Preface ........................................................ xv
Acknowledgements ............................................. xvii
Introduction .................................................... 1
Part I Relativistk field theory in Minkowski spacetime ......... 5
1 Scalar field theory and its symmetries ....................... 7
1.1 The scalar field system ................................. 7
1.2 Symmetries of the system ................................ 8
1.2.1 SO(n) internal symmetry .......................... 9
1.2.2 General internal symmetry ....................... 10
1.2.3 Spacetime symmetries - the Lorentz and
Poincare groups ................................. 12
1.3 Noether currents and charges ........................... 18
1.4 Symmetries in the canonical formalism .................. 21
1.5 Quantum operators ...................................... 22
1.6 The Lorentz group for D = 4 ............................ 24
2 The Dirac field ............................................. 25
2.1 The homomorphism of SL(2,C)→SO(3,1) .................... 25
2.2 The Dirac equation ..................................... 28
2.3 Dirac adjoint and bilinear form ........................ 31
2.4 Dirac action ........................................... 32
2.5 The spinors u(p,s) and v(p,s) for D = 4 ................ 33
2.6 Weyl spinor fields in even spacetime dimension ......... 35
2.7 Conserved currents ..................................... 36
2.7.1 Conserved U(l) current .......................... 36
2.7.2 Energy-momentum tensors for the Dirac field ..... 37
3 Clifford algebras and spinors ............................... 39
3.1 The Clifford algebra in general dimension .............. 39
3.1.1 The generating γ-matrices ....................... 39
3.1.2 The complete Clifford algebra ................... 40
3.1.3 Levi-Civita symbol .............................. 41
3.1.4 Practical γ-matrix manipulation ................. 42
3.1.5 Basis of the algebra for even dimension D =
2m .............................................. 43
3.1.6 The highest rank Clifford algebra element ....... 44
3.1.7 Odd spacetime dimension D = 2m + 1 .............. 46
3.1.8 Symmetries of γ-matrices ........................ 47
3.2 Spinors in general dimensions .......................... 49
3.2.1 Spinors and spinor bilinears .................... 49
3.2.2 Spinor indices .................................. 50
3.2.3 Fierz rearrangement ............................. 52
3.2.4 Reality ......................................... 54
3.3 Majorana spinors ....................................... 55
3.3.1 Definition and properties ....................... 56
3.3.2 Symplectic Majorana spinors ..................... 58
3.3.3 Dimensions of minimal spinors ................... 58
3.4 Majorana spinors in physical theories .................. 59
3.4.1 Variation of a Majorana Lagrangian .............. 59
3.4.2 Relation of Majorana and Weyl spinor theories ... 60
3.4.3 U(l) symmetries of a Majorana field ............. 61
Appendix 3А Details of the Clifford algebras for D = 2m ..... 62
3.4.1 Traces and the basis of the Clifford algebra .... 62
3.4.2 Uniqueness of the у-matrix representation ....... 63
3.4.3 The Clifford algebra for odd spacetime
dimensions ...................................... 65
3.4.4 Determination of symmetries of у-matrices ....... 65
3.4.5 Friendly representations ........................ 66
4 The Maxwell and Yang-Mills gauge fields ..................... 68
4.1 The abelian gauge field Aμ(x) .......................... 69
4.1.1 Gauge invariance and fields with electric
charge .......................................... 69
4.1.2 The free gauge field ............................ 71
4.1.3 Sources and Green's function .................... 73
4.1.4 Quantum electrodynamics ......................... 76
4.1.5 The stress tensor and gauge covariant
translations .................................... 77
4.2 Electromagnetic duality ................................ 77
4.2.1 Dual tensors .................................... 78
4.2.2 Duality for one free electromagnetic field ...... 78
4.2.3 Duality for gauge field and complex scalar ...... 80
4.2.4 Electromagnetic duality for coupled Maxwell
fields .......................................... 83
4.3 Non-abelian gauge symmetry ............................. 86
4.3.1 Global internal symmetry ........................ 86
4.3.2 Gauging the symmetry ............................ 88
4.3.3 Yang-Mills field strength and action ............ 89
4.3.4 Yang-Mills theory for G = SU(N) ................. 90
4.4 Internal symmetry for Majorana spinors ................. 93
5 The free Rarita-Schwinger field ............................. 95
5.1 The initial value problem .............................. 97
5.2 Sources and Green's function ........................... 99
5.3 Massive gravitinos from dimensional reduction ......... 102
5.3.1 Dimensional reduction for scalar fields ........ 102
5.3.2 Dimensional reduction for spinor fields ........ 103
5.3.3 Dimensional reduction for the vector gauge
field .......................................... 104
5.3.4 Finally ψμ(x,у) ................................ 104
6 N = 1 global supersymmetry in D = 4 ........................ 107
6.1 Basic SUSY field theory ............................... 109
6.1.1 Conserved supercurrents ........................ 109
6.1.2 SUSY Yang-Mills theory ......................... 110
6.1.3 SUSY transformation rules ...................... 111
6.2 SUSY field theories of the chiral multiplet ........... 112
6.2.1 U(l)R symmetry ................................. 115
6.2.2 The SUSY algebra ............................... 116
6.2.3 More chiral multiplets ......................... 119
6.3 SUSY gauge theories ................................... 120
6.3.1 SUSY Yang-Mills vector multiplet ............... 121
6.3.2 Chiral multiplets in SUSY gauge theories ....... 122
6.4 Massless representations of N-extended
supersymmetry ......................................... 125
6.4.1 Particle representations of N-extended
supersymmetry .................................. 125
6.4.2 Structure of massless representations .......... 127
Appendix 6A Extended supersymmetry and Weyl spinors ........ 129
Appendix 6B On-and off-shell multiplets and degrees of
freedom ............................................... 130
Part II Differential geometry and gravity ..................... 133
7 Differential geometry ...................................... 135
7.1 Manifolds ............................................. 135
7.2 Scalars, vectors, tensors, etc. ....................... 137
7.3 The algebra and calculus of differential forms ........ 140
7.4 The metric and frame field on a manifold .............. 142
7.4.1 The metric ..................................... 142
7.4.2 The frame field ................................ 143
7.4.3 Induced metrics ................................ 145
7.5 Volume forms and integration .......................... 146
7.6 Hodge duality of forms ................................ 149
7.7 Stokes' theorem and electromagnetic charges ........... 151
7.8 p-form gauge fields ................................... 152
7.9 Connections and covariant derivatives ................. 154
7.9.1 The first structure equation and the spin
connection ωμαb ................................ 155
7.9.2 The affine connection Гρμν ..................... 158
7.9.3 Partial integration ............................ 160
7.10 The second structure equation and the curvature
tensor ................................................ 161
7.11 The nonlinear σ-model ................................. 163
7.12 Symmetries and Killing vectors ........................ 166
7.12.1 σ-model symmetries ............................. 166
7.12.2 Symmetries of the Poincare plane ............... 169
8 The first and second order formulations of general
relativity ................................................. 171
8.1 Second order formalism for gravity and bosonic matter . 172
8.2 Gravitational fluctuations of flat spacetime .......... 174
8.2.1 The graviton Green's function .................. 177
8.3 Second order formalism for gravity and fermions ....... 178
8.4 First order formalism for gravity and fermions ........ 182
Part III Basic supergravity ................................... 185
9 N = 1 pure supergravity in four dimensions ................. 187
9.1 The universal part of supergravity .................... 188
9.2 Supergravity in the first order formalism ............. 191
9.3 The 1.5 order formalism ............................... 193
9.4 Local supersymmetry of N = 1, D = 4 supergravity ...... 194
9.5 The algebra of local supersymmetry .................... 197
9.6 Anti-de Sitter supergravity ........................... 199
10 D = 11 supergravity ........................................ 201
10.1 D ≤ 11 from dimensional reduction ..................... 201
10.2 The field content of D = 11 supergravity .............. 203
10.3 Construction of the action and transformation rules ... 203
10.4 The algebra of D = 11 supergravity .................... 210
11 General gauge theory ....................................... 212
11.1 Symmetries ............................................ 212
11.1.1 Global symmetries .............................. 213
11.1.2 Local symmetries and gauge fields .............. 217
11.1.3 Modified symmetry algebras ..................... 219
11.2 Covariant quantities .................................. 221
11.2.1 Covariant derivatives .......................... 222
11.2.2 Curvatures ..................................... 223
11.3 Gauged spacetime translations ......................... 225
11.3.1 Gauge transformations for the Poincare group ... 225
11.3.2 Covariant derivatives and general coordinate
transformations ................................ 227
11.3.3 Covariant derivatives and curvatures in
a gravity theory ............................... 230
11.3.4 Calculating transformations of covariant
quantities ..................................... 231
Appendix 11A Manipulating covariant derivatives ............ 233
11A.1 Proof of the main lemma .............................. 233
11A.2 Examples in supergravity ............................. 234
12 Survey of supergravities ................................... 236
12.1 The minimal superalgebras ............................. 236
12.1.1 Four dimensions ................................ 236
12.1.2 Minimal superalgebras in higher dimensions ..... 237
12.2 The R-symmetry group .................................. 238
12.3 Multiplets ............................................ 240
12.3.1 Multiplets in four dimensions .................. 240
12.3.2 Multiplets in more than four dimensions ........ 242
12.4 Supergravity theories: towards a catalogue ............ 244
12.4.1 The basic theories and kinetic terms ........... 244
12.4.2 Deformations and gauged supergravities ......... 246
12.5 Scalars and geometry .................................. 247
12.6 Solutions and preserved supersymmetries ............... 249
12.6.1 Anti-de Sitter superalgebras ................... 251
12.6.2 Central charges in four dimensions ............. 252
12.6.3 'Central charges' in higher dimensions ......... 253
Part IV Complex geometry and global SUSY ...................... 255
13 Complex manifolds .......................................... 257
13.1 The local description of complex and Kahler
manifolds ............................................. 257
13.2 Mathematical structure of Kahler manifolds ............ 261
13.3 The Kähler manifolds CPn .............................. 263
13.4 Symmetries of Kahler metrics .......................... 266
13.4.1 Holomorphic Killing vectors and moment maps ... 266
13.4.2 Algebra of holomorphic Killing vectors ......... 268
13.4.3 The Killing vectors of CP1 ..................... 269
14 General actions with N = 1 supersymmetry ................... 271
14.1 Multiplets ............................................ 271
14.1.1 Chiral multiplets .............................. 272
14.1.2 Real multiplets ................................ 274
14.2 Generalized actions by multiplet calculus ............. 275
14.2.1 The superpotential ............................. 275
14.2.2 Kinetic terms for chiral multiplets ............ 276
14.2.3 Kinetic terms for gauge multiplets ............. 277
14.3 Kahler geometry from chiral multiplets ................ 278
14.4 General couplings of chiral multiplets and gauge
multiplets ............................................ 280
14.4.1 Global symmetries of the SUSY сσ-model ......... 281
14.4.2 Gauge and SUSY transformations for chiral
multiplets ..................................... 282
14.4.3 Actions of chiral multiplets in a gauge
theory ......................................... 283
14.4.4 General kinetic action of the gauge multiplet .. 286
14.4.5 Requirements for an N = 1 SUSY gauge theory .... 286
14.5 The physical theory ................................... 288
14.5.1 Elimination of auxiliary fields ................ 288
14.5.2 The scalar potential ........................... 289
14.5.3 The vacuum state and SUSY breaking ............. 291
14.5.4 Supersymmetry breaking and the Goldstone
fermion ........................................ 293
14.5.5 Mass spectra and the supertrace sum rule ....... 296
14.5.6 Coda ........................................... 298
Appendix 14A Superspace .................................... 298
Appendix 14B Appendix: Covariant supersymmetry
transformations ....................................... 302
Part V Superconformal construction of supergravity theories ... 305
15 Gravity as a conformal gauge theory ........................ 307
15.1 The strategy .......................................... 308
15.2 The conformal algebra ................................. 309
15.3 Conformal transformations on fields ................... 310
15.4 The gauge fields and constraints ...................... 313
15.5 The action ............................................ 315
15.6 Recapitulation ........................................ 317
15.7 Homothetic Killing vectors ............................ 317
16 The conformal approach to pure N = 1 supergravity .......... 321
16.1 Ingredients ........................................... 321
16.1.1 Superconformal algebra ......................... 321
16.1.2 Gauge fields, transformations, and curvatures .. 323
16.1.3 Constraints .................................... 325
16.1.4 Superconformal transformation rules of
a chiral multiplet ............................. 328
16.2 The action ............................................ 331
16.2.1 Superconformal action of the chiral multiplet .. 331
16.2.2 Gauge fixing ................................... 333
16.2.3 The result ..................................... 334
17 Construction of the matter-coupled N = 1 supergravity ...... 337
17.1 Superconformal tensor calculus ........................ 338
17.1.1 The superconformal gauge multiplet ............. 338
17.1.2 The superconformal real multiplet .............. 339
17.1.3 Gauge transformations of superconformal
chiral multiplets .............................. 340
17.1.4 Invariant actions .............................. 342
17.2 Construction of the action ............................ 343
17.2.1 Conformal weights .............................. 343
17.2.2 Superconformal invariant action (ungauged) ..... 343
17.2.3 Gauged superconformal supergravity ............. 345
17.2.4 Elimination of auxiliary fields ................ 347
17.2.5 Partial gauge fixing ........................... 351
17.3 Projective Kahler manifolds ........................... 351
17.3.1 The example of CPn ............................. 352
17.3.2 Dilatations and holomorphic homothetic
Killing vectors ................................ 353
17.3.3 The projective parametrization ................. 354
17.3.4 The Kahler cone ................................ 357
17.3.5 The projection ................................. 358
17.3.6 Kahler transformations ......................... 359
17.3.7 Physical fermions .............................. 363
17.3.8 Symmetries of projective Kahler manifolds ...... 364
17.3.9 Г-gauge and decomposition laws ................. 365
17.3.10 An explicit example: SU(1,1)/U(1) model ...... 368
17.4 From conformal to Poincare supergravity ............... 369
17.4.1 The superpotential ............................. 370
17.4.2 The potential .................................. 371
17.4.3 Fermion terms .................................. 371
17.5 Review and preview .................................... 373
17.5.1 Projective and Köhler-Hodge manifolds .......... 374
17.5.2 Compact manifolds .............................. 375
Appendix 17A Köhler-Hodge manifolds ........................ 376
17A.1 Dirac quantization condition .................... 377
17A.2 Köhler-Hodge manifolds .......................... 378
Appendix 17B Steps in the derivation of (17.7) ............. 380
Part VI N = 1 supergravity actions and applications ........... 383
18 The physical N = 1 matter-coupled supergravity ............. 385
18.1 The physical action ................................... 386
18.2 Transformation rules .................................. 389
18.3 Further remarks ....................................... 390
18.3.1 Engineering dimensions ......................... 390
18.3.2 Rigid or global limit .......................... 390
18.3.3 Quantum effects and global symmetries .......... 391
19 Applications of N = 1 supergravity ......................... 392
19.1 Supersymmetry breaking and the super-BEH effect ....... 392
19.1.1 Goldstino and the super-BEH effect ............. 392
19.1.2 Extension to cosmological solutions ............ 395
19.1.3 Mass sum rules in supergravity ................. 396
19.2 The gravity mediation scenario ........................ 397
19.2.1 The Polonyi model of the hidden sector ......... 398
19.2.2 Soft SUSY breaking in the observable sector .... 399
19.3 No-scale models ....................................... 401
19.4 Supersymmetry and anti-de Sitter space ................ 403
19.5 R-symmetry and Fayet-Iliopoulos terms ................ 404
19.5.1 The R-gauge field and transformations ......... 405
19.5.2 Fayet-Iliopoulos terms ......................... 406
19.5.3 An example with non-minimal Kahler potential ... 406
Part VII Extended N = 2 supergravity .......................... 409
20 Construction of the matter-coupled N = 2 supergravity ...... 411
20.1 Global supersymmetry .................................. 412
20.1.1 Gauge multiplets for D = 6 ..................... 412
20.1.2 Gauge multiplets for D = 5 ..................... 413
20.1.3 Gauge multiplets for D = 4 ..................... 415
20.1.4 Hypermultiplets ................................ 418
20.1.5 Gauged hypermultiplets ......................... 422
20.2 M = 2 superconformal calculus ......................... 425
20.2.1 The superconformal algebra ..................... 425
20.2.2 Gauging of the superconformal algebra .......... 427
20.2.3 Conformal matter multiplets .................... 430
20.2.4 Superconformal actions ......................... 432
20.2.5 Partial gauge fixing ........................... 434
20.2.6 Elimination of auxiliary fields ................ 436
20.2.7 Complete action ................................ 439
20.2.8 D = 5 and D = 6, N = 2 supergravities .......... 440
20.3 Special geometry ...................................... 440
20.3.1 The family of special manifolds ................ 440
20.3.2 Very special real geometry ..................... 442
20.3.3 Special Kahler geometry ........................ 443
20.3.4 Hyper-Kähler and quaternionic-Kahler
manifolds ...................................... 452
20.4 From conformal to Poincare supergravity ............... 459
20.4.1 Kinetic terms of the bosons .................... 459
20.4.2 Identities of special Kahler geometry .......... 459
20.4.3 The potential .................................. 460
20.4.4 Physical fermions and other terms .............. 460
20.4.5 Supersymmetry and gauge transformations ........ 461
Appendix 20A SU(2) conventions and triplets ................ 463
Appendix 20B Dimensional reduction 6 → 5 → 4 ............... 464
20B.1 Reducing from D = 6 → D = 5 ......................... 464
20B.2 Reducing from Z = 5 → D = 4 ......................... 464
Appendix 20C Definition of rigid special Kahler geometry ... 465
21 The physical N = 2 matter-coupled supergravity ............. 469
21.1 The bosonic sector .................................... 469
21.1.1 The basic (ungauged) N = 2, D = 4 matter-
coupled supergravity ........................... 469
21.1.2 The gauged supergravities ...................... 471
21.2 The symplectic formulation ............................ 472
21.2.1 Symplectic definition .......................... 472
21.2.2 Comparison of symplectic and prepotential
formulation .................................... 474
21.2.3 Gauge transformations and symplectic vectors ... 474
21.2.4 Physical fermions and duality .................. 475
21.3 Action and transformation laws ........................ 476
21.3.1 Final action ................................... 476
21.3.2 Supersymmetry transformations .................. 477
21.4 Applications .......................................... 479
21.4.1 Partial supersymmetry breaking ................. 479
21.4.2 Field strengths and central charges ............ 480
21.4.3 Moduli spaces of Calabi-Yau manifolds .......... 480
21.5 Remarks ............................................... 482
21.5.1 Fayet-Iliopoulos terms ......................... 482
21.5.2 о--model symmetries ............................ 482
21.5.3 Engineering dimensions ......................... 482
Part VIII Classical solutions and the AdS/CFT correspondence .. 485
22 Classical solutions of gravity and supergravity ............ 487
22.1 Some solutions of the field equations ................. 487
22.1.1 Prelude: frames and connections on spheres ..... 487
22.1.2 Anti-de Sitter space ........................... 489
22.1.3 AdSD obtained from its embedding in  D+l ....... 490
22.1.4 Spacetime metrics with spherical symmetry ...... 496
22.1.5 AdS-Schwarzschild spacetime ................... 498
22.1.6 The Reissner-Nordström metric .................. 499
22.1.7 A more general Reissner-Nordstrцm solution ..... 501
22.2 Killing spinors and BPS solutions ..................... 503
22.2.1 The integrability condition for Killing
spinors ........................................ 505
22.2.2 Commuting and anti-commuting Killing spinors ... 505
22.3 Killing spinors for anti-de Sitter space .............. 506
22.4 Extremal Reissner-Nordström spacetimes as BPS
solutions ............................................. 508
22.5 The black hole attractor mechanism .................... 510
22.5.1 Example of a black hole attractor .............. 511
22.5.2 The attractor mechanism - real slow and
simple ......................................... 513
22.6 Supersymmetry of the black holes ...................... 517
22.6.1 Killing spinors ................................ 517
22.6.2 The central charge ............................. 519
22.6.3 The black hole potential ....................... 521
22.7 First order gradient flow equations ................... 522
22.8 The attractor mechanism - fast and furious ............ 523
Appendix 22A Killing spinors for pp-waves .................. 525
23 The AdS/CFT correspondence ................................. 527
23.1 The N = 4 SYM theory .................................. 529
23.2 Type IIB string theory and D3-branes .................. 532
23.3 The D3-brane solution of Type IIB supergravity ........ 533
23.4 Kaluza-Klein analysis on AdS5  S5 .................... 534
23.5 Euclidean AdS and its inversion symmetry .............. 536
23.6 Inversion and CFT correlation functions ............... 538
23.7 The free massive scalar field in Euclidean AdSd+1 ..... 539
23.8 AdS/CFT correlators in a toy model .................... 541
23.9 Three-point correlation functions ..................... 543
23.10 Two-point correlation functions ...................... 545
23.11 Holographic renormalization .......................... 550
23.11.1 The scalar two-point function in a CFTd ....... 554
23.11.2 The holographic trace anomaly ................. 555
23.12 Holographic RG flows ................................. 558
23.12.1 AAdS domain wall solutions .................... 559
23.12.2 The holographic c-theorem ..................... 562
23.12.3 First order flow equations .................... 563
23.13 AdS/CFT and hydrodynamics ............................ 564
Appendix A Comparison of notation ............................ 573
A.1 Spacetime and gravity ................................. 573
A.2 Spinor conventions .................................... 575
A.3 Components of differential forms ...................... 576
A.4 Covariant derivatives ................................. 576
Appendix В Lie algebras and superalgebras .................... 577
B.l Groups and representations ............................ 577
B.2 Lie algebras .......................................... 578
B.3 Superalgebras ......................................... 581
References ................................................. 583
Index ......................................................... 602
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