Gravity and Strings Tomás Ortín Index More Information

Cambridge University Press 978-0-521-82475-0 - Gravity and Strings Tomás Ortín Index More information

Index

Page numbers in italic are those on which the definition or a very ic discussion of the subject can be found. 4d means four dimensions or four-dimensional, etc.

AdSj, see anti-de Sitter (AdS) spacetime Ashtekar variables, 138 /?-brane asymptotically locally Euclidean (ALE) solutions, 272 (a\-a'i) model for intersecting branes, 570-573 Atiyah, see solution, Atiyah-Hitchin action, 570 autoparallel a-model, 514-519 curve, 7, 13 action, 574 equation, 7 't Hooft, 206 axidilaton, 359 Ansatz axion, 217, 339, 349, 359, 360, 362, 363, 478, 480, 542 for periodic SU(2) instantons, 317 axion-dilaton gravity, 359-363 for the BPS monopole, 277 and S duality, 360 Ansatz for SU(2) instantons, 274 as a truncation of N = 4, d = 4 supergravity, 361 't Hooft-Polyakov monopole, see solution, axitor, 146 't Hooft-Polyakov Banados-Teitelboim-Zanelli (BTZ) BH, see solution, BTZ Abbott-Deser approach, 106, 111, 175,176-179, 180, 372, Bekenstein-Hawking entropy, 187 394 and horizon area, see horizon, area compared with Landau-Lifshitz energy-momentum and the Euler characteristic, 245 pseudotensor, see Landau-Lifshitz energy-momentum density of BH microstates, 207 pseudotensor Bel-Robinson tensor, 171 action, 26 Belavin, see solution, BPST instanton /?-brane {a\-ai) model, see /?-brane, {a\-ai) model Belinfante tensor, 34, 35, 39, 40, 58, 59, 68, 80, 85, 130, 133 /?-brane a model, see /?-brane, a-model for a Dirac spinor, 37 /?-brane with variable tension, 502 for a spin-2 field, 82 Born-Infeld (BI), see Born-Infeld for a vector field, 35 Einstein-Hilbert, see Einstein-Hilbert action Bertotti, see solution, Robinson-Bertotti (RB) Nambu-Goto (NG), see Nambu-Goto (NG) BH,see black hole (BH) Polyakov-type, see Polyakov-type Bianchi Ramond-Neveu-Schwarz (RNS) model, see Bianchi IX gravitational instantons Ramond-Neveu-Schwarz Eguchi-Hanson solution, see solution, Eguchi-Hanson spinning particle, see Poincare, N = l,d = 1 supergravity Kaluza-Klein monopole, see solution, Kaluza-Klein supergravity theories, see supergravity theories Taub-bolt solution, see solution, Taub-bolt superparticle (Green-Schwarz), see Green-Schwarz clasification of 3d real Lie algebras, 602 ADM mass, 176,776, 189 Bianchi identities, 109, 217 of composite BH solutions, see solution, composite and gauge identities, 30, 39, 58, 59, 66, 102, 111, 114, 120, 129, 4dBHs 136, 227 of Reissner-Nordstrom BH, see solution, RN Fierz-Pauli Lagrangian, 66 of Schwarzschild's BH, see solution, Schwarzschild and Poincare dualization of the KR field, 480 AdS-CFT correspondence, 399, 536, 537, 540, 570, 573 and the dual 6-form potential of N = \,d = 11 Aharonov-Bohm phase, 254 supergravity, 452 Aichelburg, see solution, Aichelburg-Sexl shock wave contracted, 12, 13, 101, 131, 174, 214, 220, 351 angular-momentum, 268 as a gauge identity, 102, 119, 120 angular-momentum tensor, 41 for the background metric, 177 orbital, 34 for (p + 2)-form field strengths, 506, 514 spin, 34 for d-forni field strengths, 343 anholonomy coefficients, see Ricci, rotation coefficients for curvature and torsion, 8, 135, 158, 273, 602 anti-de Sitter (AdS), see de Sitter (anti-) for metric-compatible connections, 12 anti-de Sitter (AdS) spacetime, see solution, AdS for metric-compatible torsion-free connections, 13 Arnowitt, see ADM mass for embedding coordinates, 427, 440

671

672 Index

Bianchi identities (cont.) contribution to extreme BH entropy, 589 for Maxwell's field strength, 166, 167, 217, 220, 221, 225, 227, D/?-branes and O^-planes, 424 228, 257, 383 BPST instanton, see solution, BPST instanton and electric-magnetic duality, 245, 248 Brans, see Jordan-Brans-Dicke theory and electric-magnetic duality in dilaton a-model, 358 Brill, 280 and existence of a potential, 220 Brinkmann metrics, 282 and magnetic sources, 249 connection and curvature, 647 and Poincare duality, 248 BTZ, see Banados-Teitelboim-Zanelli (BTZ) BH and S duality in 4d KK theory, 311 Buscher T duality, 435-445 and S duality in dilaton-axion gravity, 360, 361 and breaking of supersymmetry, 482-484 and the Chern-Simons (CS) 3-form, 256 between Fl and W, 445 in dilaton a-model, 350, 351 between JNW and Schwarzschild solutions, 216 for RR field strengths, 458^460, 464, 487, 492 in type-II theories, 448, 485, 491, 493 and the mass parameter of Romans' theory, 465 between solutions, 551-556 for the self-dual RR 5-form, 486 transformations, 494-495 for Yang-Mills fields, 273, 276, 597 transformations of Killing spinors, 496, 562 in massive electrodynamics, 258 transformations, 437, 438 Big-Bang singularities, 195 transformations for Vielbeins, 439 Birkhoff's theorem, 189, 190, 228 black /?-branes, see solution, Schwarzschild /?-branes Cahen-Wallach symmetric spacetimes, 283, 389 black hole, 194 Cardy's formula, 578, 589 black hole (BH) Cartan axion-dilaton, see solution, axion-dilaton connection, 17, 20, 128, 132, 138 Banados-Teitelboim-Zanelli (BTZ), see solution, BTZ black first criterion (solvability), 594 hole Maurer-Cartan, see Maurer-Cartan dilaton, see solution, dilaton Riemann-Cartan spacetime, see Riemann-Cartan spacetime entropy problem, 797, 207-208 second criterion (semisimplicity), 594 information problem, 797, 206-207 Cartan-Sciama-Kibble (CSK) theory, 34, 39, 41, 113,127-140 Kerr, see solution, Kerr Cauchy horizon, see horizon, Cauchy Kerr-Newman, see solution, Kerr-Newman CFT, see conformal field theories Reissner-Nordstrom (RN), see solution, Reissner-Nordstrom Chan-Paton factors, 420, 423, 428 (RN) charge conjugation matrix, 615 Schwarzschild, see solution, Schwarzschild Chazy, see solution, Chazy-Curzon string, 573-590 Chern class SWIP, see solution, SWIP first, and the Wu-Yang monopole, 255 Bogomol'nyi (or BPS) bound, 509, 556 second, and SU(2) Yang-Mills instanton, 273, 310 and equilibrium of forces, 397, 424 Chern-Simons and supersymmetry, 373, 394-398 3-form, 132, 256, 598 in N = 1, d = 4 supergravity, 399 in N = 1, d = 10 supergravity, 498 in N = 2, d = 4 supergravity, 400 in KR field strength, 470 in N = 4, d = 4 supergravity, 400 term, 432, 530 in extended supergravities, 396 N = 1, d = 11 supergravity, 451 and the positivity of mass, 779, 398 N = 1, d = 5 supergravity, 313 and the Reissner-Nordstrom (RN) solution, 231 N = 2,d = 9 supergravity, 493 for the 't Hooft-Polyakov monopole, 276 N = 2A±T, d = 10 supergravity, 423, 487 in N = 8, d = 8 supergravity, 575 N = 2B± supergravity, 487 of string/M-theory objects, 550 in supergravity field strengths, 437, 550, 569, 570 saturated (Bogomol'nyi identity), 310 Romans' N = 2A,d = 10 supergravity, 464 Bogomol'nyi equation for SU(2) monopoles 276 Christoffel symbols, 10, 13, 20, 92, 126, 127, 210 Bogomornyi-Prasad-Sommerfield (BPS) and Weyl rescalings, 639 limit of the 't Hooft-Polyakov monopole, 276, 277, 373 Clifford Bohm, see Aharonov-Bohm phase action, 377 Boltzmann's constant, 204 algebra, 601, 611, 612, 613, 615, 616, 624, 626, 630, 631 Bonnet, see Gauss-Bonnet theorem compensator (//-), 606 Bonnor, 196, 267 Compton wavelength, 50, 115 magnetic-dipole solution, see solution, Bonnor compared with the Schwarzschild radius and Planck length, 116, Born-Infeld (BI) 206 action, 503 conformal field theories (CFTs), 573, 578, 589, 590 vector field, 435, 435, 443, 501, 564, 567 conformastationary, see metric, conformastationary dual, 549, 564 connection solitons (Blons), 549, 550, 554, 564, 566, 570 affine, 6 Boulware, 59, 78 Cartan, see Cartan, connection bound chiral, 367 Bogomol'nyi bound, see Bogomol'nyi (or BPS) bound metric-compatible, 10 BPS bound, see Bogomol'nyi (or BPS) bound spin, 78, 601 supersymmetry bound, see Bogomol'nyi (or BPS) bound continuity equation, 29 Boyer, 198 coordinate basis, 3 Boyer-Lindquist coordinates, see coordinates, Boyer-Lindquist coordinates BPS, see Bogomornyi-Prasad-Sommerfield (BPS) adapted to an isometry, 21, 99 BPS states, 370, 395-401, 408 Boyer-Lindquist, 268, 366 and the superalgebra, 395 Eddington-Finkelstein, see solution, Schwarzschild annihilated by supercharges, 394 harmonic, 199

Index 673

isotropic, 198, 216, 232, 234, 265, 271, 532-535 and the string coupling constant, 425, 487 Kruskal-Szekeres (KS), 192-193, 194, 209-211 Buscher's T-duality transformation rule, 438 Schwarzschild, see Schwarzschild, coordinates coupling to D-branes, 511 spherical, 634 coupling to fundamental branes, 511 correspondence principle, 245, 577, 578, 589 coupling to Kaluza-Klein (KK) branes, 512 cosmological constant, 102-103 coupling to solitonic branes, 511 cotangent space, 3 coupling to the D/?-brane, 434 Coulomb coupling to the string, 433 field, 200, 233 dimensional reduction on S , 436, 549 force, 218 dimensional reduction on JP, 550 covariant derivative from N = l,d = ll supergravity, 457^458, 470, 475 of world-tensor densities, 6 Dirac of world tensors, 5 conjugate, 18, 152, 468, 601, 615, 621 of Yang-Mills fields, see Yang-Mills, covariant derivative and Majorana spinors, 617 crosscaps, 412 in arbitrary signature, 628 CSK, see Cartan-Sciama-Kibble (CSK) theory conjugation matrix, 601, 613, 615, 621 curvature, see tensor, curvature in arbitrary signature, 627 curve, 7 delta function, 69, 71, 234, 518, 519 Curzon, see solution, Chazy-Curzon equation, 415, 614, 617 conjugate, 615, 616 D/?-brane from superparticle action, 412, 413 solution, see solution, Dp magnetic monopole, 248-254, 276 D/?-branes and SU(2)/U(1) coset space, 608 defined, 420 and Hopf fibrations, 636 D-instanton, see solution, D(-l) and Wu-Yang formulation, 255 de Broglie wavelength, 52 equation, 249, 249, 272, 277, 328 De Donder gauge, 57, 65, 65, 87, 282 no existence for gauge group R, 253 de Sitter (anti-) solution, 250, 252 N =l,d = 4 superalgebra, 152 magnetic monopoles gauging, 152-155 absence in massive electrodynamics, 226 N=l,d = 4 supergravity, 151, 159-160, 377 massive equation action, 754,159 from superparticle action, 413 super symmetry transformations, 160,168 matrices, see gamma matrices vacua, 384-386 quantization condition, 254, 254 N = 2,d = 4 supergravity, 151, 167-169, 378, 382 and Aharonov-Bohm phase, 254 action, 168 and the time periodicity of the Taub-NUT solution, 270 solutions, 400 and Wu-Yang formulation, 255 vacua, 389-390 for extended objects, 500 N = 4, d = 5 (SO(6)-gauged) supergravity, 540, 563 for extended objects, 508, 509, 520, 550 N = 4,d = 1 (SO(5)-gauged) supergravity, 536 for extended objects and the Newton constant, 528, 529 N = S,d = 4 (SO(8)-gauged) supergravity, 534 in Kaluza-Klein (KK) theories, 306, 312, 329, 339 (gauged) supergravities from spontaneous compactification, 547 spinor, 612 algebra, 143, 385 and Wilson lines, 347 spinorial representations, 629-633 commuting in WNI technique, 181 connection, 167 energy-momentum tensor, 36, 37, 39 group, 46, 115, 132, 142, 143, 150, 159 energy-momentum tensor coupled to gravity, 130, 131 gauging of, 143 energy-momentum tensor coupled to gravity in first-order radius, 143 formalism, 139-140 spacetime, 103, 104, 143, 159, 161, 179, 235, 286, 372 Lorentz transformations, 614 stability, 372 reducible in even d, 616 superalgebras, 563 string singularities supergroup, 150 and the wire singularities of the Taub-NUT solution, 270 Dehn twists, 336 string singularity, 251-255 Demiafiski for extended objects, 508 Plebanski-Demiafiski solutions, see solution, Plebanski Dirac-Schwinger-Zwanziger (DSZ) quantization condition, 214, Deser, 45, 176 256 Abbott-Deser approach, see Abbott-Deser approach and S duality, 362 argument for GR's self-consistency, 44, 46, 59, 78, 85, 89-93 and S duality, 258 Boulware-Deser paper, see Boulware Dirichlet (D) boundary conditions, 411 diamond invariant of E7, 587, 588 divergence Dicke, see Jordan-Brans-Dicke theory modified, 8 differential form, 3, 6, 23 Dreibein, 14 dilaton a-model, 350-358, 515 DSZ, see Dirac-Schwinger-Zwanziger (DSZ) quantization and secondary hair, 216 condition equations of motion, 351 dilaton BHs, 574 Eckart-Schrodinger representation, 600 dilaton field, 121, 149, 216, 218, 303, 307, 313, 472, 420, 426, Eguchi, see solution, Eguchi-Hanson 430-432, 439, 440, 445, 447, 453, 455-457, 464, 471, 477, Einbein, 14 480, 498, 510, 515, 530, 537, 569, 571, 576, 580 Einstein, 69, 73, 95, 99, 102 and electric-magnetic duality, 516 and the gravity energy-momentum tensor, 171 and fundamental /?-brane frames, 510 equation and S duality, 490 in axion-dilaton gravity, 361

674 Index

Einstein (cont.) and electric-magnetic duality, 245-258 equations, 45, 46,101, 114, 118, 120, 121, 123, 174, 176, 180, group, 246 182, 190, 197, 200, 202, 209-211, 213, 238, 541 equations of motion, 279, 227 cosmological, 102, 104, 106, 109, 112 equations of motion with sources, 238 from string theory, 434 generalized to N vectors, 237 in CSK theory, 133,134, 138, 140 in4d, 310 in dilaton a-model, 350, 351 and N = 2, d = 4 supergravity, 316 in Einstein-Maxwell theory, 227 oxidation of solutions, 320 in Einstein-Maxwell theory (duality-invariant form), 246 in5d in Kaluza-Klein (KK) theories, 302 and N = 1, d = 5 supergravity, 313 in purely affine theory of gravity, 126 in higher d, 263, 317 in supergravity theories, 156, 159, 165, 168 dimensional reduction, 291, 313-315 in the Einstein-scalar theory, 214 equations of motion, 263 interpretation of solutions, 187 oxidation of solutions, 320 not solved at singular points, 200 reduction of solutions, 317 ofNGT, 124 most general BH solutions, 262 vacuum, 92, 93, 96, 126, 188 with 0-term, 256-258 with a Dirac spinor, 131 Einstein-Maxwell-dilaton theory, see dilaton a-model frame, 302, 304, 305, 313, 319, 342, 346, 357, 434, 447, 464, Einstein-scalar theory 471,478,482,514,540,555 Einstein-Hilbert action, 20 manifest duality-invariance of the action, 303, 448, 466, 489, and Komar's formula, 180 490, 549 Euclideanized, 208 versus string frame, 530, 548 Einstein-Infeld-Hoffmann method, 413 frame (modified), 304, 305, 319, 322, 328, 335, 487, 536, 537, in 4d, 214-218 541, 576, 581, 585 in higher d, 262 and definition of mass, 305, 548, 583 oxidation of solutions, 326 versus string frame, 530, 548 JNW solutions, see solution, Janis-Newman-Winicour gauge, 57 Einstein-Straus-Kaufman theory, 124 medium-strong form of the PEGI, 99 Einstein-Yang-Mills theories, 215 metric, 727, 490, 516, 549, 578 Elfbein, 449 duality invariance, 586 energy-momentum tensor metric (modified), 549 Belinfante, see Belinfante non-symmetric gravity theory, see non-symmetric gravity canonical, 26, 29, 31, 39, 47, 56, 68, 80, 133, 135 theory) co variant, 129, 136, 140 scalar theories of gravity, 46 for a scalar field, 33 space, 547, 563, 637 of the FP theory, 79, 81 teleparallelism theories of gravity, 145 relation to the Vielbein energy-momentum tensor, 134 tensor, 9, 12, 13, 40, 96, 101, 133, 138 symmetrization and Belinfante tensor, 34, 35-37 cosmological, 104, 106, 177 for a (p + l)-form potential, 22, 23 for the Reissner-Nordstrom (RN) metric, 239 for the gravitational field in teleparallelism theories, 145 for the Schwarzschild metric, 202 improved, 37, 38 transformation under Weyl rescalings, 639 Rosenfeld, see Rosenfeld Einstein-Fokker theory, 55-57 super-energy-momentum tensors, 171 Einstein-Hilbert action, 46, 96, 102, 114, 775, 274 Vielbein, see Vielbein, energy-momentum tensor and Komar's formula, 177 equivalence principle of gravitation and inertia, 99 and Palatini's identity, 638 ERN, see solution, extreme Reissner-Nordstrom (RN) in dimensional reduction, 301, 333, 334, 436, 454, Euler characteristic 471 in4d, 144 and the string effective action, 431 Euler angles, 392, 603, 604 boundary terms, 118 Euler characteristic coupling to matter, 120 in 2d, 412, 425 coupling to the Maxwell field, 218 in 4d, 243, 274 dimensional reduction on S1, 299, 314 with boundary terms, 243 string frame, 436 Euler-Lagrange equations, 10, 27, 33 dimensional reduction on Tn, 332 event horizons Einstein versus string frames, 433, 530 and Killing horizons, 197 Euclideanized, 203 exterior derivative, 6, 23 first-order form, 96,123-126 from gauge formulation, 144 Fl solution, see solution, Fl in N = 1, d = 4 supergravity, 154 Ferrara, 164 in 2d and Euler characteristic, 411 Feynman, 45, 46, 59, 95 in CSK theory, 132,134 diagrams in string theory, 409, 412 for a Dirac spinor, 139 Fierz symmetries and gauge identities, 134-139 identities, 158, 159, 167, 169, 182 in teleparallel theories, 146, 148 genera, 620 Noether current, 120 in N = 2, d = 4 supergravity, 620 purely affine form, 126 in 4d, 625 symmetries and gauge identities, 119 Fierz-Pauli Lagrangian, 44-46, 58, 59, 63, 63-96, 101, 114 weak-field expansion, 107-108 and non-symmetric gravity theories (NGT), 124 Einstein-Infeld-Hoffmann method, 402 and string theory, 405 Einstein-Maxwell theory, 213, 218-227, 279, 285, 286, 350 coupling to matter, 67-76 and N = 2, d = 4 supergravity, 373 consistency, 76

Index 675

Noether method, 78 Schwarzschild /?-branes, 513 first-order form, 89 temperature, 205 in a curved background, 108, 103-112 and surface gravity, 204 in supergravity theories, 154 and the Euclidean time period, 209 in teleparallel theories, 146-149 for extreme Reissner-Nordstrom (ERN) BHs, 213 massive, 66 negative specific heat, 205 symmetries and Noether currents, 66 Reissner-Nordstrom (RN) BHs, 240 wave operator, 64 viewpoint on the BH information problem, 207 covariantized, 178 Heaviside system of units, 218 Figueroa-O'Farrill, 375 Heisenberg algebras, 283-284, 285 Fokker, see Einstein-Fokker theory spinorial representation, 389 FP, see Fierz-Pauli Higgs frame, 14 field, 276, 419, 420 Einstein, see Einstein, frame mechanism Kaluza-Klein (KK), see Kaluza-Klein (KK), frame and generalized dimensional reduction, 341 modified Einstein, see Einstein, frame (modified) in Kaluza-Klein (KK) theories, 294, 295, 295 non-holonomic, 75 Hilbert gauge, 57 Frolov, 188 Hilbert-Lorentz gauge, 57 fundamental string (Fl) solution, see solution, Fl Hirzebruch signature, 274 Hitchin, see solution, Atiyah-Hitchin GCT, see general coordinate transformation Hofava-Witten scenario, 348, 465, 470, 499, 519 Gauss, 24, 243 Hodge Gauss-Bonnet theorem, 243 dual, 22, 23, 24, 217, 221, 246, 258, 264, 316, 338, 343, 448, Gauss-Ostrogradski theorem, 24, 27, 117 458, 508, 539, 541 Gaussian system of units, 218 star, 22 general coordinate transformation (GCT), 3 Hoffmann, see Einstein-Infeld-Hoffmann method generalized dimensional reduction (GDR), see Kaluza-Klein (KK) Hopf fibrations, 392, 636-637 genus, 412 and the Kaluza-Klein (KK) monopole, 270 geodesic, 7, 13, 45, 98, 189, 190, 229 and the BPST instanton, 636 equation, 10, 98 and the Dirac monopole, 636 light-like, 192 horizon timelike, 56, 230 area Georgi-Glashow model, 276, 277 and entropy, 204 Gibbons-Hawking Euclidean approach, 208-211 and entropy in d dimensions, 212 Gibbons-Hawking multicenter metrics, see solution, Gibbons of Schwarzschild BH, see solution, Schwarzschild, horizon Ginzburg-Landau Lagrangian, 345 area Glashow, see Georgi-Glashow model Cauchy, 229, 230, 265, 323, 327 Goldstone boson, 295, 296 event, 187, 193,193, 194, 197, 200, 202, 203, 205, 212, 214, Gordon, see Klein-Gordon equation 229, 230, 232, 239, 243, 244, 265, 323, 324, 327, 350, 356, Goto, see Nambu-Goto (NG) action 365, 513, 516, 533, 535, 537, 539, 581 GR, see general relativity (GR) and no-hair conjecture, 196, 353 graviphoton, 475 and the cosmic-censorship conjecture, 195, 196 Green function, 287, 294, 648 in black /?-brane solutions, 513 Green-Schwarz (GS) Killing, 797, 198, 244 actions, 415-417 bifurcate, 198 superparticle action, 415 degenerate, 198, 231 superstring action, 416 Horowitz, 515 Gross, 272, 291, 329 Hosotani (or Wilson-line) mechanism, 347 Grossmann, 55 GSO projection, 422 ideal, see Lie, algebra, invariant subalgebra Gupta, 59 Infeld program, 78 Born-Infeld, see Born-Infeld (BI) Einstein-Infeld-Hoffmann method, see Einstein-Infeld Haag-Lopuszanski-Sohnius theorem, 163 instanton Hagedorn's temperature, 207 BPST, see solution, BPST Hamilton-Jacobi equation, 73, 75 D, see solution, D(—1) Hamiltonian, 114, 252, 253, 308 number, 598 bosonic-string theory, 418, 419 isometry, 21 fermionic-string theory, 422 group, see Lie, group, of isometries positivity in supersymmetric QFTs, 397 Israel Hanson, see solution, Eguchi-Hanson Israel-Khan solutions, see solution, Israel-Khan harmonic gauge, 65 Witten-Nester-Israel (WNI) technique, see Witten Harrison transformation, 267 IWP, see solution, Israel-Wilson-Perjes (rWP) Hartle-Hawking paper, 281 Hawking Jacobi Bekenstein-Hawking entropy, see Bekenstein-Hawking entropy Hamilton-Jacobi equation, see Hamiton-Jacobi equation Gibbons-Hawking Euclidean approach, see Gibbons identity, 5, 6, 8, 592, 593, 597, 603 Gibbons-Hawking multicent. metrics, see solution, Gibbons supersymmetric, 151, 163 Hartle-Hawking paper, see Hartle-Hawking paper theta functions, 545 non-decreasing-area theorem, 197 Jacobian, 4 radiation, 93, 187, 203, 205, 242, 590 Janis, see solution, Janis-Newman-Winicour absence in BPS limit, 398, 400 Johnson, 521

676 Index

Jordan metric, 302 vector, 294, 296, 348, 408, 426, 427, 439, 477, 522 Jordan-Brans-Dicke theory, 302, 431 and the RR 1-form, 458 and T duality, 482 Kahler and T duality, 437, 446, 466, 473, 480, 484, 491 metric, 367 in string theory, 437 Kaluza-Klein (KK) Melvin, see solution, Kaluza-Klein (KK) potential, 367, 368 Kalb-Ramon (KR) field, 12, 313 solution, see solution, Melvin Kaluza-Klein (KK) mode, see Kaluza-Klein (KK), mode Kalb-Ramond (KR) field, 124, 132, 420, 434 Kaluza-Klein (KK) monopole coupling to strings, 432, 432, 439 and SU(2) holonomy, 399 in teleparallelism theories, 147, 148 as M-theory solution reduction on S1, 436 Kaluza, 291 and the M superalgebra, 557 as M-theory solution (KK7M), 523 Kaluza-Klein (KK), 121, 348, 349, 350, 426, 436 intersections, 565 action, 358 worldvolume fields, 566 action dualized in modified Einstein frame, 311,313 as string-theory solution (KK6) action in Einstein frame, 302 and the IIB superalgebra, 559 action in KK frame, 301 as string-theory solution action in modified Einstein frame, 304 unbroken supersymmetries, 561 and the 4d dilaton a-model, 354 as string-theory solution (KK6), 521-523, 526 charges, 242 and the IIA superalgebra, 558 compactification radius and Dirac quantization, 312, intersections, 565 529 worldvolume fields, 566 decompactification, 296 n as string-theory solutions (KK6) dimensional reduction on T in BH constructions, 577-579, 588 moduli, 334 1 solution, see solution, Kaluza-Klein (KK) monopole dimensional reduction on S , 296-310 worldvolume action, 505, 512 Ansatz, 297 Kaufman, see Einstein-Straus-Kaufman AS shock wave, 321 Keplerian orbits, 72, 73, 189 ERN solutions, 317, 319 Kerr, see solution, Kerr massless-particle action, 306 KGn, see solution, Kowalski-Glikman in nd Newton constant, 305, 457 Khan, see solution, Israel-Khan of Einstein-Maxwell action, 313-316 Kibble, see Cartan-Sciama-Kibble (CSK) theory Scherk-Schwarz formalism, 291, 299-303, 453 Kaluza-Klein (KK), 290 Schwarzschild BHs, see solution, Schwarzschild modes n dimensional reduction on T , 331-335 in string theory, 500 and the modular group, 336 scalar, 511 global symmetries, 334 vector, 511 moduli, 334, 335 Kaluza-Klein (KK) mode Newton constant, 338 worldvolume action, 511 dimensional reduction on orbifolds, 348 Kaluza-Klein (KK) monopole Hofava-Witten scenario, see Hofava-Witten scenario Killing extreme electric KK BH conformal equation, 21 in higher d, 408 in Minkowski spacetime, 40 frame, 121, 302, 322, 357 energy, 777 generalized dimensional reduction (GDR), 339-347 equation, 27, 98, 111, 297, 370, 378, 504 and (d - 3)-branes, 342 in Minkowski spacetime, 36 and Wilson lines, 347 integrability condition, 21 metric, 121, 302 form, see Killing, metric modes, 282, 293, 294, 408, 426, 427, 447, 523, 649 horizon, see horizon, Killing and the DO-brane, 528, 554 versus Killing spinor, 183 Bogomol'nyi identity for, 310 metric, 43, 144, 284, 385, 498, 594, 594, 603-606 in string theory, 426, 428, 436, 442 of so(«+,«_), 600 masses and charges, 310 spinor, 176,183, 373, 378, 394-396, 399, 400, 496, 563, 608 worldline action, 309 as fermionic generator of a symmetry superalgebra, 373, 374, moduli, 299 378 and T duality, 492 bilinear and bosonic generators, 374, 379 Newton constant, 431 dependence on internal coordinates, 391, 482^483 principle, 297 of AdS4,389 scalar, 295, 296, 311, 348, 349, 426, 441, 576 of Minkowski spacetime, 384 and the dilaton, 447, 458 of the D/?-branes, 561 and T duality, 437, 491 of the F 1,562 l skew dimensional reduction on S , 330 oftheKK7M-brane,567 spectrum, 294-296, 372 oftheM2-brane, 560 massless modes, 295 of the M5-brane, 560 S duality of 4d theory, 491 of the S5-brane, 562 vacuum, 292, 372 preserved by the covariant Lie derivative, 379 and T duality, 484 versus Killing vector, 183, 374, 384 instability, 296 spinor equation, 373, 378, 379 metric, 292 and holonomy, 399 moduli problem, 292 as a paralellism equation, 380 symmetries, 292, 295 integrability equation, 380

Index 677

integrability equation in N = 2, d = 4 theories, 382-383 ofGL(rf), 15 of N = 1, d = 4 AdS supergravity, 385 ofSO(l, 2), 602 of N = l,d = 4- Poincare supergravity, 183, 398 ofSO(3), 270 of N = 1, d = 6 supergravity, 390 of SO(4) (anti-)self-dual generators, 275 of TV = 2, d = 4 Poincare supergravity, 387 of SO(n+,n-), 600 spinors of SO(«+, «_) (spinorial representation), 601 general form in maximally supersymmetric vacua, 380 of SU(2) (SO(3)), 602 T-duality tranformation rules, see Buscher T duality, in type-II of isometries, 371, 379 theories, transformations of Killing spinors of the conformal group SO(2, d - 1), 40 vector, 21, 98, 110, 111, 177-180, 183, 270, 288, 330, 370, 375, of the Lorentz group SO(l, d - 1), 40 378, 391, 504, 608 of the Lorentz group SO(l, d — 1) (spinorial representation), and symmetry of the point-particle action, 307 677,612 as bosonic generator of a symmetry superalgebra, 374, 378 of the Poincare ISO(l, d - 1), 35 as generators of the isometry algebra, 371, 504 of the Poincare ISO(l, d - 1), 143 commuting and compactification on Tn, 331 of the Poincare group ISO(l, d - 1), 36, 40, 384 conformal, 21 reductive decomposition, 605 conformal and Lie-Lorentz derivative, 376-377 semidirect sum, 605 gauge, 176 semisimple, 594, 594 null, covariantly constant and Brinkmann metrics, 282, 647 simple, 594 of coset manifolds, 606, 607 solvable, 284, 594, 594 of maximally symmetric spaces, 605 symmetric decomposition, 605 of Minkowski metric, 67 bracket, 5, 80, 592 of Minkowski spacetime, 384 and commutators of matrices, 592 of stationary axially symmetric metrics, 268 brackets of the Minkowski metric, 35, 99, 178 and Ricci rotation coefficients, 15 spacelike and Kaluza-Klein (KK) compactification, 297-299 covariant derivative, 608 timelike and definition of mass, 179 derivative, 5, 6, 8, 13, 297, 371, 374, 379, 608 timelike and Killing horizon, 244 //-covariant, 608 timelike and staticity, 188 and extrinsic curvature, 25 timelike and surface gravity, 198 and Killing vectors, 20 timelike and the Schwarzschild metric, 191 covariant, 369, 375, 375-378, 379 translational, 779 properties, 5 versus Killing spinor, 374, 384 spinorial, see Lie, Lie-Lorentz derivative KK6-brane, see Kaluza-Klein (KK) monopole, as string-theory group, 591 solution (KK6) ISO(«+,«_), 599 KK7M-brane, see Kaluza-Klein (KK) monopole, as M-theory SO(n+,n-),598 solution (KK7M) SO(«+, «_), vector representation, 599 KK8A-brane, 526 SU(2), 603 KK9A-brane, 526 affine group IGL(d, M), 17 KK9M-brane, 465, 524 and N = 1, 2, d = 6 vacua, 390 and the M superalgebra, 557 compact, 593 Klein, 290 compact (Weyl theorem), 594 Kaluza-Klein theories, see Kaluza-Klein conformal SO(2, d - 1), 40 Klein-Gordon equation, 36, 293, 294 de Sitter (anti-), see de Sitter (anti-), group Komar's formula, 180, 221 invariant subgroup, 594 compared to Abbott-Deser approach, 180 Lorentz SO(l, d - 1), 17, 32, 142 in higher d, 211 of isometries, 370 Kosmann, 375 Poincare ISO(l, d - 1), 17, 26, 32, 46, 48, 49, 52, 96, 108, Kraichnan, 59 114, 127, 132, 134, 137, 140-142, 150 Kretschmann invariant, 790 Riemannian geometry, 602-604 semisimple, 594 Lagrange, see Euler-Lagrange simple, 594 Landau translation, 145 Ginzburg-Landau Lagrangian, see Ginzburg-Landau Lie-Lorentz derivative, 375, 375-377, 384, 608 Landau-Lifshitz energy-momentum pseudo tensor, 173,174-176 and //-covariant Lie derivative, 381 compared with Abbott-Deser approach, 178, 179 Lie-Maxwell derivative, 375, 377-378, 608 Laplace equation, 272, 280, 294, 551, 648, 649 superalgebra Legendre transformation, 308, 442 N = 1, d = 4 Poincare, 151 Leibniz rule, 5, 6, 376 supergroup level operators, 419 de Sitter (anti-), see de Sitter (anti-), supergroup Lichnerowicz, 375 Poincare, 150 Lie Lie-Lorentz derivative, see Lie, Lie-Lorentz derivative algebra, 5, 43, 597, 592, 595 Lie-Maxwell derivative, see Lie, Lie-Maxwell derivative Abelian, 594 Lifshitz, see Landau-Lifshitz energy-momentum pseudotensor Bianchi classification of 3d real Lie algebras, 602 Lindquist, see coordinates, Boyer-Lindquist complexified, 593 loop quantization, 138 de Sitter (anti-), see de Sitter (anti-), algebra Lopuszanski, see Haag-Lopuszanski-Sohnius theorem derived subalgebra, 594 Lorentz Heisenberg, see Heisenberg algebras group, see Lie, group, Lorentz invariant subalgebra, 594, 605 Hilbert-Lorentz gauge, see Hilbert-Lorentz gauge nilpotent, 284, 594, 594 Lie-Lorentz derivative, see Lie, Lie-Lorentz derivative of isometry group, 183 Lovelock tensor, 101

678 Index

MacDowell-Mansouri formulation, 114, 140, 742 NGT, see non-symmetric gravity theory (NGT) Majorana Nicolai-Townsend transformations, 437 conjugate, 601, 615 nilpotent, see Lie, algebra, nilpotent in arbitrary signature, 628 Noether representation, 617 approach spinors, 617, 628 to conserved charges in GR, 179-180 Majorana-Weyl spinors, 617, 628 current, 29, 31, 36, 38, 43, 61, 66, 77, 80, 82, 85, 93, 110, 114, Maldacena, 536, 573 178, 305, 339 manifold, 3 for Lorentz transformations, 34 Mansouri, see MacDowell-Mansouri formulation for translations, 32 Maurer-Cartan in Maxwell theory, 221, 223, 227 1-form, 278, 380, 382, 385, 387, 389, 391, 592, 603, 606, 607, in Maxwell theory with #-term, 256 636 of the CSK theory, 134-136 SO(n), 609 of the Einstein-Hilbert action, 119-120 SU(2), 604 superpotential, 30, 30 horizontal, 606 method, 41^44, 46, 59 vertical, 606 for gravity, 78-88, 95 equations, 380, 592, 592, 603, 604, 607 for supergravity, 154 Maxwell method, 78 Lie-Maxwell derivative, see Lie, Lie-Maxwell derivative theorems, 26^44, 49, 61, 98, 217 Melvin non-symmetrc gravity theory (NGT), 113 Kaluza-Klein (KK) Melvin solution, see solution, Kaluza-Klein non-symmetric gravity theory (NGT), 114,124-126 (KK) Melvin Einstein-Straus-Kaufman theory, see Einstein-Straus solution, see solution, Melvin Nordstrom Mercury (precession of the perihelion), 45, 46, 57, 59, 76, 78, 85, Kaluza-Klein theories, 290 88 Reissner-Nordstrom (RN) BH, see solution, RN metric, 9 scalar theories of gravity, 46, 55 conformastationary, 280 and Einstein-Fokker's, 56 induced, 98 Nordtvedt effect, 77 Killing, see Killing, metric Novikov, 188 postulate, 10 NS, see Neveu-Schwarz Ricci-flat, see Ricci, -flat metric NS5-brane, see S5-brane Mills, see Yang-Mills NUT Misner, 176, 270 charge, 269 moduli, 218, 407 charge versus magnetic mass, 269 monopole Euclidean Taub-NUT, see solution, Euclidean Taub-NUT 't Hooft-Polyakov, see solution, 't Hooft-Polyakov monopole Taub-NUT, see solution, Taub-NUT Dirac, see Dirac, magnetic monopole Kaluza-Klein (KK), see solution, Kaluza-Klein (KK) Op planes, see orientifold (Op) planes Wu-Yang, see Wu-Yang monopole Ogievetsky, 58, 80 MP, see solution, Majumdar-Papapetrou Olive, 395 orientifold (Op) planes, 421 Nambu-Goto (NG) action, 528 Ostrogradski, see Gauss-Ostrogradski theorem for /?-branes in curved spacetime, 501, 503 wrapped on p dimensions, 501 Palatini for D/?-branes, 511 formalism, see Einstein-Hilbert action, first-order form for fundamental /?-branes, 511 identity, see Einstein-Hilbert action, and Palatini's identity for Kaluza-Klein /?-branes, 512 for the variation of the Ricci tensor, 705, 124, 132, 137 for momentum modes, 511 Papapetrou for point-particles in curved spacetime, 97-99, 224 equations for a pole-dipole singularity, 413 coupled to a scalar, 306, 309 Majumdar-Papapetrou solutions, see solution, Majumdar for point-particles in Minkowski spacetime, 51, 52 parallel for solitonic /?-branes, 511 spinor, see Killing, spinors for strings in curved spacetime, 409, 410, 413, 435 transport, 6 for winding modes, 511 Paton, see Chan-Paton factors Nambu-Goto (NG) action Pauli Nester Fierz-Pauli Lagrangian, see Fierz-Pauli Lagrangian Witten-Nester-Israel (WNI) technique, see Witten matrices, 164, 274, 275, 383, 468, 488, 496, 613, 620 Nester 2-form, 181 metric, 617 Neugebauer, 281 terms, 94 Neumann (N) boundary conditions, 410 Peccei-Quinn symmetry, 360 Neveu PEGI, see principle of equivalence of gravitation and inertia Ramond-Neveu-Schwarz (RNS) model, see Pellegrini-Plebanski Lagrangian, 145-148 Ramond-Neveu-S ch warz Penrose Neveu-Schwarz (NS) boundary conditions diagram, 193 for closed superstrings, 414 Dp p < 7 spacetime, 540 for open superstrings, 414 extreme Reissner-Nordstrom (ERN) spacetime, 231 Newman M2 spacetime, 533 Janis-Newman-Winicour solutions, see solution, naked singularity, 195 Janis-Newman-Winicour Reissner-Nordstrom (RN) spacetime, 229 Kerr-Newman solution, see solution, Kerr-Newman Schwarzschild spacetime, 193 Newman-Penrose formalism, 131 limit, 283, 286, 390, 563

Index 679

Newman-Penrose formalism, see Newman-Penrose formalism and type-IIA superstrings in a D8 background, 539 Perjes, see solution, Israel-Wilson-Perjes (rWP) N = 2A,d = lO supergravity, 303, 348, 508, 551, 554, 573 Perry, 272, 291, 329 action, 457 PGR, see principle of general covariance or relativity and type-IIA superstrings, 422, 430, 447, 523 Planck, 51 compactification on K3 and N = 1, d = 10 on T4, 481 constant, 292 compactification on S , 466-469 length, 103, 775, 116 compactified on T" and U duality, 587 and horizon entropy, 204, 582 supersymmetry transformations, 462, 463 and S duality in Kaluza-Klein (KK) theory, 312 truncation to N = l,d = 10 supergravity, 469 compared with the Compton wavelength and Schwarzschild N = 2B,d = \0 supergravity, 338, 486-491, 508, 551, 554, radius, 116,206 573 in lid, 454, 457, 457, 529 action (NSD) in Einstein frame, 490 in 1 Id versus the string length, 524 action (NSD) in string frame, 486 reduced, 775 and type-IIB superstrings, 407, 422, 430, 485 size of compact dimensions, 292, 293 compactification on S1, 491-496 versus string length, 405 compactification on S5, 540, 563 mass, 116,116, 242 Euclidean, 540 masses of Kaluza-Klein modes, 293 reduced, 116 supersymmetry transformations, 488 Plebafiski truncation to N =\,d = 10 supergravity, 497^499 Pellegrini-Plebaiiski Lagrangian, see Pellegrini-Plebanski vacua, 563 Plebanski-Demiafiski solutions, see solution, Plebafiski N = 4,d = 4 supergravity, 281, 338, 357, 362, 363, 366, 407, Poincare 541 N = (1, 0), d = 6 supergravity, 316, 320 action, 361, 482 action, 390 and heterotic strings on T , 349 supersymmetry transformation, 390 supersymmetric solutions, 400 vacua, 390-391 N = 4, d = 5 supergravity, 578 N = 1, d = 1 supergravity and U duality, 586 action, 472 N = 8, d = 4 supergravity, 575 and U duality, 586 supersymmetry transformations, 472 duality, 248 N = l,d = 10 supergravity, 300, 303, 338, 354 of Kaluza-Klein (KK) theory, 310 action, 470 gauge theory of gravity, 146 andheterotic superstrings, 348, 361, 423 group, see Lie, group, Poincare and type-I superstrings, 423 Poisson equation, 48 compactification on T4 and N = 2 A, d = 10 on K3, 481 n Polchinski, 435, 521 compactification on T , 471-482 Polubarinov, 58, 80 supersymmetry transformations, 470 Polyakov, 261, 267 N=l,d = U supergravity, 290, 300, 303, 449-451, 531, 551, 't Hooft-Polyakov monopole, see solution, 't Hooft 572 Polyakov-type action action, 449 as a-model, 99 and M theory, 348, 447, 523 for massless point-particles coupled to linearized gravity, 70 and the supermembrane, 417 for massless point-particles in curved spacetime, 97, 307 1 compactification on S , 452-458, 461-463 as source for AS shock wave, 287 compactification on S4, 536, 563 reduction on S1, 307-310 compactification on S7, 534, 563 for massless point-particles in Minkowski spacetime, 52 compactification on T2, 338 for point-particles in curved spacetime, 97 compactification on T3, 338 for point-particles in Minkowski spacetime, 52 supersymmetry transformations, 457 for strings in curved spacetime, 470 vacua, 563 topological term, 411 N =l,d = 2 supergravity supersymmetric, see Poincare, N = l,d = 2 supergravity action, 413 Pontrjagin class (first), 274 and superstrings, 412-414 Prasad, see Bogomol'nyi-Prasad-Sommerfield (BPS) supersymmetry transformations, 413 principle of correspondence, see correspondence principle N=l,d = 4 supergravity, 134, 151,155-159, 377 principle of equivalence of gravitation and inertia (PEGI), 56, 69 action, 754, 755 and the principle of general covariance or relativity (PGR), 93 super symmetric solutions, 398-399 strong form and self-coupling of the gravitational field, 77, supersymmetry transformations, 156 78 vacua, 383-384 weak form and identity between gravitational and inertial N=l,d = 5 supergravity, 291, 320, 393 masses, 72 action, 313 principle of general covariance of relativity (PGR), 96, 99-102 supersymmetry transformation, 392 and point-particle actions, 96 vacua, 392-393 and the equations of motion of the gravitational field, 101 N = 2,d = 4 supergravity, 151, 164-167, 374, 375, 382, 393, and the Lovelock tensor, 101 396, 397 Proca Lagrangian, 60, 225-227 action, 164 dualization, 258-259 from N = 1, d = 5 supergravity, 316 Pythagoras, 434 super symmetric solutions, 399-401 supersymmetry transformations, 166 QFT, see quantum field theory vacua, 386-389 quantum field theory (QFT), 143, 188, 213, 369, 371, 372, 374, N = 2A, d = 10 massive (Romans') supergravity, 463-465, 502 397, 573, 676 action, 464 Quinn, see Peccei-Quinn symmetry

680 Index

R, see Ramond Ramond-Neveu-Schwarz (RNS) model, see Ramond Ramond-Neveu-Schwarz model Kalb-Ramond (KR) field, see Kalb-Ramond Scherk-Schwarz formalism for dimensional reduction, see Ramond (R) boundary conditions Kaluza-Klein (KK) for closed superstrings, 414 Schwarzschild for open superstrings, 414 coordinates, 176, 188, 189, 228, 269 Ramond-Neveu-Schwarz (RNS) model, 414 problems of, 191 Randall-Sundrum scenarios, 519 observer, 192, 193 Rarita-Schwinger spinor, 134, 752, 154 radius, 74, 116,116, 189-191, 589 RB, see solution, Robinson-Bertotti (RB) at the correspondence point, 577 reduction theorem, 273 compared with the Compton wavelength and Planck length, Regge slope, 409, 547 116,206 Reissner-Nordstrom (RN) BH, see solution, Reissner-Nordstrom solutions, see solution, Schwarzschild rheonomic approach, see supergravity theories, rheonomic time, 192, 193 Ricci Sch winger -flat metric, 93, 277, 287, 547 Dirac-Schwinger-Zwanziger (DSZ) quantization condition, see 1-form, 25 Dirac-Schwinger-Zwanziger (DSZ) quantization condition identities, 8 pair creation, 205 identity, 12,18, 21, 109, 111, 158, 182, 602 Rarita-Schwinger spinor, see Rarita-Schwinger spinor for GL(d, R) connection, 16 Sciama, see Cartan-Sciama-Kibble (CSK) theory for Yang-Mills fields, 597 semisimple algebra (group), see Lie, algebra (group) rotation coefficients, 75, 145, 333 Sen argument, 245, 434 scalar, 9, 12, 13, 40, 56, 102, 106, 114, 115, 132, 301, 305, 333, Sexl, see solution, Aichelburg-Sexl shock wave 454,471,482 Simons, see Chern-Simons linearized, 94 simple algebra (group), see Lie, algebra (group) tensor, 8, 12, 13, 91, 92, 94, 107, 123, 126, 273 Smarr formula of group manifolds with bi-invariant metrics, 603 for RN BHs, 240 Riemann for Schwarzschild BHs, 203, 205 2-form, 25 in higher d, 212 space, 13-19 Sohnius, see Haag-Lopuszaiiski-Sohnius theorem tensor, 7, 7, 12, 14, 107, 126, 605 solution of a maximally symmetric space, 610 2 AdS2 x S , 235 relation with the curvature of the GL(d, R) connection, 16 d 2 AdS2 x S ~ , 266 Riemann-Cartan spacetime, 10-12, 144 3 RN, see solution, Reissner-Nordstrom (RN) AdS3 x S , 390 7 RNS model, see Ramond-Neveu-Schwarz AdS4 xS ,5J4, 560, 563 5 Robinson AdS5 xS , 540,563 4 Bel-Robinson tensor, see Bel-Robinson tensor AdS7 x S , 536, 560, 563 Robinson-Bertotti (RB) solution, see solution, AdSd,534 Robinson-Bertotti as coset spaces, 610 Romans, see Poincare, N = 2A, d = 10 massive S1 Rosenfeld energy-momentum tensor, 34, 35, 38, 39-41, 58, 59, 68, as a Hopf fibration, 636 73, 80, 82, 93, 130 Sn, 634-635 and gauge identities, 102, 172 Sn as coset spaces, 609-610 for a massive point-particle, 50 p-brane a-model, 575 for the first-order FP theory, 90, 93 with extra isometries, 516 for the FP theory, 83, 85, 86 pq-5, 556 and corrections to the point-particle solution, 87 pq-strings, 555 relation to the one predicted by GR, 84 AdSj of embedding coordinates, 410 as coset spaces, 384 Rosenfeld energy-momentum tensor, 172 't Hooft-Polyakov monopole, 267, 276 Rumpf Lagrangian, 146 BPS limit, see Bogomornyi-Prasad-Sommerfield (BPS) Aichelburg-Sexl (AS) shock wave, 71, 286 SRFT, see special-relativistic field theory dimensional reduction, see Kaluza-Klein (KK), dimensional SRQFT, see special-relativistic quantum-field theory reduction on S1 S5-brane Atiyah-Hitchin, 279 solution, see solution, S5 axion-dilaton BH, 363 Bianchi IX gravitational instantons, 277-279 Sagnac connection, 283 Bonnor's magnetic dipole, 196 Scherk-Schwarz BPST instanton, 267, 272, 274-275 formalism for dimensional reduction, see Kaluza-Klein (KK) and the KK monople, 277 Schoen-Yau positive-energy theorem, 181 and the round S7, 636 Schrodinger and the S5-brane, 562 Eckart-Schrodinger representation, see Eckart-Schrodinger BTZ black hole, 211,590 representation Chazy-Curzon, 324 Schrodinger composite 4d BHs, 574 equation, 252, 253 N = 8, d = 4 central charges, 575 Schur's lemma, 616 ADM mass, 575 Schwarz (A.S.), see solution, BPST instanton horizon area, 575 Schwarz (J.H.) relation with dilaton BHs, 575 Green-Schwarz (GS) actions, see Green-Schwarz (GS) actions D(-l), 541 Neveu-Schwarz (NS) boundary conditions, see Neveu D3, 539

Index

Dp, 538 in higher d, 263,351 compactified on T6, 577 massless, 216 compactified on JP, 576 scalar hair, 216 Dp (black), 532 Kaluza-Klein (KK) Melvin, 331 Dp (black, p < 7), 538 Kaluza-Klein (KK) monopole, 267, 270, 272, 329 D7, 543 and the D6-brane, 554 D7 (another form), 544 and T duality, 555 D7 (modular-covariant form), 544 as a Bianchi IX gravitational instanton, 278, 279 dilaton a = 1 BH (dyonic), 357 as M-theory solution (KK7M), 460 dilaton a = 1, d = 4 BH (electric), 355 worldvolume action, 522 dilaton a = 1/V3, rf = 4BH (electric), 355 Kaluza-Klein (KK) monopole, 501 dilaton a = V3, d = 4 BH (electric), 354, 576, 577 Kerr, 180, 196,268 dilaton BHs, 350 and cosmic censorship, 268 thermodynamics, 356 interior solution not known, 268 dilaton general a, d BH (electric), 352 Kerr-Newman, 267, 279 dilaton general a, d = 4 BH (electric), 354 boosting, 286 Eguchi-Hanson, 272, 277 in IWP class, 281 as a Bianchi IX gravitational instanton, 278 in SWIP class, 366 as a Bianchi IX gravitational instanton with self-dual supersymmetric, 399 connection, 279 Kerr-Newman, 195, 196 Euclidean Taub-NUT, see solution, Kaluza-Klein monopole Kowalski-Glikman Hpp-waves, 381 multicenter, 272 Kowalski-Glikman in lOd (KG10), 434, 563 extreme electric KK BH Kowalski-Glikman in 1 Id (KG11), 563 in higher d, 322, 357 Kowalski-Glikman in 4d (KG4), 286, 382, 386, 393, extreme Reissner-Nordstrom (ERN), 231, 291, 310, 397, 532, 399 533 supersymmetry algebra, 389 4d entropy, Euclidean calculation, 244 Kowalski-Glikman in 5d (KG5), 287, 393 and equilibrium of forces, 234 Kowalski-Glikman in 6d (KG6), 391 as a supersymmetric solution, 398 dimensional reduction, 393 as an interpolating soliton, 235 Mp (black), 531 as supersymmetric solution, 399, 400 M2, 533 dimensional reduction, see Kaluza-Klein (KK), dimensional M2 (black), 533 reduction M5, 535 in 4d Euclidean, 242, 244 M5 (black), 535 in 4d isotropic coordinates, 232 Majumdar-Papapetrou (MP), 397 in IWP class, 281 dyonic in 4d isotropic coordinates, 260 near-horizon limit, 235 in 4d isotropic coordinates, 233 oxidation, 320 in higher d, 317 reduced on S1, 319 in higher-d isotropic coordinates, 265 reduction on S1, 317-320, 322 Melvin, 267, 331 Extreme Reissner-Nordstrom (ERN) Plebanski-Demiafiski, 400 in 4d as composite BHs, 575 Reissner-Nordstrom (RN), 268, 356 near-horizon limit, 519 4d dyonic in Schwarzschild coordinates, 260 Penrose diagram, 231 4d entropy, 240, 244 Fp (black), 531 4d magnetic, 260 Fl,445 4d temperature, 240 compactified on S1, 576 ADM mass, 228 compactified on T6, 576 and the cosmic-censorship conjecture, 230 Fl (black), 536, 536 as Weyl's static axisymmetric electrovacuum solution, Godel-like solution of N = \,d = 5 SUGRA, 381, 393, 393 228 Gibbons-Hawking multicenter metrics, 272 Cauchy horizon, 229 and self-duality, 273 derived, 227 wire singularities, 272 discharge, 205 intersecting branes horizon area, 229 /?-brane (a\ — a'l) model, 571 in 4d, 280 D5 || Dl || W (black), 580 in 4d alternative coordinate system, 236 D5 || Dl || W (black, reduced to 5d), 581 in 4d dressed-Schwarschild form, 236 D5 || Dl || S5A || W (black), 584 in 4d Euclidean, 242, 273 D5 || Dl || S5A || W (black, reduced to 4d), 585 in 4d Schwarzschild coordinates, 228 D5 || Dl (black), 579 in 4d, as a string solution, 585 M2 _L M2(0), 572 in 5d as a string solution, 581 M2_LM5(1), 572 in dilaton a-model, 351, 352, 358 Israel-Khan, 323, 324 in higher d, 265 and periodic arrays of Schwarzschild BHs, 325 Penrose diagram, 229 Israel-Wilson-Perjes (IWP), 280, 281 reduction on Sl, 325 as solutions of N = 2,d = 4 supergravity, 399, 400 sources, 238-239 as solutions of N = 4, d = 4 supergravity, 400 specific heat, 241 connection and curvature, 642 thermodynamics, 240-242 included in SWIP class, 366 with N electric charges, 237 Neugebauer, see Neugebauer Reissner-Nordstrom-Kaluza-Klein (RNKK) dyon, Janis-Newman-Winicour 330, 350 in 4d, 275, 351 Reissner-Nordstrom-(anti-)de Sitter, 779

682 Index

solution (cont.) Weyl's static axisymmetric electrovacuum, 228 Robinson-Bertotti (RB), 235, 286, 382, 399, 519 solvable, see Lie, algebra, solvable Reissner-Nordstrom (RN), 196, 203 Sommerfield, see Bogomornyi-Prasad-Sommerfield (BPS) as a vacuum ofN = 2,d = 4 SUEGRA, 386 Sorkin, 272, 291, 329 dyonic in 4d, 394 space electric, in 4d, 235 affinely connected, 6 electric, in higher d, 266 cotangent, see cotangent space in 4d Euclidean, 242 maximally symmetric, 605 symmetry superalgebra, 386-388, 608 Riemann, see Riemann, space Sp (black), 531 Riemann-Cartan, see Riemann-Cartan spacetime S5, 537 tangent, see tangent space S5 (black), 537 Weitzenbock, see Weitzenbock, spacetime Schwarzschild, 187-212, 215, 216, 225, 228, 230, 234, 236, 238, special geometry, 367-368 240, 242, 578 special-relativistic field theory (SRFT), 26, 32, 33, 45-47, 49, 51, 4d entropy, 204 53, 55, 57-61, 63, 65, 67-73, 75, 77-79, 81, 83, 85, 87, 89, 4d temperature, 204 91, 93, 95, 96, 100, 112, 171, 173, 200 ADM mass, 189 special-relativistic quantum-field theory (SRQFT), 95, 112 and supersymmetry, 398 specific heat as an eternal BH, 194 dilaton a-model BH, 356 as Weyl's axisymmetric vacuum solution, 324 Reissner-Nordstrom (RN) BH, 213, 247 boosting, 286 Schwarzschild BH, 205 derived, 188 spin-energy potential, 34, 36,41, 131, 135, 136 dimensional oxidation of, 326 spin-energy potnetial, 133 dimensional reduction of, 323-326 spinning particle, 413 entropy in higher d, 212 spinor, 601 horizon area, 197 Dirac, see Dirac, spinor horizon area in higher d, 212 Majorana, see Majorana, spinor in 4d Eddington-Finkelstein coordinates, 191 Majorana-Weyl, see Majorana-Weyl, spinor in 4d Euclidean, 209-210, 242, 273 symplectic-Majorana, 618, 629 in 4d Euclidean action, 211 Weyl, see Weyl, spinor in 4d harmonic coordinates, 200 squashed in 4d isotropic coordinates, 198 S3 and S7, 637 in 4d Kruskal-Szekeres coordinates, 193, 194 spheres, 609 in 4d Schwarzschild coordinates, 176, 179,188, 268, 269 Stiickelberg in higher d, 211, 263, 264, 352, 512, 578 field, 226, 227, 258, 259, 296, 314, 341, 342, 345, 465 in string theory, 405 Stokes' theorem, 24, 29, 31, 32, 175, 178, 181, 222, 223, 250, 255, interior, 190 257, 276, 306, 507, 508 Penrose diagram, 193 and the Gauss-Ostrogradski theorem, 24 periodic arrays of, 325 Straus, see Einstein-Straus-Kaufman perturbations of, 195 string perturbative expansion, 88 USp(32) theory, 498 singularities, 190-192, 230, 238 coupling constant, 116, 412, 420, 425, 431, 456, 487 sources, 200-202 heterotic theories, 423 specific heat, 205 length, 116, 292, 409, 430, 431, 520, 527, 547, 577, 582 stability, 189 versus lid Planck length, 524 surface gravity, 198 versus Planck length, 405 temperature in higher d, 212 mass, 409 thermodynamics, 202-211, 241, 356 metric, 121 with negative mass, 195 oriented, 411 Schwarzschild /?-branes, 326, 331, 512-514, 515 singularity, see Dirac, string singularity instability of, 513-514 type-I theories, 423 tension, 513 type-l' theory, 499 SWIP, 363-368, 544, 588 type-IIA theories, 422 as N = 2, d = 4 supergravity solutions, 367-368 type-IIB theories, 422 as solutions of N = 4, d = 4 supergravity, 400 unoriented, 411 duality properties, 366 Strominger, 515 entropy and N = 4, d = 4 central charges, 401 Strominger-Vafa paper, 579 general, 363-365 SUEGRA, see supergravity theories, extended horizon area, 365 SUGRA, see supergravity theories supersymmetric, 365-366 Sundrum, see Randall-Sundrum scenarios Taub-bolt, 273 mechanism, 346, 465 as a Bianchi IX gravitational instanton, 278 supergravity theories, 113, 126, 132, 313, 349, 354, 358, 359, Taub-NUT, 267-271 369-371, 373, 375, 380, 405, 413, 417, 424 charged, 268 N = 2, d = 4 supergravity and special geometry, 367 electrically charged, 280 and the WNI technique, 181, 395 extreme electrically charged, 280 as effective string theories, 290, 406-408, 432, 435 in Schwarzschild coordinates, 269 as supergroup gauge theories, 140, 142,151-155 interior solutions, 271 and supersymmetric solutions, 557-559 Weyl's axisymmetric vacuum, 323 Bogomol'nyi (or BPS) bounds, see Bogomol'nyi (or BPS) boosting, 286 bounds higher-d generalizations, 326 bosonic truncation, 373 Schwarzschild solution, 324 de Sitter (anti-), 143, see de Sitter (anti-)defined, 150

Index 683

extended, 121, 150,160-163, 379, 381 Thome, 194 quasi-central charges, 397 Tod, 399, 400 gauged, see de Sitter (anti-) torsion, 6, 8 hidden symmetries and dualities, 448 modified, 8 manifest duality invariance in the Einstein frame, see Einstein, Townsend, 188,437 frame, manifest duality invariance Nicolai-Townsend transformations, see Nicolai-Townsend massive, 342 translation group, see Lie, group, translation Romans', see Poincare, N = 2A, d = 10 massive (Romans') trator, 146 supergravity twisted sectors, 421 matter fields, 316 type-I superstring theories, see string, type II Poincare, see Poincare type-I' superstring theory, see string, type Ir rheonomic approach, 139 type-II superstring theories, see string, type II super symmetric solutions, 396 tyupkin, see solution, BPST instanton and string states, 408 T duality, 247, 426-588 vacua, 374, 379 and D-branes, 443-445 supergravity theory, 374 and O-planes, 424 superparticle (Green-Schwarz), see Green-Schwarz and string BH solutions, 587 superpotential, 31 between Eg x Eg and SO(32) heterotic superstrings on S1, 478 superstring (Green-Schwarz), see Green-Schwarz between massive N = 2A, d = 10 and N = 25, d = 10 supersymmetric states, see BPS states supergravity, 466 surface gravity, 198, 212 between type-I and type-I/ theories, 499 and temperature, 204 between type-IIA and type-IIB theories, 491, 522 and temperature in d dimensions, 212 and extended objects, 522-523, 524-526, 541 of Schwarzschild solution, see solution, Schwarzschild, surface and intersections of extended objects, 565 gravity and the Chern-Simons term, 451, 487 Susskind, 206 and the superalgebra, 559 S duality, 247, 262, 291, 338, 500, 586, 588 Buscher's rules, see Buscher T duality, in type-II theories and generation of solutions, 329 between winding and momentum modes, 441-442 group, 338 Buscher's rules, see Buscher T duality inN = 4,d = 4 supergravity, 361, 362, 363, 366, 401 group for the heterotic string on Tn in 4d axion-dilaton gravity, 359-361 and pure N = 4, d = 4 supergravity, 482 in 4d dilaton a-model, 357-358 group for the heterotic superstring on Tn, 448, 473, 477, 480 in 4d Kaluza-Klein (KK) theory, 310-313, 408 inN = 4,d = 4 supergravity, 361, 362, 363, 366, 401 and generation of solutions, 328 self dual radius and symmetry enhancement, 313 in axion-dilaton gravity, 349 in string theory, 407, 407 invariance of the DSZ quantization condition, 258 U duality, 247, 312, 407, 523, 578, 586 of N = 2B,d = 10 supergravity and string BH solutions, 587 solutions, 555-556, 564 diamond-invariant, 587 superalgebra, 558 groups in various dimensions, 449 of N = 2B,d = 10 supergravity and the type-IIB superstring, 488-491 Vafa, see Strominger-Vafa paper of heterotic superstring theory on T6, 448, 480, 482 van Nieuwenhuizen, 164 of type-IIB superstring theory, 447, 485, 494, 497, 521-522, vector, 3 523-525, 539, 542, 549 world, 4 and its relation to M theory on T , 493 Vielbein, 11,74 Ansatz for dimensional reduction from d = lltod = 10, 456 Ansatz for dimensional reduction on S1, 300 tadpoles, 424 n tangent space, 3 Ansatz for dimensional reduction on T , 332 Taub-bolt, see solution, Taub-bolt energy-momentum tensor, 128, 133, 135, 136 Taub-NUT, see solution, Taub-NUT conservation, 129 Teitelboim, see solution, BTZ black hole for a Dirac spinor, 131 tensor, 3 relation to the canonical energy-momentum tensor, angular momentum, see angular-momentum tensor 130, 134 Bel-Robinson, see Bel-Robinson tensor first postulate, 16,16, 17,17, 18, 20 Belinfante, see energy-momentum tensor, Belinfante formalism, 19, 113 curvature for the Einstein-Hilbert action, 136 of GL(d, R) connection, 16 second postulate, 16 of affine connection, see Riemann, tensor Vierbein, 14 of Yang-Mills fields, 597 von der Heyde model, 148 density, 4 Einstein, see Einstein, tensor Wald, 80, 203 energy-momentum, see energy-momentum tensor Wallach, see Cahen-Wallach symmetric spacetimes Lovelock, see Lovelock tensor Weinberg, 59, 78 non-metricity, 9 Weitzenbock Ricci, see Ricci, tensor connection, 11,19, 145 spin-energy potential, see spin-energy potential invariants, 145 torsion, see torsion spacetime, 11,79-20, 145 Weyl, see Weyl, tensor and teleparallelism, 20, 145 tentor, 146 Wess-Zumino (WZ) term tetrad, 14 for /?-branes, 528 Thirring, 46, 59, 70, 82, 87 for a point-particle, 224, 225, 306

684 Index

Wess-Zumino (cont.) Wilczek, 144 in the string worldsheet action, 432 Wilson (G. A.), see solution, Israel-Wilson-Perjes (IWP) supersymmetric, 417 Wilson line, 347 for D/?-branes, 435, 443 and D-branes, 428, 429, 499 in the superstring GS action, 416 mechanism, see Hosotani (or Wilson-line) mechanism Wess-Zumino (WZ) term Winicour, see solution, Janis-Newman-Winicour for /?-branes, 507-509 wire singularities Weyl, 57 of Gibbons-Hawking multicenter metrics, see solution, Gibbons axisymmetric vacuum solutions, see solution, Weyl of Taub-NUT solution, see Dirac, string singularities basis, 162 Witten, 395 canonical cordinates, 323 condition, 183 formalism for fermions in curved spacetime, 18, 37, 727, 128, effect, 256-258, 261,362 145, 375 Hofava-Witten scenario, see Hofava-Witten scenario and the CSK theory, 131 Witten-Nester-Israel (WNI) technique, 181, 395, 398 non-metricity tensor, 431 WNI, see Witten-Nester-Israel (WNI) technique representation, 396, 617, 624, 625 worldsheet parity, 420 rescalings, 40, 727, 122, 264, 417, 418 Wu-Yang monopole, 254-255 and conformal reference frames, 510 WZ, see Wess-Zumino term and tracelessness of the Maxwell energy-momentum tensor, 219 Yang effect on the curvature, 639 Wu-Yang monopole, see Wu-Yang monopole invariance of string action, 410, 433 Yang-Mills invariance of the string action, 432 Bianchi identity, see Bianchi identities, for Yang-Mills fields supersymmetric, 414 Cherns-Simons 3-form, see Chern-Simons, 3-form rescalings and massless fields, 123 covariant derivative, 596 spinors, 612, 616, 628 equation of motion, 597 equivalent to Majorana in 4d, 151 field, 274, 391, 438, 470, 471, 478, 596 static axisymmetric electrovacuum solutions, see solution, strength, 273, 597 Weyl theories, 127, 134, 137, 139, 595-598 tensor in 4d, 14, 383, 384, 386 and gravity, 140-144, 148 and the Einstein-Fokker theory, 56 instantons, see solution, BPST instanton tensor in higher d, 14 solutions, 391 theorem (compactness of Lie groups), 594 supersymmetric, 373, 564 Wick rotation, 209, 243, 285 Yang-Mills theories, 273 not commuting with electric-magnetic duality, 262 Yau, see Schoen-Yau positive-energy theorem of gamma matrices, 626 of Taub-NUT solution, 271 of the SL(2, M)/SO(2) a-model Zanelli, see solution, BTZ black hole and the D-instanton, 540 Zehnbein, 462 Wigner's theorem, 595 Zumino, see Wess-Zumino term Wigner-Inomi contraction, 115, 142, 152, 153, 159, 161, Zwanziger, see Dirac-Schwinger-Zwanziger 162, 167 Zweibein, 14