Gravity and Strings

Half-title......Page 2
Series-title......Page 4
Title......Page 6
Copyright......Page 7
Dedication......Page 8
Contents......Page 10
Preface......Page 20
Part I Introduction to gravity and supergravity......Page 22
1.1 World tensors......Page 24
1.2 Affinely connected spacetimes......Page 26
1.3 Metric spaces......Page 30
1.3.1 Riemann–Cartan spacetime Ud......Page 32
1.3.2 Riemann spacetime Vd......Page 34
1.4 Tangent space......Page 35
1.4.1 Weitzenböck spacetime Ad......Page 40
1.5 Killing vectors......Page 41
1.6 Duality operations......Page 42
1.7 Differential forms and integration......Page 44
1.8 Extrinsic geometry......Page 46
2.1 Equations of motion......Page 47
2.2 Noether's theorems......Page 48
2.3 Conserved charges......Page 52
2.4 The special-relativistic energy–momentum tensor......Page 53
2.4.1 Conservation of angular momentum......Page 54
2.4.2 Dilatations......Page 58
2.4.3 Rosenfeld's energy–momentum tensor......Page 60
3 A perturbative introduction to general relativity......Page 66
3.1 Scalar SRFTs of gravity......Page 67
3.1.1 Scalar gravity coupled to matter......Page 68
3.1.2 The action for a relativistic massive point-particle......Page 69
3.1.3 The massive point-particle coupled to scalar gravity......Page 71
3.1.4 The action for a massless point-particle......Page 72
3.1.6 Self-coupled scalar gravity......Page 74
3.1.7 The geometrical Einstein–Fokker theory......Page 76
3.2 Gravity as a self-consistent massless spin-2 SRFT......Page 78
3.2.1 Gauge invariance, gauge identities, and charge conservation in the SRFT of a spin-1 particle......Page 81
3.2.2 Gauge invariance, gauge identities, and charge conservation in the SRFT of a spin-2 particle......Page 84
3.2.3 Coupling to matter......Page 88
3.2.4 The consistency problem......Page 97
3.2.5 The Noether method for gravity......Page 99
3.2.6 Properties of the gravitational energy–momentum tensor…......Page 106
3.2.7 Deser's argument......Page 110
3.3 General relativity......Page 117
3.4 The Fierz–Pauli theory in a curved background......Page 124
3.4.1 Linearized gravity......Page 125
3.4.2 Massless spin-2 particles in curved backgrounds......Page 129
3.5 Final Comments......Page 133
4 Action principles for gravity......Page 135
4.1 The Einstein–Hilbert action......Page 136
4.1.1 Equations of motion......Page 138
4.1.2 Gauge identity and Noether current......Page 140
4.1.3 Coupling to matter......Page 141
4.2 The Einstein–Hilbert action in different conformal frames......Page 142
4.3 The first-order (Palatini) formalism......Page 144
4.3.1 The purely affine theory......Page 147
4.4 The Cartan–Sciama–Kibble theory......Page 148
4.4.1 The coupling of gravity to fermions......Page 149
4.4.2 The coupling to torsion: the CSK theory......Page 152
4.4.3 Gauge identities and Noether currents......Page 155
4.4.4 The first-order Vielbein formalism......Page 157
4.5 Gravity as a gauge theory......Page 161
4.6 Teleparallelism......Page 165
4.6.1 The linearized limit......Page 167
5 N = 1, 2, d = 4 supergravities......Page 171
5.1 Gauging N = 1, d = 4 superalgebras......Page 172
5.2 N = 1, d = 4 (Poincaré) supergravity......Page 176
5.2.1 Local supersymmetry algebra......Page 179
5.3 N = 1, d = 4 AdS supergravity......Page 180
5.4 Extended supersymmetry algebras......Page 181
5.4.1 Central extensions......Page 184
5.5 N = 2, d = 4 (Poincaré) supergravity......Page 185
5.6 N = 2, d = 4 "gauged" (AdS) supergravity......Page 188
5.7 Proofs of some identities......Page 190
6 Conserved charges in general relativity......Page 192
6.1 The traditional approach......Page 193
6.1.1 The Landau–Lifshitz pseudotensor......Page 195
6.1.2 The Abbott–Deser approach......Page 197
6.2 The Noether approach......Page 200
6.3 The positive-energy theorem......Page 201
Part II Gravitating point-particles......Page 206
7 The Schwarzschild black hole......Page 208
7.1 Schwarzschild's solution......Page 209
7.1.1 General properties......Page 210
7.2 Sources for Schwarzschild’s solution......Page 221
7.3 Thermodynamics......Page 223
7.4 The Euclidean path-integral approach......Page 229
7.4.1 The Euclidean Schwarzschild solution......Page 230
7.4.2 The boundary terms......Page 231
7.5 Higher-dimensional Schwarzschild metrics......Page 232
7.5.1 Thermodynamics......Page 233
8 The Reissner–Nordström black hole......Page 234
8.1 Coupling a scalar field to gravity and no-hair theorems......Page 235
8.2 The Einstein–Maxwell system......Page 239
8.2.1 Electric charge......Page 242
8.2.2 Massive electrodynamics......Page 246
8.3 The electric Reissner–Nordström solution......Page 248
8.4 The Sources of the electric RN black hole......Page 259
8.5 Thermodynamics of RN black holes......Page 261
8.6 The Euclidean electric RN solution and its action......Page 263
8.7 Electric-magnetic duality......Page 266
8.7.2 Magnetic charge: the Dirac monopole and the Dirac quantization condition......Page 269
8.7.3 The Wu–Yang monopole......Page 275
8.7.4 Dyons and the DSZ charge-quantization condition......Page 277
8.7.5 Duality in massive electrodynamics......Page 279
8.8 Magnetic and dyonic RN black holes......Page 280
8.9 Higher-dimensional RN solutions......Page 283
9 The Taub–NUT solution......Page 288
9.1 The Taub–NUT solution......Page 289
9.2 The Euclidean Taub–NUT solution......Page 292
9.2.1 Self-dual gravitational instantons......Page 293
9.2.2 The BPST instanton......Page 295
9.2.3 Instantons and monopoles......Page 296
9.2.5 Bianchi IX gravitational instantons......Page 298
9.3 Charged Taub–NUT solutions and IWP solutions......Page 300
10.1 pp-Waves......Page 303
10.1.1 Hpp-waves......Page 304
10.2 Four-dimensional pp-wave solutions......Page 306
10.3 Sources: the AS shock wave......Page 308
11 The Kaluza–Klein black hole......Page 311
11.1 Classical and quantum mechanics on…......Page 312
11.2 KK dimensional reduction on a circle S......Page 317
11.2.1 The Scherk–Schwarz formalism......Page 320
11.2.2 Newton's constant and masses......Page 324
11.2.3 KK reduction of sources: the massless particle......Page 327
11.2.4 Electric–magnetic duality and the KK action......Page 331
11.2.5 Reduction of the Einstein–Maxwell action and N = 1, d = 5 SUGRA......Page 334
11.3 KK reduction and oxidation of solutions......Page 337
11.3.1 ERN black holes......Page 338
11.3.2 Dimensional reduction of the AS shock wave: the extreme electric KK black hole......Page 342
11.3.3 Non-extreme Schwarzschild and RN black holes......Page 344
11.3.4 Simple KK solution-generating techniques......Page 347
11.4 Toroidal (Abelian) dimensional reduction......Page 352
11.4.1 The 2-torus and the modular group......Page 357
11.4.2 Masses, charges and Newton’s constant......Page 359
11.5 Generalized dimensional reduction......Page 360
11.5.1 Example 1: a real scalar......Page 362
11.5.2 Example 2: a complex scalar......Page 366
11.5.3 Example 3: an SL(2,R)/SO(2) δ-model......Page 367
11.5.4 Example 4: Wilson lines and GDR......Page 368
11.6 Orbifold compactification......Page 369
12 Dilaton and dilaton/axion black holes......Page 370
12.1 Dilaton black holes: the a-model......Page 371
12.1.1 The a-model solutions in four dimensions......Page 375
12.2 Dilaton/axion black holes......Page 379
12.2.1 The general SWIP solution......Page 384
12.2.2 Supersymmetric SWIP solutions......Page 386
12.2.3 Duality properties of the SWIP solutions......Page 387
12.2.4 N = 2, d = 4 SUGRA solutions......Page 388
13 Unbroken supersymmetry......Page 390
13.1 Vacuum and residual symmetries......Page 391
13.2 Supersymmetric vacua and residual (unbroken) supersymmetries......Page 394
13.2.1 Covariant Lie derivatives......Page 396
13.2.2 Calculation of supersymmetry algebras......Page 399
13.3 N = 1, 2, d = 4 vacuum supersymmetry algebras......Page 400
13.3.1 The Killing-spinor integrability condition......Page 403
13.3.2 The vacua of N = 1, d = 4 Poincar supergravity......Page 404
13.3.3 The vacua of N = 1, d = 4 AdS4 supergravity......Page 405
13.3.4 The vacua of N = 2, d = 4 Poincar supergravity......Page 407
13.3.5 The vacua of N = 2, d = 4 AdS supergravity......Page 410
13.4.1 N = (1,0), d 6 supergravity......Page 411
13.4.2 N = 1, d = 5 supergravity......Page 412
13.4.3 Relation to the N = 2, d = 4 vacua......Page 414
13.5 Partially supersymmetric solutions......Page 415
13.5.1 Partially unbroken supersymmetry, supersymmetry bounds, and the superalgebra......Page 416
13.5.2 Examples......Page 419
Part III Gravitating extended objects of string theory......Page 424
14 String theory......Page 426
14.1 Strings......Page 430
14.1.1 Superstrings......Page 433
14.1.2 Green–Schwarz Actions......Page 436
14.2.1 Quantization of free-bosonic-string theories......Page 438
14.2.2 Quantization of free-fermionic-string theories......Page 443
14.2.3 D-Branes and O-planes in superstring theories......Page 445
14.2.4 String interactions......Page 446
14.3.1 Closed bosonic strings on S1......Page 447
14.3.2 Open bosonic strings on S1 and D-branes......Page 448
14.3.3 Superstrings on S1......Page 450
15.1 Effective actions and background fields......Page 451
15.1.1 The D-brane effective action......Page 455
15.2 T duality and background fields: Buscher's rules......Page 456
15.2.1 T duality in the bosonic-string effective action......Page 457
15.2.2 T duality in the bosonic-string worldsheet action......Page 460
15.2.3 T duality in the bosonic Dp-brane effective action......Page 464
15.3 Example: the fundamental string (F1)......Page 466
16 From eleven to four dimensions......Page 468
16.1.1 11-dimensional supergravity......Page 470
16.1.2 Reduction of the bosonic sector......Page 473
16.1.3 Magnetic potentials......Page 479
16.1.4 Reduction of fermions and the supersymmetry rules......Page 482
16.2 Romans’ massive N = 2A, d = 10 supergravity......Page 484
16.3.1 Dimensional reduction of the bosonic RR sector......Page 487
16.3.2 Dimensional reduction of fermions and supersymmetry rules......Page 488
16.4 The effective-field theory of the heterotic string......Page 490
16.5.1 Reduction of the action of pure N = 1, d = 10 supergravity......Page 492
16.5.2 Reduction of the fermions and supersymmetry rules of N = 1, d = 10 SUGRA......Page 496
16.5.3 The truncation to pure supergravity......Page 498
16.5.4 Reduction with additional U(1) vector fields......Page 499
16.5.5 Trading the KR 2-form for its dual......Page 501
16.6 T duality, compactification, and supersymmetry......Page 503
17 The type-IIB superstring and type-II T duality......Page 506
17.1 N = 2B, d = 10 supergravity in the string frame......Page 507
17.1.1 Magnetic potentials......Page 508
17.2 Type-IIB S duality......Page 509
17.3 Dimensional reduction of N = 2B, d = 10 SUEGRA and type-II T duality......Page 512
17.3.1 The type-II T-duality Buscher rules......Page 515
17.4 Dimensional reduction of fermions and supersymmetry rules......Page 516
17.5 Consistent truncations and heterotic/type-I duality......Page 518
Introduction......Page 521
18.1.1 Worldvolume actions......Page 522
18.1.2 Charged branes and Dirac charge quantization for extended objects......Page 527
18.1.3 The coupling of p-branes to scalar fields......Page 530
18.2.1 Schwarzschild black p-branes......Page 533
18.2.2 The p-brane a-model......Page 535
18.2.3 Sources for solutions of the p-brane a-model......Page 538
19 The extended objects of string theory......Page 541
19.1 String-theory extended objects from duality......Page 542
19.1.1 The masses of string- and M-theory extended objects from duality......Page 545
19.2 String-theory extended objects from effective-theory solutions......Page 550
19.2.1 Extreme p-brane solutions of string and M-theories and sources......Page 553
19.2.2 The M2 solution......Page 554
19.2.3 The M5 solution......Page 556
19.2.4 The fundamental string F1......Page 557
19.2.5 The S5 solution......Page 558
19.2.6 The Dp-branes......Page 559
19.2.7 The D-instanton......Page 561
19.2.8 The D7-brane and holomorphic (d – 3)-branes......Page 563
19.2.9 Some simple generalizations......Page 567
19.3.1 Masses......Page 568
19.3.2 Charges......Page 571
19.4.1 N = 2A, d = 10 SUEGRA solutions from d = 11 SUGRA solutions......Page 572
19.4.2 N 2A/B, d = 10 SUEGRA T-dual solutions......Page 575
19.4.3 S duality of N = 2B, d = 10 SUEGRA solutions: pq-branes......Page 576
19.5 String-theory extended objects from superalgebras......Page 578
19.5.1 Unbroken supersymmetries of string-theory solutions......Page 580
19.6 Intersections......Page 584
19.6.1 Brane-charge conservation and brane surgery......Page 587
19.6.2 Marginally bound supersymmetric states and intersections......Page 588
19.6.3 Intersecting-brane solutions......Page 589
19.6.4 The (a1–a2) model for p1- and p2-branes and black intersecting branes......Page 591
20 String black holes in four and five dimensions......Page 594
20.1 Composite dilaton black holes......Page 595
20.2.1 Black holes from single wrapped branes......Page 597
20.2.2 Black holes from wrapped intersecting branes......Page 599
20.2.3 Duality and black-hole solutions......Page 607
20.3 Entropy from microstate counting......Page 609
A.1 Generalities......Page 612
A.2.1 Fields and covariant derivatives......Page 616
A.2.2 Kinetic terms......Page 618
A.2.3 SO(n…n–) gauge theory......Page 619
A.3 Riemannian geometry of group manifolds......Page 623
A.3.1 Example: the SU(2) group manifold......Page 624
A.4 Riemannian geometry of homogeneous and symmetric spaces......Page 625
A.4.1 H-covariant derivatives......Page 629
A.4.2 Example: round spheres......Page 630
B.1 Generalities......Page 632
B.1.1 Useful identities......Page 639
B.1.2 Fierz identities......Page 640
B.1.3 Eleven dimensions......Page 641
B.1.4 Ten dimensions......Page 643
B.1.6 Eight dimensions......Page 644
B.1.9 Four dimensions......Page 645
B.1.10 Five dimensions......Page 646
B.2 Spaces with arbitrary signatures......Page 647
B.2.1 AdS4 gamma matrices and spinors......Page 650
Appendix C n-Spheres......Page 655
C.1 S3 and S7 as Hopf fibrations......Page 657
C.2 Squashed S3 and S7......Page 658
Appendix D Palatini's identity......Page 659
Appendix E Conformal rescalings......Page 660
F.1.1 General static, spherically symmetric metrics (I)......Page 661
F.1.2 General static, spherically symmetric metrics (II)......Page 662
F.1.3 d = 4 IWP-type metrics......Page 663
F.2.1 d > 4 General static, spherically symmetric metrics......Page 664
F.2.2 A general metric for (single, black) p-branes......Page 665
F.2.3 A general metric for (composite, black) p-branes......Page 666
F.2.4 A general metric for extreme p-branes......Page 667
F.2.5 Brinkmann metrics......Page 668
Appendix G The harmonic operator on R3 × S1......Page 669
References......Page 671
Index......Page 692