String Theory and M-Theory

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Poincaré coordinates, 645, 646, 686 antibrane, 211 anticommutation relations fermionic oscillators, 124, 164 heterotic string fermions, 256 world-sheet fermion fields, 120 area law, 661, 662 attractor equation, 590, 593 attractor mechanism, 587 auxiliary field in point-particle action, 19, 22, 145, 185 in world-sheet supermultiplet, 113–115 PST, 314 world-sheet metric, 7, 26, 27, 31 axion, 316, 401, 492, 498, 541 axion–dilaton field, 316, 476, 481, 484, 485, 524 background fields, 81, 89, 227, 228, 230, 237, 267, 275, 281, 295, 628, 629 NS–NS, 227, 237, 245 R–R, 229, 237 background-field gauge condition, 627 Bekenstein–Hawking entropy formula, 14, 563, 603, 609, 648 leading correction to, 585 Beltrami differential, 92 Bertotti–Robinson metric, 562 Betti numbers, 363, 442 of S N , 443 Bianchi identity for antisymmetric tensor field, 245 for four-form in heterotic M-theory, 521 for Maxwell field, 560 for Riemann tensor, 384 with H flux, 511, 516 with D3-brane source, 483, 492 with M5-brane source, 353 Big Bang, 16 biholomorphic function, 61 Birkhoff’s theorem, 552 black p-branes, 14, 551, 613 nonextremal, 625 black D-branes nonextremal, 686 black holes, 14, 408, 549 challenges posed by, 550 dyonic, 590 entropy of, 14, 563, 570, 604 extremal, 558 with four charges for D = 4, 574 with three charges for D = 5, 567 inner <strong>and</strong> outer horizons, 557 multi-center solutions, 594 nonextremal, 572 D = 4, 581 D = 5, 586 production at accelerators, 552 Reissner–Nordström, 557 entropy of, 564 extremal, 558 Index 727 extremal for D = 5, 559 temperature of, 565 rotating <strong>and</strong> BPS for D = 5, 573 Schwarzschild, 552, 554 in D dimensions, 553 supermassive, 549 temperature of, 562 thermodynamic properties, 14, 562 black ring, 596, 608 blow-up, 367, 368, 373, 435, 455 BMN limit, 680 BMN operator, 680 anomalous dimension, 680 correlation function, 681 Born–Infeld action, 231, 246, 687 bosonic ghost fields, 142 bosonic string, 17 number operator, 43 bosonization, 67 of ghosts, 80 bound state of D0-branes, 331, 626 of strings, 328 threshold, 331 boundary condition Dirichlet, 33, 164, 187, 193–195, 201, 202, 210, 622, 623 Neumann, 164, 202, 209 boundary state, 171 BPS (p, q) string, 327 bound, 151, 297, 300, 353, 408, 485, 655 D-string, 325 branes, 296, 421 tensions, 307, 341 D-branes, 208, 209, 213, 228, 230, 621, 638 type IIA, 405 type IIB, 328, 405 F-string, 221 M-branes, 307, 332, 614 M-theory solutions, 617 states, 298, 328, 408 in nine dimensions, 340 wrapped D-branes, 414 tensions, 427 brane–antibrane annihilation, 212, 587 brane–antibrane inflation, 537, 540 brane–brane inflation, 539 brane-world scenario, 11, 354, 459, 493 Brink–Schwarz superparticle, 185 BRST charge, 77 for RNS string, 143 mode expansion, 77 nilpotency, 78, 107 cohomology, 79, 108 quantization, 75, 108 symmetry, 77 of RNS string, 142 bubbling AdS, 655
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This is the first comprehensive tex
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cambridge university press Cambridg
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vi An Ode to the Unity of Time and
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viii Contents 4.5 Canonical quantiz
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Preface String theory is one of the
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Preface xiii stein of Caltech for t
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Preface xv NL, NR left- and right-m
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1 Introduction There were two major
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1.2 General features 3 theory fell
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1.2 General features 5 is only cons
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1.3 Basic string theory 7 world she
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1.4 Modern developments in superstr
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2 The bosonic string This chapter i
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2.1 p-brane actions 19 choice of pa
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particle, namely 2.1 p-brane action
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SOLUTION 2.1 p-brane actions 23 The
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where ˙X µ = ∂Xµ ∂τ 2.2 The
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Tαβ, that is, 2.2 The string acti
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2.2 The string action 29 where we h
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2.3 String sigma-model action: the
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2.4 Canonical quantization 37 compo
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2.4 Canonical quantization 39 These
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finds that Eq. (2.85) is solved by
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where N = 2.4 Canonical quantizatio
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In fact, any such state can be reca
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2.4 Canonical quantization 47 Criti
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2.5 Light-cone gauge quantization 4
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2.5 Light-cone gauge quantization 5
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Homework Problems 53 The closed str
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(i) Dirichlet boundary conditions a
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Homework Problems 57 PROBLEM 2.11 T
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3.1 Conformal field theory 59 and r
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3.1 Conformal field theory 61 rotat
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symmetry, this tensor is also conse
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3.1 Conformal field theory 65 This
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3.1 Conformal field theory 67 Such
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or the equivalent commutation relat
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3.1 Conformal field theory 71 An im
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3.1 Conformal field theory 73 is pr
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SOLUTION 3.2 BRST quantization 75 A
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where TX is given in Eq. (3.23) and
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3.2 BRST quantization 79 in differe
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SOLUTION 3.3 Background fields 81 U
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3.3 Background fields 83 unoriented
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3.4 Vertex operators 85 3.4 Vertex
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3.4 Vertex operators 87 must act on
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3.5 The structure of string perturb
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SOLUTION 3.5 The structure of strin
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3.6 The linear-dilaton vacuum and n
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3.7 Witten’s open-string field th
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form Homework Problems 107 Φ(z) =
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4 Strings with world-sheet supersym
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4.1 Ramond-Neveu-Schwarz strings 11
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4.2 Global world-sheet supersymmetr
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4.3 Constraint equations and confor
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EXERCISES 4.3 Constraint equations
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4.4 Boundary conditions and mode ex
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4.5 Canonical quantization of the R
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4.6 Light-cone gauge quantization o
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are given by 4.6 Light-cone gauge q
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4.7 SCFT and BRST 141 which now has
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4.7 SCFT and BRST 143 These contrib
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Homework Problems 145 needs to incl
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Homework Problems 147 PROBLEM 4.14
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5.1 The D0-brane action 149 5.1 The
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5.1 The D0-brane action 151 It turn
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Thus δ(S1 + S2) = −2m 5.1 The D
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SOLUTION 5.2 The supersymmetric str
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5.2 The supersymmetric string actio
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5.2 The supersymmetric string actio
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5.3 Quantization of the GS action 1
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5.4 Gauge anomalies and their cance
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and ω3 = (ω3L − ω3Y)/4, where
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Homework Problems 185 PROBLEM 5.2 I
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6 T-duality and D-branes String the
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6.1 The bosonic string and Dp-brane
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6.2 D-branes in type II superstring
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associated with the Fock-space stat
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6.3 Type I superstring theory 223 O
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6.3 Type I superstring theory 225 T
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6.4 T-duality in the presence of ba
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6.5 World-volume actions for D-bran
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SOLUTION Because 6.5 World-volume a
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Homework Problems 245 theory for ra
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PROBLEM 6.11 Homework Problems 247
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7 The heterotic string The precedin
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7.1 Nonabelian gauge symmetry in st
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7.2 Fermionic construction of the h
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where 7.2 Fermionic construction of
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7.3 Toroidal compactification 265 E
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7.3 Toroidal compactification 267 T
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where 4 or the inverse 7.3 Toroidal
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7.3 Toroidal compactification 271 t
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7.3 Toroidal compactification 273 N
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7.3 Toroidal compactification 275 w
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7.3 Toroidal compactification 277 I
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• Using Eq. (7.66), one obtains 7
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7.3 Toroidal compactification 281 N
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7.3 Toroidal compactification 283 E
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7.3 Toroidal compactification 285 F
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7.4 Bosonic construction of the het
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7.4 Bosonic construction of the het
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Homework Problems 291 To derive thi
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Homework Problems 293 in each case?
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Homework Problems 295 where the coo
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M-theory and string duality 297 tha
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M-theory and string duality 299 exp
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8.1 Low-energy effective actions 30
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In terms of the elfbein E A M 8.1 L
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8.2 S-duality 323 with the correspo
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8.2 S-duality 325 This leads to the
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8.2 S-duality 327 Type IIB S-dualit
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8.3 M-theory 329 since an SL(2, ) t
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8.3 M-theory 331 states, and carry
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8.3 M-theory 333 answering this que
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spectrum, while the zero mode of 8.
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8.3 M-theory 337 the weakly coupled
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8.4 M-theory dualities 339 An M-the
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8.4 M-theory dualities 341 the type
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8.4 M-theory dualities 343 which co
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that 8.4 M-theory dualities 345 T (
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8.4 M-theory dualities 347 d5 = dim
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SOLUTION 8.4 M-theory dualities 349
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Homework Problems 351 M2-brane wrap
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Homework Problems 353 (i) Use the r
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String geometry 355 is typically gi
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String geometry 357 erotic string r
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9.1 Orbifolds 359 four-dimensional
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9.1 Orbifolds 361 they are called o
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9.2 Calabi-Yau manifolds: mathemati
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9.2 Calabi-Yau manifolds: mathemati
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9.3 Examples of Calabi-Yau manifold
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9.4 Calabi-Yau compactifications of
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SOLUTION 9.5 Deformations of Calabi
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9.5 Deformations of Calabi-Yau mani
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9.5 Deformations of Calabi-Yau mani
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9.6 Special geometry 391 9.6 Specia
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9.6 Special geometry 393 In analogy
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9.6 Special geometry 395 Equations
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9.6 Special geometry 397 Kähler po
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which implies that 9.7 Type IIA and
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9.7 Type IIA and type IIB on Calabi
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9.8 Nonperturbative effects in Cala
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9.9 Mirror symmetry 411 Ωabc = e
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9.9 Mirror symmetry 413 1/R Fig. 9.
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9.10 Heterotic string theory on Cal
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9.10 Heterotic string theory on Cal
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9.11 K3 compactifications and more
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9.12 Manifolds with G2 and Spin(7)
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Appendix: Some basic geometry and t
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with Appendix: Some basic geometry
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HOMEWORK PROBLEMS Homework Problems
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Flux compactifications 457 on a new
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Flux compactifications 459 Some mod
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10.1 Flux compactifications and Cal
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SOLUTION 10.1 Flux compactification
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10.2 Flux compactifications of the
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where 10.2 Flux compactifications o
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which descends from the four-form a
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10.3 Moduli stabilization 501 Super
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since the superpotential transforms
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V(σ) V0 10.3 Moduli stabilization
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is 10.3 Moduli stabilization 507 V0
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10.4 Fluxes, torsion and heterotic
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SOLUTION 10.4 Fluxes, torsion and h
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10.5 The strongly coupled heterotic
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10.5 The strongly coupled heterotic
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10.6 The landscape 523 observed val
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The superpotential takes the simple
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10.7 Fluxes and cosmology 527 sente
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10.7 Fluxes and cosmology 529 are t
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10.7 Fluxes and cosmology 531 Howev
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10.7 Fluxes and cosmology 533 The i
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10.7 Fluxes and cosmology 535 of pa
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10.7 Fluxes and cosmology 537 satis
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10.7 Fluxes and cosmology 539 Here
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10.7 Fluxes and cosmology 541 Infla
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Appendix: Dirac matrix identities 5
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In general, [γm1...mp, γ n1...nq
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11 Black holes in string theory Bla
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Black holes in string theory 551 di
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11.1 Black holes in general relativ
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The solution to these equations is
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the period of τ is 11.2 Black-hole
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11.2 Black-hole thermodynamics 565
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11.3 Black holes in string theory 5
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symmetry is 11.3 Black holes in str
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of Eq. (11.46) is where 11.3 Black
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11.4 Statistical derivation of the
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11.4 Statistical derivation of the
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11.5 The attractor mechanism 587 cl
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11.5 The attractor mechanism 589 Th
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11.5 The attractor mechanism 591 Fi
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Equation (11.119) implies that 11.5
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11.5 The attractor mechanism 595 Th
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11.5 The attractor mechanism 597 in
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11.6 Small BPS black holes in four
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One then reproduces the desired ent
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Gauge theory/string theory dualitie
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12.1 Black-brane solutions in strin
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12.2 Matrix theory 625 the volume o
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12.2 Matrix theory 627 To test this
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12.2 Matrix theory 629 (1PI) graphs
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12.2 Matrix theory 631 (a) (b) (c)
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12.2 Matrix theory 633 The other gr
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12.2 Matrix theory 635 • The abse
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12.2 Matrix theory 637 in the type
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12.3 The AdS/CFT correspondence 639
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where 12.3 The AdS/CFT corresponden
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12.4 Gauge/string duality for the c
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12.5 Plane-wave space-times and the
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SOLUTION 12.5 Plane-wave space-time
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12.6 Geometric transitions 685 rela
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Homework Problems 689 where σ i ar
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Bibliographic discussion 691 harmon
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Bibliographic discussion 693 emissi
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Bibliographic discussion 695 gauge
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Bibliographic discussion 697 (1990)
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Bibliographic discussion 699 dacena
Page 1436: Bibliography 701 Angelantonj, C., a
Page 1440: Bibliography 703 angles. Nucl. Phys
Page 1444: Bibliography 705 Relativity. San Fr
Page 1448: Bibliography 707 Dixon, L.J., Fried
Page 1452: Bibliography 709 Commun. Math. Phys
Page 1456: Bibliography 711 Green, M. B., and
Page 1460: Bibliography 713 theories and strin
Page 1464: Bibliography 715 D64, 123522. E-pri
Page 1468: Bibliography 717 Maldacena, J. M. (
Page 1472: Bibliography 719 Nielsen, H. B. (19
Page 1476: Bibliography 721 Schwarz, J. H. (19
Page 1480: Bibliography 723 Phys. Rev. Lett.,
Page 1484: Bibliography 725 B155, 151. Witten,
Page 1490: 728 Index Calabi-Yau four-fold, 458
Page 1494: 730 Index with E6,6 symmetry, 570 c
Page 1498: 732 Index Friedmann equation, 528 F
Page 1502: 734 Index dual Coxeter number, 69 e
Page 1506: 736 Index homogeneity equation, 603
Page 1510: 738 Index superconformal symmetry,
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