String Theory and M-Theory

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open, 164, 166 world-sheet topologies, 178 type I ′ theory, 224, 246 type II superstring theory GS world-sheet action, 155 kappa symmetry, 156 spectrum, 167 type IIA superstring theory Calabi–Yau compactification, 400 D0-branes, 330 D6-brane, 332 K3 compactification, 424 mirror symmetry, 425 S-duality, 426 strong coupling limit, 330 type IIB superstring theory, 175 (p, q) 5-branes, 329 (p, q) strings, 327 7-branes, 329, 428, 429 anomaly cancellation, 175 Calabi–Yau compactification, 400 D3-brane, 329 K3 compactification, 426 string webs, 329 three-string junction, 329, 687 U-duality, 345, 636 E3(¢ ), 347 En(¢ ), 346 E6,6, 567, 570 E7,7, 576 ultraviolet divergences, 2, 7 absence of, 83 cancellation, 83 unitarity bound, 672 universal hypermultiplet, 401 unoriented string, 83, 196, 220, 221 vacuum energy density, 15, 84, 527 vacuum selection problem of, 14 vanishing cycle, 404 vector supermultiplet, 204 D = 10, 257, 261 D = 3, 473 D = 4, N = 4, 400 D = 4, N = 2, 400 massive, 300 E8, 181 SO(32), 258 U(N), 239 Verma module, 68 vertex operator, 85 integrated, 87 of tachyon, 85 unintegrated, 87 with ghosts, 86 vielbein, 444 Virasoro algebra, 40, 57 classical, 61 generators, 39, 40 Index 739 operator product expansions, 140 Volkov–Akulov theory, 236 volume of unit n-sphere, 554 Wald’s formula, 603, 609 warp factor, 462 of warped conifold, 493 warped compactification, 355, 456 of the heterotic theory, 508 of the type IIB theory, 480 warped conifold geometry of, 492 warped geometry, 457, 461, 612 warped metric, 461 type IIB compactification, 481 wave equation, 32 web of dualities, 12, 296 weighted complex projective space, 371 Wess–Zumino consistency conditions, 173 Wess–Zumino–Witten model, 69 Weyl symmetry, 89 Weyl transformations, 30 white hole, 555 Wigner–Inönü contraction, 679 Wilson line, 198, 288, 295, 418 Wilson-loop operator, 661 winding number, 188 fractional, 199 WMAP, 538 world line, 6 world-sheet action with ghost fields, 76 cosmological constant term, 28, 31, 99 light-cone coordinates, 33 metric, 7, 26 conformally flat, 89 with fermions, 110 with local supersymmetry, 120 world-sheet parity, 220, 226 projection, 221 world-sheet topology, 6, 8 Yang–Mills theory, 4, 11 anomalies, 173 field strength, 102, 172 gauge field, 172 gauge transformation, 103, 172 in three dimensions, 662 finite temperature, 663 large-N expansion, 639 multi-trace operators, 656 planar approximation, 641 single-trace operators, 656 with N = 4 supersymmetry, 15, 298, 323, 636, 640, 641, 657, 687 zero-norm spurious states, 45, 129 zero-point energy, 133, 146, 257 zeta-function regularization, 50, 146
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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
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Bibliography 701 Angelantonj, C., a
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Bibliography 703 angles. Nucl. Phys
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Bibliography 705 Relativity. San Fr
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Bibliography 707 Dixon, L.J., Fried
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Bibliography 709 Commun. Math. Phys
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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 1488: Poincaré coordinates, 645, 646, 68
Page 1492: 211, 214, 234, 244, 271, 329, 330,
Page 1496: anomaly analysis, 171 breakdown of,
Page 1500: hypermultiplet, 300, 400, 670 infin
Page 1504: naked singularity, 551, 558 Nambu-G
Page 1508: scalar curvature, 447 scalar potent
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