Index

Abelian differential, 139 Dirichlet, 12, 19, 188 A-D-E classification, 351 mixed, 29, 189, 315 singularities, 716 Neumann, 12, 19, 188 ADM tension, 694 Boundary state, 101, 164 AdS/CFT correspondence fermionic, 232 strong, 748 formalism, 93 weak, 748 fractional D-brane, 540 Almost complex structure, 454 BPS, 241 Anomaly bound, 696 gauge, 652 state, 560, 684 gravitational, 250 Brane world scenario, 641 holographic, 754 Breitenlohner-Freedman bound, 765 Anthropic principle, 3, 663 BRST Anti-de-Sitter, 741 charge, 385, 406 global coordinates, 742 current, 113, 387 metric, 735 quantization, 110 Poincare« coordinates, 742 Bulk, 641 A-roof genus, 633 Bundle Atlas, 450 canonical, 474 Axion, 629 line, 469 stable, 503 vector, 453 Background charge, 395 Buscher rules, 434 Beltrami differential, 129 equation, 129 Betti numbers, 448, 461 Calabi-Yau in Kahler¬ manifolds, 468 four-fold, 478 Bianchi identity, 498, 625, 657 Hodge numbers, 476 Bosonic string holomorphic n-form, 475 map, 561, 567, 568 manifold, 385, 444, 473 propagators, 37 rigid, 532 Bosonization, 341, 391 warped, 662 Boundary changing operator, 98 Calibrated submanifold, 648 Boundary condition Canonical quantization, 35 A-type, 380 Cardy condition, 165, 170 B-type, 380 Cardy formula, 93

R. Blumenhagen et al., Basic Concepts of String Theory, Theoretical and Mathematical 775 Physics, DOI 10.1007/978-3-642-29497-6, © Springer-Verlag Berlin Heidelberg 2013 776 Index

Cartan group, 63, 737 matrix, 329 Killing spinor, 182 subalgebra, 328 Killing vector, 18, 129, 182, 737 Cartan-Weyl basis, 328 spin, 66 Casimir, 325 structure, 129 Center of mass position, 24, 29 Ward identity, 84 Central charge, 39, 323 weight, 66 Central extension, 39, 323, 720 Conformal field theory, 63 Chan-Paton factor, 49, 244 boundary, 93 Character, 153, 344 rational, 79, 164, 167, 344 Charge conjugation, 153, 211, 348 super, 355 Chern Conifold, 482 character, 473 Conjugacy class, 331, 392 class, 471 Constraint connection, 457, 470 first class, 9 Chern-Simons primary, 8, 12 3-form, 628 secondary, 9 term, 624 second class, 9, 186, 279 Chiral field, 66 Contorsion, 667 N D 2 superconformal, 377 Coordinates Chirality matrix, 212 homogeneous, 451 Chirality operator, 211 inhomogeneous, 452 Chiral ring, 383, 577 Coset, 331 Christoffel symbols, 509 lattice, 328 Clebsch-Gordan coefficients, 219 Covariant Clifford algebra, 198, 210 derivative, 510 Closed-open string duality, 99 lattice, 416, 578 Closed string, 19 Coxeter element, 524 classical solution, 32 Critical dimension fermionic mode expansion, 189 bosonic string, 47 fermionic spectrum, 207 fermionic string, 203 mode expansion, 23 Critical string, 431 partition function, 148 Crosscap, 124 spectrum, 50 gluing conditions, 167 Cocycle factor, 339 Ishibashi states, 167 Coherent state, 101 state, 166, 248 Cohomology, 461 Crossing symmetry, 83 de Rham, 461 Cross ratio, 82 Dolbeault, 463 Current algebra, 324 Compactification, 264, 440 families, 325 brane world, 641 Cylinder amplitude, 157 flux, 658 heterotic on Calabi-Yau, 497 supersymmetric, 441 type II on Calabi-Yau, 487 Complex D-brane, 30 manifold, 130 fractional, 540 structure, 130, 450 intersecting, 310, 646 moduli, 485 magnetized, 317 Conformal R-R charge, 240 bootstrap, 83 SUGRA solution, 701 coordinates, 129 tension, 163, 171, 240 family, 76 Dedekind eta-function, 147 gauge, 16 Dehn twist, 135 Index 777

Descendant metric, 658 field, 69, 741 NS-NS three-form, 657 state, 75 R-R p-form, 658 Dilation, 64, 740 Fractional D-brane, 540 Dimensional reduction, 435, 445 Frenkel-Kac-Segalˇ construction, 327 double, 719 F-theory, 704, 724 Dirac-Born-Infeld action, 436, 630 Fubini-Study metric, 458 Dirac bracket, 186, 279 Fundamental form, 456 Dirac matrix, 210 Fundamental region of torus, 137 Dirichlet boundary condition, 12, 19, 188 Fundamental string Discrete torsion, 300 SUGRA solution, 699 Discriminant, 725 tension, 10, 240 Disk, 94, 101, 145 Fusion rules, 78, 344 Dolbeault cohomology, 384, 463 N D 2 superconformal, 381 Doubling trick, 28, 96, 191 sbu.2/k , 351 Dual Coxeter number, 325 lattice, 327 Gauge hierarchy problem, 439 resonance model, 601 Gauss-Bonnet term, 15 Duality, 688 Gepner model, 567 heterotic-type IIA in 6D, 713 heterotic, 576 S- heterotic-type I in 10D, 710 partition function, 573 strong-weak, 676 type II, 573 S- type IIB in 10D, 705 Ghost, 36, 142 T-, 269, 432 action, 107 target-space, 678 field, 107 U-, 712 number operator, 111 picture Eddington-Finkelstein coordinates, 762 canonical, 404 Effective action, 615 charge, 403 D-brane, 630 system, 394 supergravity, 622 Gluing Effective field theory, 586 automorphism, 164 Einstein frame, 172, 619, 626, 690 conditions, 102, 164, 233 modified, 690 Graviphoton, 489 Einstein spaces, 736 Gravitino, 489, 546 Elliptic curve, 724 two-dimensional, 177 Elliptic fibration, 724 Green-Schwarz mechanism, 499, 629, 651 Energy-momentum tensor, 13, 67 Green-Schwarz superstring, 176 holographic, 757 GSO projection, 203, 230, 551 Euler Beta-function, 607, 636 Euler number, 431, 468, 478 Hagedorn temperature, 93 Harmonic form, 466 Faddeev-Popov ghosts, 53 Hawking temperature, 759 Fefferman-Graham Hermitian expansion, 752 conjugate, 36, 70 gauge, 751 connection, 457 First order system, 393 metric, 456 Fischler-Susskind mechanism, 319 Yang Mills, 470, 503 Fixed point, 528 Heterotic string, 284 Flux, 657 compactification, 497 compactification, 658 covariant lattice, 422, 578 imaginary self-dual, 660 Gepner model, 576 778 Index

non-supersymmetric, 425 Kac-Moodyˇ algebra, 321, 323 orbifold, 525 representation, 343 S-duality, 710 SO.2n/, 341 Hirzebruch L-polynomial, 634 sbu.2/k , 349, 381 Hodge Kaluza-Klein decomposition, 384, 462, 463, 467 Ansatz, 435 diamond, 464, 477 modes, 264 numbers, 463 monopole, 720 operator, 465 Kawai-Lewellen-Tye relations, 614 Holography, 749 Killing Holonomy group, 298, 443, 460 group, 132 G2, 445 spinor, 442, 522 SU.2/, 445 equations, 664 SU.3/, 444 vector, 432 Homology conformal, 18, 129, 737 class, 461 Klein bottle, 167, 242 group, 461 amplitude, 168, 242 Horava-Wittenˇ theory, 723 K-theory, 251, 648 Hydrodynamics, 760 K3 surface, 445, 523, 530, 713 Hypermultiplet, 489 universal, 554, 574 Hypersurface, 452 Lagrangian submanifold, 648 Laplacian, 448, 467 Large extra dimension scenario, 642 Large-N expansion, 748 Instanton Lattice, 327 D-, 706 covariant, 416, 578 world-sheet, 487 D5, 392 Yang-Mills, 678 D , 416 Intersecting D-brane 5;1 E5;1, 421 boundary conditions, 310 even, 273, 328 models, 544, 646 Lorentzian, 272 Intersection number, 313, 463 Narain, 273, 293, 715 triple, 496 odd, 328 Ishibashi state, 164, 233, 284 rational, 273 Isometry, 432 self-dual, 287, 328, 423 unimodular, 328 Leech lattice, 335 Jacobi Lefschetz decomposition, 466 theta-function, 227 Level triple identity, 228 Kac-Moodyˇ algebra, 323 Jacobian, 128 matching, 40 j-function, 703 Lichnerowicz operator, 483 Lie algebra, 322 E6, 330 E7, 330 Kahler¬ E8, 330 cone, 484 lattice, 328, 332 complexification, 486 simply laced, 329 form, 457 SO.2n/, 329 manifold, 457 SU.n/, 330 metric, 457 Lie derivative, 432 moduli, 484 Lie group, 321 potential, 457 simply laced, 326 Index 779

Light-cone Modular coordinates group, 136 space-time, 42 invariance, 138, 148 world-sheet, 16 S-transformation, 137, 153 gauge, 42, 200 T-transformation, 137, 153 Line bundle, 469, 726 Moduli, 269, 449 Little group, 46, 178 bundle, 534 Loop group, 321 Calabi-Yau, 483Ð487 Loop-channel, 159, 165, 236 complex structure, 276, 485 annulus, 245, 539 Kahler,¬ 276, 484 Klein-bottle, 242, 537 space Riemann surface, 140 Mobius¬ strip, 246, 542 stabilization, 643 Loop-channel Ð tree-channel equivalence, 159, 2-torus, 276, 449 238, 242, 540 Momentum angular, 22 canonical, 12 M-theory, 688, 719 conjugate, 8 M2-brane, 719 space-time, 24, 27 M5-brane, 720, 721 Monodromy charge, 155 Mobius¬ strip, 169, 246 loop-channel, 170, 246 Nambu-Goto action, 10 tree-channel, 169, 247 Narain lattice, 273, 293, 715 transformations, 72 Navier-Stokes equation, 762 Magnetic monopole, 240 Near-horizon Maldacena limit, 745 geometry, 736 Mandelstam variables, 600 limit, 735 Manifold Neumann boundary condition, 12, 19, 188 Calabi-Yau, 473 Neveu-Schwarz sector, 188, 225 complex, 450 Newton’s constant, 434 Hermitian, 457 Niemeier lattices, 424 Kahler,¬ 457 Nijenhuis tensor, 455 Kahler-Einstein,¬ 459 Non-BPS D-branes, 251 orientable, 451 Non-linear sigma model, 315, 428 Ricci-flat, 443 No-scale structure, 661 Mapping class group, 140 NS5-brane, 689 Marginal operator, 560 SUGRA solution, 700 Massless spectrum Gepner model, 574, 577 orientifold, 537 4 T =Z2 orbifold, 529 Open string, 19 4 T =Z2 orientifold, 543 classical solution, 31 6 T =Z3 orbifold, 531 fermionic mode expansion, 191 Membrane, 719 fermionic spectrum, 203 Mirror mode expansion, 27 map, 493 partition function, 157 symmetry, 277, 385, 449, 491, 560 spectrum, 46 transformation, 560 Operator product expansion, 69 Mittag-Leffler theorem, 609 Orbifold, 138 Mode expansion circle S 1, 294 closed string, 24 heterotic, 525 fermionic closed string, 189 supersymmetric, 523 4 fermionic open string, 191 T =Z2, 528 6 open string, 27 T =Z3, 530 780 Index

Orientifold, 51, 317 Ramond compactification, 534 ground state, 198, 379 construction, 241 sector, 188, 225 plane, 52, 169, 243, 317 Reflection coefficients, 165 type IIA on CY, 646 Regge 4 type IIB on T =Z2, 535 slope, 11 type IIB on CY, 645 trajectory, 32 Resolution of singularity, 523 Ricci P-transformation, 170, 247, 252 form, 458 Partition function, 80 scalar, 509 torus, 146 tensor, 430, 458, 509 4 T =Z2, 529 Riemann 6 T =Z3, 531 identity, 228, 257 Peccei-Quinn shift symmetry, 490, 629 generalized, 257 Penrose-Brown-Henneaux (PBH) sphere, 72 transformations, 751 surface, 122 Period, 462 tensor, 509 matrix, 139 zeta-function, 44 Physical state, 90, 115 Riemannian conditions, 40 connection, 457 Picard-Lefschetz theorem, 728 geometry, 508 Picture structure, 129 changing operation, 405 Riemann-Roch theorem, 132, 232, 397 changing operator, 408 RNS-superstring, 176 ghost, 404 Root, 328 Planar limit, 748 lattice, 328, 329 Planck length, 3 positive, 329 Planck mass, 1, 52, 642 simple, 329 Plasma, 759 R-symmetry, 371, 744 Poincare« duality, 461 polynomial, 384 Scalar potential, 643, 659 Point group, 298 Scattering amplitude Poisson closed string bracket, 8, 21 1-closed string, 611 resummation formula, 229, 257, 286 3-gauge bosons, 596 Polyakov action, 12, 19, 176 3-gravitons, 598 Prepotential, 494 3-tachyon, 596 Primary field, 65, 69, 324, 741 4-tachyons, 600 Primitive p-form, 466 one loop, 123 Projective space open string complex, 451, 458 3-gluon, 606 weighted, 480 4-gauge bosons, 609 Pure spinor formulation, 176, 429, 615 2-open string, 610 3-tachyon, 606 4-tachyons, 608 Quasi-primary field, 72 tree level, 121, 144 Quintic, 480, 486, 576 Schur’s lemma, 685 Schwarzian derivative, 73 Schwinger term, 324 Radial S-duality ordering, 64 heterotic-type I, 710 quantization, 64, 741 type IIB, 705 Index 781

Secondary field, 66 transverse, 452 Sen-limit, 730 Sugawara construction, 325 Shapiro-Virasoro model Sugimoto model, 319 extended, 51 Super restricted, 51 charge, 411 Siegel’s upper-half plane, 139 current, 183 Simple current, 154, 570 derivative, 356 extension, 156 interval, 356 orbit, 156 N D 2, 371 Singularity, 452 Kac-Moodyˇ algebra, 556 Soliton, 265, 678, 689, 701 operator product expansion, 359 Space Virasoro algebra, 191, 197, 358 anti-de-Sitter, 736 Superconformal conformally flat, 736 algebra, 358 Einstein, 736 gauge, 180 group, 297 Hilbert space, 361 maximally symmetric, 736 transformation, 356 Special Superfield coordinates, 494 chiral, 356, 372, 377 holonomy, 444 primary, 357 Kahler¬ geometry, 493 quasi-primary, 361 Kahler¬ manifolds, 493 Supergravity, 622 Lagrangian, 497, 648 democratic formulation, 627 Spectral flow, 379, 552 eleven-dimensional, 623, 717 Spin gauged N D 8, 750 connection, 509, 697 heterotic, 629 field, 364, 412 type I, 627 manifolds, 223 type IIA/B, 624 structure, 223 Supermultiplet, 489, 531, 537 Spinor short, 685 covariantly constant, 442 Superspace, 356 Dirac, 213 N D 2, 371 Majorana, 216 Majorana-Weyl, 217 representation, 213 Tachyon, 47, 90, 162, 251 Weyl, 217 Tadpole, 95, 101, 243 St¬uckelberg cancellation, 247, 541, 647, 732 mass, 499 local, 317 mechanism, 656 Tangent bundle, 452 Standard embedding, 499 holomorphic, 453 State-operator correspondence, 74, 741 T-duality, 269, 432 String Teichmuller ¬ closed, 14, 23 parameter, 134 frame, 172, 620, 690 space, 134 functions, 350, 381 Tension landscape, 662 ADM, 694 length scale, 11 D-brane, 163, 171, 240 mass scale, 11 M2-brane, 719 open, 14, 27 M5-brane, 719 scale, 449 NS5-brane, 695 tension, 11 string, 11 Structure constants, 322 Theta-function, 227, 254 Submanifold ’t Hooft coupling, 748 complex, 452 Three-algebra, 721 782 Index

Throat, 735 graviton, 592 Topological field theory, 385 open string gauge boson, 604 Torelli group, 141 open string tachyon, 604 Toric variety, 482 R-R p-form, 595 Toroidal compactification Virasoro algebra, 39, 59, 70, 323 D-dimensional, 270 centerless, 26 spectrum, 266 N D 1 supersymmetric, 360 Torsion, 455, 504 N D 2 supersymmetric, 375, 551 intrinsic, 666 Virasoro constraints, 40 Trace anomaly, 84 Virasoro-Shapiro amplitude, 601 Tree-channel, 159, 165 Volume form, 475 annulus, 245, 540 Klein-bottle, 243, 538 Mobius¬ strip, 247, 542 W-algebra, 71 Twisted sector, 265, 296, 300 Ward identity Type I superstring, 208, 241 boundary, 94 S-duality, 710 conformal, 83 Type I’ superstring, 317 Warp factor, 440, 662 Type IIA supergravity Weierstrass form, 725 action, 624 Weight vector, 331 10D-field content, 488 Weyl 4D on CY, 488 anomaly, 53, 429, 754 Type IIA superstring, 207 fermion, 178 orientifold on CY, 646 invariance, 16 Type IIB supergravity rescaling, 15 action, 624 spinor, 217 10D-field content, 488 transformation, 179 4D on CY, 490 Weyl-Kacˇ character formula, 350 Type IIB superstring, 206 Wick orientifold on CY, 645 rotation, 64 theorem, 88 Wilson line, 290, 310, 527 Uhlenbeck-Yau theorem, 503 continuous, 284 Unit cell, 327 discrete, 504 Unitarity, 78, 343, 601 moduli, 715 Unitary representation Wilson loop, 766 Kac-Moodyˇ algebra, 343 Winding N D 2 super Virasoro algebra, 380 number, 265 sbu.2/ , 349 k state, 265 super Virasoro algebra, 362 Witt algebra, 26 Virasoro algebra, 79 World-line, 7 Upstairs geometry, 318, 634 World-sheet, 10 parity, 50, 207, 241 Vector bundle T-duality, 317 Hermitian, 469 holomorphic, 453, 468 stable, 503 Yau’s theorem, 473 tangent, 452 Vectormultiplet, 489 Veneziano amplitude, 608 Zero mode, 18, 24 Verlinde formula, 154, 166 dimensional reduction, 447 Verma module, 75 fermionic, 196 Vertex operator, 90, 392 Zeta-function gaugino, 594 regularization, 44, 202, 259 gravitino/dilatino, 594 Riemann, 44