Gravity and Strings

integrability equation in N = 2, d = 4 theories, 382–383
of N = 1, d = 4 AdS supergravity, 385
of N = 1, d = 4 Poincaré supergravity, 183, 398
of N = 1, d = 6 supergravity, 390
of N = 2, d = 4 Poincaré supergravity, 387
spinors
general form in maximally supersymmetric vacua, 380
T-duality tranformation rules, see Buscher T duality, in type-II
theories, transformations of Killing spinors
vector, 21, 98, 110, 111, 177–180, 183, 270, 288, 330, 370, 375,
378, 391, 504, 608
and symmetry of the point-particle action, 307
as bosonic generator of a symmetry superalgebra, 374, 378
as generators of the isometry algebra, 371, 504
commuting and compactification on T n , 331
conformal, 21
conformal and Lie–Lorentz derivative, 376–377
gauge, 176
null, covariantly constant and Brinkmann metrics, 282, 647
of coset manifolds, 606, 607
of maximally symmetric spaces, 605
of Minkowski metric, 67
of Minkowski spacetime, 384
of stationary axially symmetric metrics, 268
of the Minkowski metric, 35, 99, 178
spacelike and Kaluza–Klein (KK) compactification, 297–299
timelike and definition of mass, 179
timelike and Killing horizon, 244
timelike and staticity, 188
timelike and surface gravity, 198
timelike and the Schwarzschild metric, 191
translational, 179
versus Killing spinor, 374, 384
KK6-brane, see Kaluza–Klein (KK) monopole, as string–theory
solution (KK6)
KK7M-brane, see Kaluza–Klein (KK) monopole, as M–theory
solution (KK7M)
KK8A-brane, 526
KK9A-brane, 526
KK9M-brane, 465, 524
and the M superalgebra, 557
Klein, 290
Kaluza-Klein theories, see Kaluza–Klein
Klein–Gordon equation, 36, 293, 294
Komar’s formula, 180, 221
compared to Abbott–Deser approach, 180
in higher d, 211
Kosmann, 375
Kraichnan, 59
Kretschmann invariant, 190
Lagrange, see Euler–Lagrange
Landau
Ginzburg–Landau Lagrangian, see Ginzburg–Landau
Landau–Lifshitz energy–momentum pseudotensor, 173, 174–176
compared with Abbott–Deser approach, 178, 179
Laplace equation, 272, 280, 294, 551, 648, 649
Legendre transformation, 308, 442
Leibniz rule, 5, 6, 376
level operators, 419
Lichnerowicz, 375
Lie
algebra, 5, 43, 591, 592, 595
Abelian, 594
Bianchi classification of 3d real Lie algebras, 602
complexified, 593
de Sitter (anti-), see de Sitter (anti-), algebra
derived subalgebra, 594
Heisenberg, see Heisenberg algebras
invariant subalgebra, 594, 605
nilpotent, 284, 594, 594
of isometry group, 183
Index 677
of GL(d),15
of SO(1, 2), 602
of SO(3), 270
of SO(4) (anti-)self-dual generators, 275
of SO(n+, n−), 600
of SO(n+, n−) (spinorial representation), 601
of SU(2) (SO(3)), 602
of isometries, 371, 379
of the conformal group SO(2, d − 1),40
of the Lorentz group SO(1, d − 1),40
of the Lorentz group SO(1, d − 1) (spinorial representation),
611, 612
of the PoincaréISO(1, d − 1),35
of the PoincaréISO(1, d − 1), 143
of the Poincaré group ISO(1, d − 1), 36, 40, 384
reductive decomposition, 605
semidirect sum, 605
semisimple, 594, 594
simple, 594
solvable, 284, 594, 594
symmetric decomposition, 605
bracket, 5, 80, 592
and commutators of matrices, 592
brackets
and Ricci rotation coefficients, 15
covariant derivative, 608
derivative, 5,6,8,13, 297, 371, 374, 379, 608
H-covariant, 608
and extrinsic curvature, 25
and Killing vectors, 20
covariant, 369, 375, 375–378, 379
properties, 5
spinorial, see Lie, Lie–Lorentz derivative
group, 591
ISO(n+, n−), 599
SO(n+, n−), 598
SO(n+, n−),vector representation, 599
SU(2), 603
affine group IGL(d, R),17
and N = 1, 2, d = 6 vacua, 390
compact, 593
compact (Weyl theorem), 594
conformal SO(2, d − 1),40
de Sitter (anti-), see de Sitter (anti-), group
invariant subgroup, 594
Lorentz SO(1, d − 1), 17, 32, 142
of isometries, 370
Poincaré ISO(1, d − 1), 17, 26, 32, 46, 48, 49, 52, 96, 108,
114, 127, 132, 134, 137, 140–142, 150
Riemannian geometry, 602–604
semisimple, 594
simple, 594
translation, 145
Lie–Lorentz derivative, 375, 375–377, 384, 608
and H-covariant Lie derivative, 381
Lie-Maxwell derivative, 375, 377–378, 608
superalgebra
N = 1, d = 4 Poincaré, 151
supergroup
de Sitter (anti-), see de Sitter (anti-), supergroup
Poincaré, 150
Lie–Lorentz derivative, see Lie, Lie–Lorentz derivative
Lie–Maxwell derivative, see Lie, Lie–Maxwell derivative
Lifshitz, see Landau–Lifshitz energy–momentum pseudotensor
Lindquist, see coordinates, Boyer–Lindquist
loop quantization, 138
Lopuszanski, see Haag–Lopuszański–Sohnius theorem
Lorentz
group, see Lie, group, Lorentz
Hilbert–Lorentz gauge, see Hilbert–Lorentz gauge
Lie–Lorentz derivative, see Lie, Lie–Lorentz derivative
Lovelock tensor, 101