String Theory and M-Theory

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scalar curvature, 447 scalar potential flux compactification M-theory on a Calabi–Yau four-fold, 502 type IIB on a Calabi–Yau three-fold, 502 no-scale type, 503 with nonperturbative terms, 504 scale factor, 528, 535 scale invariance, 649 scale transformation, 60 Schwarzian derivative, 66 Schwarzschild black hole, 552 in D dimensions, 553 in Kruskal–Szekeres coordinates, 555 temperature of, 563 Schwarzschild radius, 553 second law of thermodynamics, 563 second superstring revolution, 10, 403 Seiberg duality, 674 self-dual charge, 208 self-dual field strength, 313 self-dual radius, 209 short supermultiplet, 297 Siegel upper half plane, 96 sigma-model action for p-brane, 29 for string, 26, 27, 30 simplicial homology, 443 simply-connected manifold, 364 simply-laced Lie groups, 69, 422 singularities ADE classification, 360 AN , 360, 436 DN , 373 blow-up, 367, 368, 373, 435 slow-roll approximation, 535 conditions, 535 inflation, 540 parameters, 535, 539, 540 small black holes, 599 source graviton, 632 special conformal transformation, 60 special coordinates, 392 special geometry, 391, 394, 589 homogeneous coordinates, 393 inhomogeneous coordinates, 393 special Lagrangian cycle, 404, 589 special Lagrangian submanifold, 407 special-Kähler manifold, 400, 401, 589 spin connection, 172, 305, 445, 446 embedding in the gauge group, 417 spin manifold, 378, 447 Spin(1,1), 111 Spin(8), 162, 169 Clebsch–Gordon coefficients, 166 Dynkin diagram, 163 spinors, 133, 136, 162 triality symmetry, 163 vector, 136 spurious states, 44 Index 737 standard big-bang model (SBB), 530 standard model, 4, 355 standard-model brane, 495 state–operator correspondence, 67, 85, 655 static gauge, 27, 233, 236 stationary metric, 553 Stokes’ theorem, 215, 217, 443, 451 string charge, 227 string coupling constant, 11, 85, 90, 227, 324 in six dimensions, 426 nonperturbative effects, 357 T-duality mapping, 210 type IIA, 310 string field theory, 100 Chern–Simons action, 104 cubic string interaction, 105 gauge transformation, 103 in light-cone gauge, 682 string geometry, 411 string length scale, 6, 34, 310 string perturbation theory, 89 string sigma-model action, 26, 27, 30 string tension, 34 string theory landscape, 459, 522 string world sheet, 6 topology, 8 stringy exclusion principle, 660 strong nuclear force, 2 submanifold special Lagrangian, 408 super Yang–Mills theory N = 2 for D = 4, 670 N = 4 for D = 4, 636, 638, 640 θ term, 650 in terms of N = 1 superfields, 666 multi-trace operators, 656 planar approximation, 641 S-duality, 650 single-trace operators, 656 in 0 + 1 dimensions, 627 in ten dimensions, 157, 176, 626 super-Poincaré algebra, 112, 149 super-Virasoro algebra constraints, 118 generators, 126 in the NS sector, 127 in the R sector, 127 operator-product expansions, 141 superspace formulation, 141 superalgebra OSp(1 | 2), 128 OSp(6, 2|4), 644 OSp(8|4), 644 P SU(2, 2|4), 640, 667 SU(1, 1 | 1), 128 supercharges, 114 superconformal algebra SU(2, 2|1), 671 superconformal field theory, 612 chiral primary operator, 655 superconformal gauge theory, 670, 671
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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
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 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 1510: 738 Index superconformal symmetry,
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