Supergravity

Cover
......Page 1
Supergravity......Page 2
Our conventions......Page 3
Title......Page 4
Copyright......Page 5
Contents......Page 6
Preface......Page 16
Acknowledgements......Page 18
Introduction......Page 20
PART I: RELATIVISTIC FIELD THEORY IN
MINKOWSKI SPACETIME......Page 24
1.1 The scalar field system......Page 26
1.2 Symmetries of the system......Page 27
1.2.1 SO(n) internal symmetry......Page 28
1.2.2 General internal symmetry......Page 29
1.2.3 Spacetime symmetries – the Lorentz and Poincaré groups......Page 31
1.3 Noether currents and charges......Page 37
1.4 Symmetries in the canonical formalism......Page 40
1.5 Quantum operators......Page 41
1.6 The Lorentz group for D = 4......Page 43
2.1 The homomorphism of SL(2, C) → SO(3, 1)......Page 44
2.2 The Dirac equation......Page 47
2.3 Dirac adjoint and bilinear form......Page 50
2.4 Dirac action......Page 51
2.5 The spinors u(p,s) and v(p,s) for D = 4 			......Page 52
2.6 Weyl spinor fields in even spacetime dimension......Page 54
2.7.1 Conserved U(1) current......Page 55
2.7.2 Energy–momentum tensors for the Dirac field......Page 56
3.1.1 The generating γ-matrices......Page 58
3.1.2 The complete Clifford algebra......Page 59
3.1.3 Levi-Civita symbol......Page 60
3.1.4 Practical γ-matrix manipulation......Page 61
3.1.5 Basis of the algebra for even dimension D = 2m......Page 62
3.1.6 The highest rank Clifford algebra element......Page 63
3.1.7 Odd spacetime dimension D = 2m + 1......Page 65
3.1.8 Symmetries of γ-matrices......Page 66
3.2.1 Spinors and spinor bilinears......Page 68
3.2.2 Spinor indices......Page 69
3.2.3 Fierz rearrangement......Page 71
3.2.4 Reality......Page 73
3.3 Majorana spinors......Page 74
3.3.1 Definition and properties......Page 75
3.3.3 Dimensions of minimal spinors......Page 77
3.4.1 Variation of a Majorana Lagrangian......Page 78
3.4.2 Relation of Majorana and Weyl spinor theories......Page 79
3.4.3 U(1) symmetries of a Majorana field......Page 80
3A.1 Traces and the basis of the Clifford algebra......Page 81
3A.2 Uniqueness of the γ-matrix representation......Page 82
3A.4 Determination of symmetries of  γ-matrices......Page 84
3A.5 Friendly representations......Page 85
4: The Maxwell and Yang–Mills gauge fields......Page 87
4.1.1 Gauge invariance and fields with electric charge......Page 88
4.1.2 The free gauge field......Page 90
4.1.3 Sources and Green's function......Page 92
4.1.4 Quantum electrodynamics......Page 95
4.2 Electromagnetic duality......Page 96
4.2.2 Duality for one free electromagnetic field......Page 97
4.2.3 Duality for gauge field and complex scalar......Page 99
4.2.4 Electromagnetic duality for coupled Maxwell fields......Page 102
4.3.1 Global internal symmetry......Page 105
4.3.2 Gauging the symmetry......Page 107
4.3.3 Yang–Mills field strength and action......Page 108
4.3.4 Yang–Mills theory for G = SU(N)......Page 109
4.4 Internal symmetry for Majorana spinors......Page 112
5: The free Rarita–Schwinger field......Page 114
5.1 The initial value problem......Page 116
5.2 Sources and Green's function......Page 118
5.3.1 Dimensional reduction for scalar fields......Page 121
5.3.2 Dimensional reduction for spinor fields......Page 122
5.3.4 Finally Ψμ(x,y)......Page 123
6: N = 1 global supersymmetry in D = 4......Page 126
6.1.1 Conserved supercurrents......Page 128
6.1.2 SUSY Yang–Mills theory......Page 129
6.1.3 SUSY transformation rules......Page 130
6.2 SUSY field theories of the chiral multiplet......Page 131
6.2.1 U(1)R symmetry......Page 134
6.2.2 The SUSY algebra......Page 135
6.2.3 More chiral multiplets......Page 138
6.3 SUSY gauge theories......Page 139
6.3.1 SUSY Yang–Mills vector multiplet......Page 140
6.3.2 Chiral multiplets in SUSY gauge theories......Page 141
6.4.1 Particle representations of N-extended supersymmetry......Page 144
6.4.2 Structure of massless representations......Page 146
Appendix 6A  Extended supersymmetry and Weyl spinors......Page 148
Appendix 6B  On- and off-shell multiplets and degrees of freedom......Page 149
PART II: DIFFERENTIAL GEOMETRY AND GRAVITY......Page 152
7.1 Manifolds......Page 154
7.2 Scalars, vectors, tensors, etc.......Page 156
7.3 The algebra and calculus of differential forms......Page 159
7.4.1 The metric......Page 161
7.4.2 The frame field......Page 162
7.4.3 Induced metrics......Page 164
7.5 Volume forms and integration......Page 165
7.6 Hodge duality of forms......Page 168
7.7 Stokes' theorem and electromagnetic charges......Page 170
7.8 p-form gauge fields......Page 171
7.9 Connections and covariant derivatives......Page 173
7.9.1 The first structure equation and the spin connection ωμab......Page 174
7.9.2 The affine connection ρμν......Page 177
7.9.3 Partial integration......Page 179
7.10 The second structure equation and the curvature tensor......Page 180
7.11 The nonlinear σ-model......Page 182
7.12.1 σ-model symmetries......Page 185
7.12.2 Symmetries of the Poincaré plane......Page 188
8: The first and second order formulations of general relativity......Page 190
8.1 Second order formalism for gravity and bosonic matter......Page 191
8.2 Gravitational fluctuations of flat spacetime......Page 193
8.2.1 The graviton Green's function......Page 196
8.3 Second order formalism for gravity and fermions......Page 197
8.4 First order formalism for gravity and fermions......Page 201
PART III: BASIC SUPERGRAVITY......Page 204
9: N =1 pure supergravity in four dimensions......Page 206
9.1 The universal part of supergravity......Page 207
9.2 Supergravity in the first order formalism......Page 210
9.3 The 1.5 order formalism......Page 212
9.4 Local supersymmetry of N = 1, D = 4 supergravity......Page 213
9.5 The algebra of local supersymmetry......Page 216
9.6 Anti-de Sitter supergravity......Page 218
10.1 D ≤ 11 from dimensional reduction......Page 220
10.3 Construction of the action and transformation rules......Page 222
10.4 The algebra of D = 11 supergravity......Page 229
11.1 Symmetries......Page 231
11.1.1 Global symmetries......Page 232
11.1.2 Local symmetries and gauge fields......Page 236
11.1.3 Modified symmetry algebras......Page 238
11.2 Covariant quantities......Page 240
11.2.1 Covariant derivatives......Page 241
11.2.2 Curvatures......Page 242
11.3.1 Gauge transformations for the Poincaré group......Page 244
11.3.2 Covariant derivatives and general coordinate transformations......Page 246
11.3.3 Covariant derivatives and curvatures in a gravity theory......Page 249
11.3.4 Calculating transformations of covariant quantities......Page 250
11A.1 Proof of the main lemma......Page 252
11A.2 Examples in supergravity......Page 253
12.1.1 Four dimensions......Page 255
12.1.2 Minimal superalgebras in higher dimensions......Page 256
12.2 The R-symmetry group......Page 257
12.3.1 Multiplets in four dimensions......Page 259
12.3.2 Multiplets in more than four dimensions......Page 261
12.4.1 The basic theories and kinetic terms......Page 263
12.4.2 Deformations and gauged supergravities......Page 265
12.5 Scalars and geometry......Page 266
12.6 Solutions and preserved supersymmetries......Page 268
12.6.1 Anti-de Sitter superalgebras......Page 270
12.6.2 Central charges in four dimensions......Page 271
12.6.3 `Central charges' in higher dimensions......Page 272
PART IV: COMPLEX GEOMETRY AND GLOBAL SUSY......Page 274
13.1 The local description of complex and Kähler manifolds......Page 276
13.2 Mathematical structure of Kähler manifolds......Page 280
13.3 The Kähler manifolds CPn......Page 282
13.4.1 Holomorphic Killing vectors and moment maps......Page 285
13.4.2 Algebra of holomorphic Killing vectors......Page 287
13.4.3 The Killing vectors of CP1......Page 288
14.1 Multiplets......Page 290
14.1.1 Chiral multiplets......Page 291
14.1.2 Real multiplets......Page 293
14.2.1 The superpotential......Page 294
14.2.2 Kinetic terms for chiral multiplets......Page 295
14.2.3 Kinetic terms for gauge multiplets......Page 296
14.3 Kähler geometry from chiral multiplets......Page 297
14.4 General couplings of chiral multiplets and gauge multiplets......Page 299
14.4.1 Global symmetries of the SUSY σ-model......Page 300
14.4.2 Gauge and SUSY transformations for chiral multiplets......Page 301
14.4.3 Actions of chiral multiplets in a gauge theory......Page 302
14.4.5 Requirements for an N = 1 SUSY gauge theory......Page 305
14.5.1 Elimination of auxiliary fields......Page 307
14.5.2 The scalar potential......Page 308
14.5.3 The vacuum state and SUSY breaking......Page 310
14.5.4 Supersymmetry breaking and the Goldstone fermion......Page 312
14.5.5 Mass spectra and the supertrace sum rule......Page 315
Appendix 14A  Superspace......Page 317
Appendix 14B  Appendix: Covariant supersymmetry transformations......Page 321
PART V: SUPERCONFORMAL CONSTRUCTION
OF SUPERGRAVITY THEORIES......Page 324
15: Gravity as a conformal gauge theory......Page 326
15.1 The strategy......Page 327
15.2 The conformal algebra......Page 328
15.3 Conformal transformations on fields......Page 329
15.4 The gauge fields and constraints......Page 332
15.5 The action......Page 334
15.7 Homothetic Killing vectors......Page 336
16.1.1 Superconformal algebra......Page 340
16.1.2 Gauge fields, transformations, and curvatures......Page 342
16.1.3 Constraints......Page 344
16.1.4 Superconformal transformation rules of a chiral multiplet......Page 347
16.2.1 Superconformal action of the chiral multiplet......Page 350
16.2.2 Gauge fixing......Page 352
16.2.3 The result......Page 353
17: Construction of the matter-coupled N = 1 supergravity......Page 356
17.1.1 The superconformal gauge multiplet......Page 357
17.1.2 The superconformal real multiplet......Page 358
17.1.3 Gauge transformations of superconformal chiral multiplets......Page 359
17.1.4 Invariant actions......Page 361
17.2.2 Superconformal invariant action (ungauged)......Page 362
17.2.3 Gauged superconformal supergravity......Page 364
17.2.4 Elimination of auxiliary fields......Page 366
17.3 Projective Kähler manifolds......Page 370
17.3.1 The example of CPn......Page 371
17.3.2 Dilatations and holomorphic homothetic Killing vectors......Page 372
17.3.3 The projective parametrization......Page 373
17.3.4 The Kähler cone......Page 376
17.3.5 The projection......Page 377
17.3.6 Kähler transformations......Page 378
17.3.7 Physical fermions......Page 382
17.3.8 Symmetries of projective Kähler manifolds......Page 383
17.3.9 T-gauge and decomposition laws......Page 384
17.3.10 An explicit example: SU(1,1)/U(1) model......Page 387
17.4 From conformal to Poincaré supergravity......Page 388
17.4.1 The superpotential......Page 389
17.4.3 Fermion terms......Page 390
17.5 Review and preview......Page 392
17.5.1 Projective and Kähler–Hodge manifolds......Page 393
17.5.2 Compact manifolds......Page 394
Appendix 17A  Kähler–Hodge manifolds......Page 395
17A.1 Dirac quantization condition......Page 396
17A.2 Kähler–Hodge manifolds......Page 397
Appendix 17B  Steps in the derivation of (17.7)......Page 399
PART VI: N = 1 SUPERGRAVITY ACTIONS AND
APPLICATIONS......Page 402
18: The physical N = 1 matter-coupled supergravity......Page 404
18.1 The physical action......Page 405
18.2 Transformation rules......Page 408
18.3.2 Rigid or global limit......Page 409
18.3.3 Quantum effects and global symmetries......Page 410
19.1.1 Goldstino and the super-BEH effect......Page 411
19.1.2 Extension to cosmological solutions......Page 414
19.1.3 Mass sum rules in supergravity......Page 415
19.2 The gravity mediation scenario......Page 416
19.2.1 The Polónyi model of the hidden sector......Page 417
19.2.2 Soft SUSY breaking in the observable sector......Page 418
19.3 No-scale models......Page 420
19.4 Supersymmetry and anti-de Sitter space......Page 422
19.5 R-symmetry and Fayet–Iliopoulos terms......Page 423
19.5.1 The R-gauge field and transformations......Page 424
19.5.3 An example with non-minimal Kähler potential......Page 425
PART VII: EXTENDED N = 2 SUPERGRAVITY......Page 428
20: Construction of the matter-coupled N = 2 supergravity......Page 430
20.1.1 Gauge multiplets for D = 6......Page 431
20.1.2 Gauge multiplets for D = 5......Page 432
20.1.3 Gauge multiplets for D = 4......Page 434
20.1.4 Hypermultiplets......Page 437
20.1.5 Gauged hypermultiplets......Page 441
20.2.1 The superconformal algebra......Page 444
20.2.2 Gauging of the superconformal algebra......Page 446
20.2.3 Conformal matter multiplets......Page 449
20.2.4 Superconformal actions......Page 451
20.2.5 Partial gauge fixing......Page 453
20.2.6 Elimination of auxiliary fields......Page 455
20.2.7 Complete action......Page 458
20.3.1 The family of special manifolds......Page 459
20.3.2 Very special real geometry......Page 461
20.3.3 Special Kähler geometry......Page 462
20.3.4 Hyper-Kähler and quaternionic-Kähler manifolds......Page 471
20.4.2 Identities of special Kähler geometry......Page 478
20.4.4 Physical fermions and other terms......Page 479
20.4.5 Supersymmetry and gauge transformations......Page 480
Appendix 20A  SU(2) conventions and triplets......Page 482
20B.2 Reducing from D = 5 → D = 4......Page 483
Appendix 20C  Definition of rigid special Kähler geometry......Page 484
21.1.1 The basic (ungauged) N = 2, D = 4 matter-coupled supergravity......Page 488
21.1.2 The gauged supergravities......Page 490
21.2.1 Symplectic definition......Page 491
21.2.3 Gauge transformations and symplectic vectors......Page 493
21.2.4 Physical fermions and duality......Page 494
21.3.1 Final action......Page 495
21.3.2 Supersymmetry transformations......Page 496
21.4.1 Partial supersymmetry breaking......Page 498
21.4.3 Moduli spaces of Calabi–Yau manifolds......Page 499
21.5.3 Engineering dimensions......Page 501
PART VIII: CLASSICAL SOLUTIONS AND THE
AdS/CFT CORRESPONDENCE......Page 504
22.1.1 Prelude: frames and connections on spheres......Page 506
22.1.2 Anti-de Sitter space......Page 508
22.1.3 AdSD obtained from its embedding in RD+1......Page 509
22.1.4 Spacetime metrics with spherical symmetry......Page 515
22.1.5 AdS–Schwarzschild spacetime......Page 517
22.1.6 The Reissner–Nordström metric......Page 518
22.1.7 A more general Reissner–Nordström solution......Page 520
22.2 Killing spinors and BPS solutions......Page 522
22.2.2 Commuting and anti-commuting Killing spinors......Page 524
22.3 Killing spinors for anti-de Sitter space......Page 525
22.4 Extremal Reissner–Nordström spacetimes as BPS solutions......Page 527
22.5 The black hole attractor mechanism......Page 529
22.5.1 Example of a black hole attractor......Page 530
22.5.2 The attractor mechanism – real slow and simple......Page 532
22.6.1 Killing spinors......Page 536
22.6.2 The central charge......Page 538
22.6.3 The black hole potential......Page 540
22.7 First order gradient flow equations......Page 541
22.8 The attractor mechanism – fast and furious......Page 542
Appendix 22A  Killing spinors for pp-waves......Page 544
23: The AdS/CFT correspondence......Page 546
23.1 The N = 4 SYM theory......Page 548
23.2 Type IIB string theory and D3-branes......Page 551
23.3 The D3-brane solution of Type IIB supergravity......Page 552
23.4 Kaluza–Klein analysis on AdS5 S5......Page 553
23.5 Euclidean AdS and its inversion symmetry......Page 555
23.6 Inversion and CFT correlation functions......Page 557
23.7 The free massive scalar field in Euclidean AdSd+1......Page 558
23.8 AdS/CFT correlators in a toy model......Page 560
23.9 Three-point correlation functions......Page 562
23.10 Two-point correlation functions......Page 564
23.11 Holographic renormalization......Page 569
23.11.1 The scalar two-point function in a CFTd......Page 573
23.11.2 The holographic trace anomaly......Page 574
23.12 Holographic RG flows......Page 577
23.12.1 AAdS domain wall solutions......Page 578
23.12.2 The holographic c-theorem......Page 581
23.12.3 First order flow equations......Page 582
23.13 AdS/CFT and hydrodynamics......Page 583
A.1 Spacetime and gravity......Page 592
A.2 Spinor conventions......Page 594
A.4 Covariant derivatives......Page 595
B.1 Groups and representations......Page 596
B.2 Lie algebras......Page 597
B.3 Superalgebras......Page 600
References......Page 602
Index......Page 621