Gravity and Strings

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3.5 Final Comments 113 theory as well as other alternative theories of gravity. Some of these elements are more or less trivial extensions of GR (for instance, its reformulation in the Vielbein formalism which allows the coupling to spinors) and, in fact, it is always (or usually) possible to see the new theories as GR coupled to different fields. In the next few chapters we are going to review these elements and theories that contain them: the Cartan–Sciama–Kibble theory, non-symmetric theories of gravity, theories of teleparallelism, and supergravity theories. The simplest way to introduce most of them is through a minimal action principle and the formulation of the minimal action principle for GR will be the first step in this direction.
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Gravity and Strings One appealing f
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Gravity and Strings TOMÁS ORTÍN S
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To Marimar, Diego, and Tomás, the
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x Contents 3.1.1 Scalar gravity cou
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xii Contents 8.7.4 Dyons and the DS
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xiv Contents 14.3.2 Open bosonic st
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xvi Contents Appendix A Lie groups,
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Preface String theory has lived for
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Part I Introduction to gravity and
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4 Differential geometry function 1
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6 Differential geometry and on weig
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8 Differential geometry We can also
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10 Differential geometry where ρ
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12 Differential geometry a metric-c
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14 Differential geometry scalar R,
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16 Differential geometry Local GL(d
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18 Differential geometry In the sec
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20 Differential geometry where ω(e
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22 Differential geometry We define
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24 Differential geometry Since ⋆d
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2 Noether’s theorems In the next
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28 Noether’s theorems where x ′
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30 Noether’s theorems First, obse
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32 Noether’s theorems Sometimes i
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34 Noether’s theorems According t
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36 Noether’s theorems which is th
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38 Noether’s theorems Furthermore
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40 Noether’s theorems The Rosenfe
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42 Noether’s theorems is invarian
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44 Noether’s theorems In this way
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46 A perturbative introduction to g
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Page 270: 4.1 The Einstein-Hilbert action 115
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4.4 The Cartan-Sciama-Kibble theory
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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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