Freedman D.Z. Supergravity (Cambridge, 2012). - ОГЛАВЛЕНИЕ / CONTENTS

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ОбложкаFreedman D.Z. Supergravity / D.Z.Freedman, A. Van Proeyen. - Cambridge: Cambridge university press, 2012. - xvii, 607 p.: tab. - Bibliogr.: p.583-601. - Ind.: p.602-607. - ISBN 978-0-521-19401-3
 

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Оглавление / Contents
 
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 fig.2D+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 fig.3 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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