Gravity and Strings

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Contents xvii F.2.4 A general metric for extreme p-branes 646 F.2.5 Brinkmann metrics 647 Appendix G The harmonic operator on R 3 × S 1 648 References 650 Index 671
Page 2: Gravity and Strings One appealing f
Page 10: Gravity and Strings TOMÁS ORTÍN S
Page 14: To Marimar, Diego, and Tomás, the
Page 20: x Contents 3.1.1 Scalar gravity cou
Page 24: xii Contents 8.7.4 Dyons and the DS
Page 28: xiv Contents 14.3.2 Open bosonic st
Page 32: xvi Contents Appendix A Lie groups,
Page 40: xx Preface such as differential geo
Page 46: 1 Differential geometry The main pu
Page 50: 1.2 Affinely connected spacetimes 5
Page 54: 1.2 Affinely connected spacetimes 7
Page 58: 1.3 Metric spaces 9 1.3 Metric spac
Page 62: T = 0 1.3 Metric spaces 11 Affinely
Page 66: 1.3 Metric spaces 13 1.3.2 Riemann
Page 70: 1.4 Tangent space 15 We can now rel
Page 74: 1.4 Tangent space 17 The significan
Page 78: 1.4 Tangent space 19 For more gener
Page 82: 1.6 Duality operations 21 we say th
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1.7 Differential forms and integrat
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1.8 Extrinsic geometry 25 The Vielb
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2.2 Noether’s theorems 27 The var
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where 2.2 Noether’s theorems 29 j
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2.3 Conserved charges 31 which is a
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2.4 The special-relativistic energy
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in any dimension: T µ ν = 2.4 The
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2.5 The Noether method 41 We may ex
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2.5 The Noether method 43 which can
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3 A perturbative introduction to ge
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3.1 Scalar SRFTs of gravity 47 Afre
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3.1 Scalar SRFTs of gravity 49 via
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Eq. (3.3) becomes S[φ(x), X µ (ξ
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3.1 Scalar SRFTs of gravity 53 3.1.
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3.1 Scalar SRFTs of gravity 55 wher
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3.2 Gravity as a self-consistent ma
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Using η µνρ AD 3.2 Gravity as a
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In a complete revolution 3.2 Gravit
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ϕ µν ,wefind 3.2 Gravity as a se
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3.3 General relativity 97 the Nambu
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3.3 General relativity 99 which can
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3.3 General relativity 101 local an
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3.4 The Fierz-Pauli theory in a cur
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3.5 Final Comments 113 theory as we
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4.1 The Einstein-Hilbert action 115
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4.2 The Einstein-Hilbert action in
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4.3 The first-order (Palatini) form
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4.3 The first-order (Palatini) form
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4.4 The Cartan-Sciama-Kibble theory
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we find the Bianchi identity 4.4 Th
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and using 4.4 The Cartan-Sciama-Kib
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4.5 Gravity as a gauge theory 141 i
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4.5 Gravity as a gauge theory 143 M
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4.6 Teleparallelism 145 the Riemann
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4.6 Teleparallelism 147 Lagrangian,
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4.6 Teleparallelism 149 this result
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5.1 Gauging N = 1, d = 4 superalgeb
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5.1 Gauging N = 1, d = 4 superalgeb
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5.2 N = 1, d = 4(Poincaré) supergr
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5.2 N = 1, d = 4(Poincaré) supergr
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5.3 N = 1, d = 4 AdS supergravity 1
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5.4 Extended supersymmetry algebras
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5.4 Extended supersymmetry algebras
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5.5 N = 2, d = 4 (Poincaré) superg
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5.6 N = 2, d = 4 “gauged” (AdS)
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5.7 Proofs of some identities 169 T
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6 Conserved charges in general rela
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6.1 The traditional approach 173 en
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6.1 The traditional approach 175 Ha
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6.1 The traditional approach 177 th
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6.2 The Noether approach 179 which
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6.3 The positive-energy theorem 181
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6.3 The positive-energy theorem 183
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7 The Schwarzschild black hole With
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7.1 Schwarzschild’s solution 189
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where dρ 2 + ρ 2 d 2 (2) 7.1 Schw
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7.2 Sources for Schwarzschild’s s
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7.3 Thermodynamics 203 This relatio
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7.3 Thermodynamics 205 and so the f
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7.3 Thermodynamics 207 If no inform
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7.4 The Euclidean path-integral app
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7.5 Higher-dimensional Schwarzschil
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8 The Reissner-Nordström black hol
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8.1 Coupling a scalar field to grav
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8.1 Coupling a scalar field to grav
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8.2 The Einstein-Maxwell system 219
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8.3 The electric Reissner-Nordströ
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8.4 The Sources of the electric RN
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8.5 Thermodynamics of RN black hole
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8.6 The Euclidean electric RN solut
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8.7 Electric-magnetic duality 245 I
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of whose dual over S 2 ∞ is 16πG
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8.7 Electric-magnetic duality 249 d
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8.7 Electric-magnetic duality 251 x
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8.7 Electric-magnetic duality 253 n
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8.7 Electric-magnetic duality 255 N
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8.7 Electric-magnetic duality 257 c
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8.8 Magnetic and dyonic RN black ho
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8.8 Magnetic and dyonic RN black ho
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8.9 Higher-dimensional RN solutions
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8.9 Higher-dimensional RN solutions
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9 The Taub-NUT solution The asympto
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9.1 The Taub-NUT solution 269 More
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9.2 The Euclidean Taub-NUT solution
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where g(x) is the SU(2)-valued func
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9.3 Charged Taub-NUT solutions and
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9.3 Charged Taub-NUT solutions and
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10.1 pp-Waves 283 in the positive d
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10.2 Four-dimensional pp-wave solut
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10.3 Sources: the AS shock wave 287
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10.3 Sources: the AS shock wave 289
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11.1 Classical and quantum mechanic
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11.2 KK dimensional reduction on a
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connection ˆω â ˆbĉ : 11.2 KK
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11.3 KK reduction and oxidation of
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11.4 Toroidal (Abelian) dimensional
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11.5 Generalized dimensional reduct
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leaving the action in the form S =
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12 Dilaton and dilaton/axion black
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12.1 Dilaton black holes: the a-mod
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The inverse relations are, for x =
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12.2 Dilaton/axion black holes 359
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13 Unbroken supersymmetry In our st
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13.1 Vacuum and residual symmetries
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13.2 Supersymmetric vacua and resid
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13.2 Supersymmetric vacua and resid
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3. 13.2 Supersymmetric vacua and re
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13.3 N = 1, 2, d = 4 vacuum supersy
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13.4 The vacua of d = 5, 6 supergra
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13.4 The vacua of d = 5, 6 supergra
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13.5 Partially supersymmetric solut
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14 String theory In this chapter we
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14 String theory 407 generically ca
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14.1 Strings 409 As advertised, in
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Neumann (N) boundary conditions: 14
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14.1 Strings 413 and the equations
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14.1 Strings 415 14.1.2 Green-Schwa
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14.2 Quantum theories of strings 41
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14.3 Compactification on S 1 :Tdual
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14.3 Compactification on S 1 :Tdual
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15.1 Effective actions and backgrou
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15.1 Effective actions and backgrou
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Nambu-Goto action [647]: 15.2 T dua
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15.2 T duality and background field
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15.3 Example: the fundamental strin
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16 From eleven to four dimensions I
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16.1 Dimensional reduction from d =
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which leads to 16.1 Dimensional red
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and, finally, the action becomes ˆ
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16.2 Romans’ massive N = 2A, d =
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16.2 Romans’ massive N = 2A, d =
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16.3 Further reduction of N = 2A, d
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For the fermions: 16.4 The effectiv
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16.5 Toroidal compactification of t
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16.6 T duality, compactification, a
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17 The type-IIB superstring and typ
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17.1 N = 2B, d = 10 supergravity in
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17.2 Type-IIB S duality 489 SU(1,1)
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17.3 Dimensional reduction of N = 2
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17.3 Dimensional reduction of N = 2
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RR fields: 17.4 Dimensional reducti
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17.5 Consistent truncations and het
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17.5 Consistent truncations and het
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general families of solutions. 18.1
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18.1 Generalities 503 Another, more
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18.1 Generalities 505 their equatio
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18.1 Generalities 507 The conservat
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multiple of 2π: (−1) (p+1) q B
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18.1 Generalities 511 string in d =
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18.2 General p-brane solutions 513
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19.1 String-theory extended objects
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19.3 The masses and charges of the
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19.3 The masses and charges of the
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19.4 Duality of string-theory solut
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KK9M (9,1,1) KK7M (7,1,3) M5-3 (6,3
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19.4 Duality of string-theory solut
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19.6 Intersections 563 Maximally su
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19.6 Intersections 565 Table 19.4.
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19.6 Intersections 567 outside the
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19.6 Intersections 569 should exist
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19.6 Intersections 571 The coordina
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20 String black holes in four and f
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20.1 Composite dilaton black holes
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20.2 Black holes from branes 577 If
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20.2 Black holes from branes 579 si
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20.2 Black holes from branes 581 Th
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20.2 Black holes from branes 583 Th
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20.2 Black holes from branes 585 (y
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20.2 Black holes from branes 587 wh
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20.3 Entropy from microstate counti
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Appendix A Lie groups, symmetric sp
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Lie groups, symmetric spaces, and Y
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G/H isreductive if Lie groups, symm
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Appendix B Gamma matrices and spino
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Gamma matrices and spinors 613 wher
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Gamma matrices and spinors 615 or,
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Gamma matrices and spinors 617 Usin
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For instance, we can obtain Gamma m
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Gamma matrices and spinors 621 It i
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Gamma matrices and spinors 623 We w
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Gamma matrices and spinors 625 and,
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Gamma matrices and spinors 627 Thus
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Gamma matrices and spinors 629 Tabl
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Gamma matrices and spinors 631 Thes
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Gamma matrices and spinors 633 labe
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Appendix C 635 For some purposes, s
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Appendix C 637 C.2 Squashed S 3 and
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Appendix E Conformal rescalings If
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Connections and curvature component
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and the Ricci scalar is given by We
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Connections and curvature component
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The Ricci scalar is Connections and
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The harmonic operator on R 3 × S 1
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[35] E. Álvarez, L. Álvarez-Gaum
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[128] E. Bergshoeff, R. Kallosh, T.
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[222] L. Castellani, R. D’Auria,
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[316] S. Deser and B. Zumino, Phys.
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[411] D. V. Gal’tsov and O. V. Ke
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[506] S. F. Hassan, Nucl. Phys. B58
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References 663 [600] J. M. Izquierd
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[696] R. R. Metsaev and A. A. Tseyt
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[793] H. Quevedo, Fortschr. Phys. 3
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[889] G. ’t Hooft, Nucl. Phys. B6
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Index Page numbers in italic are th
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isotropic, 198, 216, 232, 234, 265,
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Noether method, 78 first-order form
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integrability equation in N = 2, d
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Newman-Penrose formalism, see Newma
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Dp, 538 compactified on T 6 , 577 c
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extended, 121, 150, 160-163, 379, 3
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Gravity and Strings

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