String Theory and M-Theory

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644 Gauge theory/string theory dualities Eq. (12.87), this corresponds to a flow to the infrared in the gauge theory. It also corresponds to the radius of the circular eleventh dimension increasing giving an 11-dimensional M-theory geometry in the limit. Therefore, in the limit, the coupling becomes infinite, <strong>and</strong> one reaches the conformallyinvariant fixed-point theory that describes a collection of coincident M2branes in 11 dimensions. This theory should have an SO(8) R symmetry corresponding to rotations in the eight dimensions that are transverse to the M2-branes in 11 dimensions. The AdS4 × S 7 metric has the isometry group SO(3, 2) × SO(8) ≈ Sp(4) × Spin(8). (12.95) As before, the first factor is the symmetry of the AdS space, which corresponds to the conformal symmetry group of the dual gauge theory. Also, the second factor is the symmetry of the sphere, which corresponds to the R symmetry of the dual gauge theory. This solution is maximally supersymmetric, which means that there are 32 conserved supercharges. In the dual gauge theory 16 supersymmetries are realized linearly, <strong>and</strong> the other 16 are conformal supersymmetries. Including the supersymmetries, the complete isometry superalgebra is OSp(8|4). This contains 32 fermionic generators (the supercharges) transforming as (8, 4) under Spin(8) × Sp(4). The M5-brane case Similar remarks apply to the six-dimensional CFT associated with a stack of M5-branes that is dual to M-theory with an AdS7 × S 4 geometry. The AdS7 × S 4 metric has the isometry group SO(6, 2) × SO(5) ≈ Spin(6, 2) × USp(4). (12.96) Including the supersymmetries, the complete isometry superalgebra in this case is OSp(6, 2|4). This superalgebra contains 32 fermionic generators transforming as (8, 4) under Spin(6, 2) × USp(4). The problem of defining the conformal field theory on the M5-branes is more severe than in the M2-brane case. To define a field theory, a weakcoupling description in the UV is required. Unlike the M2-brane case, there is no such description in the M5-brane case, because it is a six-dimensional theory. Still, there must be a CFT associated with the M5-brane system. The problem is that we don’t know how to describe it other than via the AdS/CFT duality.
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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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670 Gauge theory/string theory dual
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Bibliographic discussion In the fol
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692 Bibliographic discussion The BR
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694 Bibliographic discussion derive
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696 Bibliographic discussion which
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698 Bibliographic discussion Maldac
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Bibliography Abouelsaood, A., Calla
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702 Bibliography Singapore: World S
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704 Bibliography Brink, L., and Nie
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706 Bibliography E-print hep-th/000
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708 Bibliography hep-th/0105203. Eg
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710 Bibliography Phys., 71, 983-108
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712 Bibliography hep-th/9802109. Gu
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714 Bibliography Johnson, C. V. (20
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716 Bibliography Lerche, W., Schell
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718 Bibliography Montonen, C. (1974
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720 Bibliography Polchinski, J., an
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722 Bibliography Sen, A. (1999). No
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724 Bibliography Tseytlin, A. A. (1
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acceleration equation, 528 action p
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728 Index Calabi-Yau four-fold, 458
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730 Index with E6,6 symmetry, 570 c
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732 Index Friedmann equation, 528 F
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734 Index dual Coxeter number, 69 e
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736 Index homogeneity equation, 603
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738 Index superconformal symmetry,
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