Supergravity

Supergravity, together with string theory, is one of the most significant developments in theoretical physics. Written by two of the most respected workers in the field, this is the first-ever authoritative and systematic account of supergravity. The book starts by reviewing aspects of relativistic field theory in Minkowski spacetime. After introducing the relevant ingredients of differential geometry and gravity, some basic supergravity theories (D=4 and D=11) and the main gauge theory tools are explained. In the second half of the book, complex geometry and N=1 and N=2 supergravity theories are covered. Classical solutions and a chapter on AdS/CFT complete the book. Numerous exercises and examples make it ideal for Ph.D. students, and with applications to model building, cosmology and solutions of supergravity theories, it is also invaluable to researchers. A website hosted by the authors, featuring solutions to some exercises and additional reading material, can be found at (http://www.cambridge.org/supergravity) www.cambridge.org/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 "Supergravity, together with string theory, is one of the most significant developments in theoretical physics. Although there are many books on string theory, this is the first-ever authoritative and systematic account of supergravity. Written by two of the most respected workers in the field, it provides a solid introduction to the fundamentals of supergravity. It starts by reviewing aspects of relativistic field theory in Minkowski spacetime. After introducing the relevant ingredients of differential geometry and gravity, some basic supergravity theories (D=4 and D=11) and the main gauge theory tools are explained. In the second half of the book, complex geometry and N=1 and N=2 supergravity theories are covered. Classical solutions and a chapter on AdS/CFT complete the book. Numerous exercises and examples make it ideal for Ph. D. students and with applications to model building, cosmology and solutions of supergravity theories, it is also invaluable to researchers"-- Provided by publisher