......Page 617
7.2.1. Basic chiral model......Page 618
7.2.2. Covariantly chiral Feynman superpropagator......Page 621
7.2.3. The chiral d’Alembertian......Page 623
7.2.4. Covariantly chiral Schwinger\'s superkernel......Page 625
7.2.5. aT(z, z ) and a;(z, z )......Page 627
7.2.6. One-loop divergences......Page 628
7.2.7. Switching on an external Yang-Mills superfield......Page 629
7.3.1. Quantization of the massless vector multiplet model......Page 630
7.3.2. Connection between GLy) and GcY)......Page 633
7.3.3. Scalar Schwinger\'s superkernel......Page 634
7.4. Super Weyl anomaly......Page 636
7.4.1. Super Weyl anomaly in a massless chiral scalar model......Page 637
7.4.2. Anomalous effective action......Page 640
7.4.3. Solution of effective equations of motion in conformally flat superspace......Page 643
7.5.1. Problem of quantum equivalence......Page 644
7.5.2. Gauge antisymmetric tensor field......Page 647
7.5.3. Quantization of the chiral spinor model......Page 650
7.5.4. Analysis of quantum equivalence......Page 654
Bibliography......Page 657