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Index

12 BPS operator of SU (N) Super Yang–Mills, 142,

190, 519SU(2) Einstein–Yang–Mills, 483su(4) representation, 519USp(N ), 120U(1)A symmetry , 404�QCD, 412η/s, 437, 455ρ meson, 417N = 4 Super Yang–Mills theory, 135, 138, 180

action, 136adding fundamental matter, 211β function, 138confining deformation of, 410from dimensional reduction, 137

Abelian Higgs model, 479ABJ anomaly, 47ABJM theory, 279, 281additive renormalisation, 23adjoint representation, 35AdS/CFT correspondence, 179–204AdS5/CFT4, 179–193

comparison of symmetries, 189correlation function tests, 220for generating functionals, 196strong form, 181, 182strongest form, 180, 182weak form, 182

AdS/QCD, 426AdS–Schwarzschild black hole, 356advanced Green’s function, 351Affleck–Dine–Seiberg superpotential, 315anomalous dimension, 23anomaly, 46Anti-de Sitter(AdS) spacetime, 70–76area law

confining Wilson loop, 403Entanglement entropy, 498, 500

asymptotic freedom, 26, 38, 400axial anomaly, 395axial vector current, 29axial vector meson, 425

backreaction, 331of fermions, 492

bare Lagrangian, 21baryon, 401baryon number symmetry, 283baryon vertex, 326Bekenstein bound, 180Bekenstein–Hawking entropy, 86Belinfante energy-momentum tensor, 8Bern–Dixon–Smirnov (BDS) conjecture, 262beta function, 26, 38, 40

for gauge coupling, 400holography, 306

beta-deformed theory, 299Bethe ansatz, 246Bianchi identity, 62black brane, 88, 344, 352, 390, 394, 464black hole, 125

asymptotically AdS, 86–88BTZ, 475charged, 463dyonic, 474event horizon, 76extremal, 484in holographic hydrodynamics, 376, 379, 390, 393Kerr–Newman, 86planar, see black braneReissner–Nordström, 84–86, 393, 464, 466, 471,

475, 485, 493Schwarzschild, 76–79small and large, 87thermodynamics, 86

BMN limit, 240Bogolyubov transformation, 80Bogomolnyi–Prasad–Sommerfield (BPS) bound, 124bottom-up approach, 426, 463, 480, 490BPS multiplet, 124BPS operator, 142Breitenlohner–Freedman bound, 195

violation of, 480, 484bulk viscosity, 383bulk-to-boundary propagator, 198bulk-to-bulk propagator, 198Buscher rules, 167

C-theoremholography, 304quantum field theory, 296

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528 Index

Calabi–Yau manifold, 163, 276Calabi–Yau theorem, 163canonical ensemble, 344–345canonical quantisation, 9–10Cartan subalgebra, 510Cartan–Weyl form, 510causality, 370, 379central charge, 120, 123, 124, 135

conformal symmetry, 107, 112, 114, 116Chan–Paton factor, 170, 274charge density, 463, 464charged fluid, 383, 393Chern class, 164Chern–Simons term, 394, 467, 474, 483Chern–Simons theory, 280, 281chiral anomaly, 395, 474chiral perturbation theory, 405chiral primary operator, 142chiral symmetry breaking, 413, 415, 421

explicit breaking of, 404holographic, 455in QCD, 403R-symmetry, 314, 315spontaneous breaking of, 404

chiral vortical effect, 394Clifford algebra, 28Coleman–Mandula theorem, 91, 143collinear anomalous dimension, 263, 269colour, 401colour ordering, 259compact symplectic group, 120composite operator, 23compressible system, 496condensed matter physics, 461

holographic, 463conductivity

AC, 371, 482DC, 478, 482holographic, 468, 481, 488

confined phaseAdS/QCD, 426holographic, 444, 455

confinement, 309, 313, 401, 405, 407in QCD, 401

confinement–deconfinement transitionholographic, 443, 454

conformal algebra, 102, 103, 491conformal anomaly, 115–117, 233, 234, 300

holographic, 498relation to entanglement entropy, 498

conformal diagram, 68AdS space, 75

conformal field theory(CFT), 107–123one-dimensional CFT, 466

conformal structure, 74, 209conformal symmetry, 102

coordinate transformations, 103correlation functions, 110energy-momentum tensor, 108field transformations, 107operator product expansion, 112two dimensions, 106Ward identity, 115

conifold, 276connected Green’s function, 16connection, 31, 55

Christoffel, 55spin, 56

consistent truncation, 307, 389, 393Constable–Myers flow, 411Cooper pair, 478correlation function, 11

12 BPS operator, 221

coset space, 70cosmological constant, 65

of Anti-de Sitter spacetime, 73Coulomb phase, 139counterterm, 20covariant derivative, 55–56critical exponent, 462cusp anomalous dimension, 263, 269cycle, 164

D-brane, 125, 149, 166, 168, 172, 182closed string perspective, 182, 186, 188D3-branes, 182D3/D5-brane system, 340–342D3/D7-brane system, 328–339, 363fluctuations, 335–339open string perspective, 182, 184–186, 188regular, 274

D-branes at singularity, 273DBI (Dirac–Born–Infeld) action, 169

for D3-brane, 185for D7-brane, 332, 336

deconfined phase, 444holographic, 455

defect field theory, 340derivative expansion, 382, 390, 393

vorticity contribution, 394descendant

conformal, 107superconformal, 140

differential form, 57–59Hodge dual, 60–61volume form, 59–60

differential geometry, 50diffusion constant, 372, 386dilatino, 160dilaton in holographic RG flows, 411dilaton flow, see Constable–Myers flowdimensional regularisation, 17, 19






529 Index

Dirac field, 28Dirac matrix, 28Dirichlet boundary condition, 148dissipation, 379, 380

dissipative fluid, 382, 395holographic, 379, 390, 393

domain wall, 326domain wall ansatz, 408domain wall flow, 302double line notation, 42Dp-brane, 171Drude peak, 471D-term, 128dual AdS space, 264dual superconformal symmetry, 269duality, 179

string theory, 164duality cascade, 313dynamical scaling exponent, 462Dynkin diagram, 511

Eckart frame, 382effective action, 16Einstein frame, 156Einstein static universe, 72Einstein tensor, 65Einstein’s field equations, 65–66Einstein–Hilbert action, 66Einstein–Maxwell theory, 463Einstein–Maxwell dilaton theory, 494electron star, 492emergent quantum criticality, 466, 488energy condition

dominant, 89null, 88strong, 89weak, 88

energy-momentum tensor, 8entanglement entropy, 496

holographic, 498in two-dimensional CFT, 498replica trick, 498strong subadditivity, 501

entropy current, 381, 382, 395Euclidean action, 18Euler characteristic, 42Euler number, 154exterior derivative, 58extremal correlation function, 230extrinsic curvature, 81

factorisation, 45Fefferman–Graham coordinates, 204Fefferman–Graham metric, 75Fefferman–Graham theorem, 208

Fermi liquid theory, see also Fermi surface,Luttinger’s theorem, non-Fermi liquid

fractionalisation, 494marginal Fermi liquid, 489

Fermi surface, 488, 492, 494, 496hidden, 496

fermionsholographic, 485

Feynman propagator, 12Feynman rules, 13Fick’s law of diffusion, 383field-operator map, 189–196fixed point, 26flavour holographic, 326–342flavour brane, 326–342flavour symmetry, 329fluid-gravity correspondence, 389flux tube, 326, 401, 407Fock space, 15four-gluon scattering, 265four-point function, 229fractional brane, 310free energy

in canonical ensemble, 345in grand canonical ensemble, 346

F-term, 128fundamental representation, 33

gauge field, 31Gell-Mann–Oakes–Renner relation, 429, 417generalised coupling, 24, 27generating functional, 9–12, 15, 18, 24

fermions, 29genus, 42

string theory, 153geodesic, 61ghost field, 37Gibbons–Hawking term, 81, 354, 469Gibbs free energy, 465Ginzburg–Landau theory, 478global transformation, 9gluon, 400Goldstone boson, 416, 424, see also

pseudo-Goldstone bosonin QCD, 404

Goldstone’s theorem in QCD, 404gradient flow, 302grand canonical ensemble, 345–346, 465Grassmann number, 30gravitino, 160Green function, small ω expansion, 471Green–Kubo relation, 384

charge diffusion, 384shear viscosity, 385

Green–Schwarz formalism, 248GSO projection, 159






530 Index

Hamiltonian, 7hard probe, 437, 440hard wall model, 407, 427Hawking radiation, 80–84Hawking temperature, 80Hawking–Page transition, 357, 443heavy-ion collisions, 369holographic Green’s function, 357–362holographic principle, 179, 181holographic renormalisation, 204–210

of scalar field, 204holographic RG flow, see warp factor, domain wall

ansatzholonomy, 163hydrodynamics, 369

hydrodynamic expansion, 382hydrodynamic limit, 380, 384hydrodynamic pole, 374,Quark–gluon plasma, 372, 436

hyperscaling violation, 494, 496

ideal fluid, 381holographic, 390

imaginary time formalism, 347, 349–350infalling boundary condition, 358integrating out, 126internal symmetry, 9interpolating flow, 296, 306IR fixed point, 26irrelevant operator, 26, 27isotropic fluid, 381

jet quenching parameter, 440

Kähler manifold, 163Kähler potential, 132Kalb–Ramond field, 153, 155Kaluza–Klein expansion, of IIB SUGRA fields, 191Killing horizon, 73Killing vector, 57Klebanov–Strassler flow, 311Klein–Gordon equation, 5Konishi current, 301Konishi operator, 142Kramers–Kronig relation, 470

Lagrangian, 4Landau frame, 382Landau pole, 26, 331large N limit, 40–43Large Hadron Collider, 435lattice gauge theory, 400Lax connection, 253Lax pair, 253Leigh–Strassler flow, 306Lie algebra, 508

Lie derivative, 56–57Lie group, 508Lie representation, 515Lifshitz algebra, 491Lifshitz space, 490, 493linear response theory, 370, 384, 468, 481

holographic, 374, 387little group, 99Lorentz algebra

tensor representations, 92spinor representations, 94

Lorentz symmetry, 91Lorentz transformation, 4low-energy limit, 186, 187LSZ reduction formula, 15Luttinger’s theorem, 494

M2-brane, 278M5-brane, 287magnetic brane, 475magnetic field, 474magnetisation density, 474magnetisation susceptibility, 474magnon, 246Majorana field, 29Maldacena limit, 327, see also low-energy limitMaldacena–Núñez solution, 315Manifold, 50–51

complex, 63Kähler, 163Lorentzian, 53Riemannian, 53

marginal deformation, 298marginal operator, 26, 27mass gap, 293Matsubara frequency, 349maximally symmetric spacetime, 66–68M-brane, 173mesino, holographic, 336, 341meson, 330, 401, 413, 416, 417, 425

holographic, 332, 335, 341, 448unstable, 448

metric, 53–54minimal surface, holographic Wilson loop, 406minimal coupling, 31, 63minimal subtraction, 21minimal surface, 498Minkowski spacetime, 3, 68–69M-theory, 285multi-particle state, 10multiplet shortening, 142multiplicative renormalisation, 20multi-trace operator, 143

Nambu–Goto action, 145holographic Wilson loop, 406






531 Index

near-extremal brane, 172near-horizon limit, 188near-horizon region, 187negative frequency mode, 360Neumann boundary condition, 148Neveu–Schwarz (NS) sector, 158, 159Noether’s theorem, 6, 45Non-Abelian gauge theory, see asymptotic freedom,

confinementnon-Fermi liquid, 477, 486, 489non-relativistic systems, 489non-renormalisation theorem, 222NS-NS sector, 160NS5-brane, 174NSVZ beta function, 135, 300null energy condition, 306

Ohm’s law, 371, 468on-shell action, 201one-half BPS operator, 142, 520one-particle irreducible Feynman diagram, 16orbifold, 273order parameter, 481outgoing boundary condition, 358

parallel transport, 61–63particle scattering, 15partition function

string theory, 154thermodynamics, 28, 84

path integral quantisation, 10path integral, String theory, 153path ordering, 402Penrose diagram, 68Penrose–Brown–Henneaux (PBH) transformation,

235, 237perfect fluid, 493perturbation theory, 13phase diagram, 461, 462, 475, 477

of QCD, 435phase transition, see also critical exponent, order

parameter, quantum phase transitionchiral, 454first order, 446, 452holographic, 446, 452, 454second order, 461, 462

phonon, 478pion, 424pion decay constant, 429planar diagram, 43planar limit, 181Planck length, 401Poincaré group, 4Poincaré symmetry, 91

particle states, 98Polchinski–Strassler flow, 320

Polyakov action, 146, 147, 252positive frequency mode, 360primary operator, 111

conformal, 107superconformal, 140, 141

probe braneblack hole embedding, 447D5-brane, 340–342D7-brane, 331–339, 363, 413, 445, 448, 450D8-brane, 421, 423, 454embedding function, 415finite density, 450finite temperature, 445, 446, 448in holographic chiral symmetry breaking, 413, 421Minkowski embedding, 447

probe Limit, 201, 331, 423projective space, 286pseudo-Goldstone boson, 405, 416

QCD, 399at finite temperature and density, 435low energy, 455, see also baryon, chiral symmetry

breaking, confinement, flux tube, mesonquantum chromodynamics, see QCDquantum critical point, 461, 477quantum critical region, 462, 477quantum critical theory, 462quantum entanglement, 496quantum phase transition, 460, see also quantum

critical point, quantum critical region, quantumcritical theory

emergent quantum criticality, 462holographic, 463, 475

quark condensate, 415, 424holographic, 446, 448

quark density, 363quark mass, 415

holographic, 329, 446quark–gluon plasma, 369, 394, 435, 435

energy density, 439Holographic, 439

quarks, 400in baryons, 401in mesons, 401in the Standard Model, 399

quasi-primary field, 109quasi-primary operator, 112, 114quasinormal modes, 379, 449quenched approximation, 331quenching, holographic, 423quiver gauge theory, 274

Ramond (R) sector, 158–159real scalar field, 3real time formalism, 347, 350–351reduced density matrix, 497






532 Index

Regge trajectory, 401relevant deformation, 298, 300relevant operator, 26, 27remainder function, 269renormalisation group, 22renormalisation group (RG) flow 24, 296

holography, 302quantum field theory, 296

renormalisation group (RG) equation, 22retarded Green’s function, 12, 351, 370

condensed matterelectric current, 468Fermi liquid, 486heat current, 468

energy-momentum tensor, 371, 372, 386, 387holographic, 374, 376, 378, 386, 387poles, 373, 379R-symmetry current, 371, 372, 376, 378

RHIC accelerator, 435Ricci scalar, 62Ricci tensor, 62Riemann tensor, 62root, 510R-sector, see Ramond sectorR-symmetry, 120

saddle point approximation, 198Sakai–Sugimoto model, 421, 444

finite temperature, 454holographic, 454

scalar electrodynamics, 32scattering amplitude, 241

partial, 259Schrödinger algebra, 491Schrödinger spacetime, 475, 490Schur’s lemma, 307Schwinger–Dyson equation, 45Schwinger–Keldysh contour, 438Schwinger–Keldysh formalism, 347, 350, 351Schwinger–Keldysh propagator, 351S-duality, 138, 143, 167Seiberg duality, 313semi-simple Lie algebra, 510

compact, 510shear mode, 386shear viscosity, 374, 383, 385

holographic calculation, 387, 388, 393shooting technique, 446single-particle state, 10single-trace operator, 44, 141, 190S-matrix, 15soft wall model, 429, 430soliton, 124sound mode, 386spatially modulated phase, 483spectral function, 370, 371, 380

as energy change rate, 380R-symmetry current, 373

spherical harmonics, 190spin chain, 245spinor

Dirac, 400Majorana, 29, 97Weyl, 97

standard model, 399static gauge, 332stationary spacetime, 57string frame, 156string theory, 145

bosonic, 145conformal gauge, 147

closed string, 148quantisation, 152

coupling constant, 154, 156fermions, 157left- and right-moving modes, 147mode expansion, 148negative norm states, 150open string, 148

quantisation, 152perturbation theory, 153spectrum, 147type IIA theory, 160type IIB theory, 160unoriented strings, 170

strong–weak duality, 179subadditivity, 498

strong, 498superconducting density, 478superconductor,

high Tc, 477holographic, 479, 483Meissner–Ochsenfeld effect, 482p-wave superconductor, 483s-wave superconductor, 479

superconformal operator, 141superconformal symmetry, 139

algebra, 139representations, 140

superfield, 128chiral, 129vector, 130

superfluid, 477holographic, 467, 494

supergravity, 160eleven-dimensional, 162, 279gauged, 307M-theory, 161, 279type IIA, 162type IIB, 161

superpotential, 132superspace, 127






533 Index

superstring theory, 156closed superstring, 159open superstring, 157

supersymmetry, 118algebra, 118extended algebra, 119field theory, 125, 132maximal, 120, 135representation, 120, 121, 123

symmetrised trace, 141

tachyon, 151tadpole, 16target spacetime, 145T-duality, 164temperature, 463, 474tensor, 51–52tensor propagator, 231thermal AdS space, 356thermal equilibrium, 462thermal Green’s function, 346–350thermalisation, quark–gluon plasma, 436three-point function, 220, 225throat, 187time ordering, 11

of thermal Green’s function, 347’t Hooft coupling, 41–44, 181

for D-branes, 327’t Hooft limit, 41, 181torsion tensor, 62trace anomaly, 115transport coefficient, 370, 382, 384, 389triangle graph, 46

quark loop, 404TsT transformation, 309twist, 316type IIB supergravity, 182type IIB superstring theory, 180

unitarity bound, 107universality, 460UV fixed point, 26, 27

in non-Abelian gauge theory, 400

vectorcotangent space, 50–51Killing vector, 57tangent space, 50–51

vector current, 29vector meson, 417, 425

holographic, 453unstable, 453

vector propagator, 231Verma module, 141vertex operator, 155, 264vielbein, 53–54Virasoro constraints, 147, 249von Neumann entropy, 497vorticity, holographic, 394

Ward identity, 45, 110Lorentz symmetry, 100

warp factor, 303, 407wedge product, 58Wess–Zumino model, 127Weyl fermion, 29Weyl tensor, 62, 117Weyl transformation, 235white hole, 79Wick rotation, 18Wilson line

holographic, 441jet quenching parameter, 437

Wilson loop, 39, 405N = 4 Super Yang–Mills, 211–216for non-supersymmetric gauge theory,

402Wilsonian effective action, 25Wilsonian renormalisation group, 24Witten diagram, 197, 201

Feynman rules, 198worldsheet, 145wrapped brane, 310

Yangian, 269

Zeeman level splitting, 421






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