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Tasi Lectures on Branes, Black Holes and Anti-De Sitter Space1

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Tasi Lectures on Branes, Black Holes and Anti-De Sitter Space1 View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CERN Document Server UM-TH-99-07 hep-th/9912164 TASI LECTURES ON BRANES, BLACK HOLES AND ANTI-DE SITTER SPACE1. M. J. Duff2 Randall Laboratory, Department of Physics, University of Michigan, Ann Arbor, MI 48109-1120 ABSTRACT In the light of the duality between physics in the bulk of anti-de Sitter space and a conformal field theory on the boundary, we review the M2, D3andM5 branes and how their near-horizon geometry yields the compactification of D =11supergravityonS7, Type IIB supergravity on S5 and D =11supergravityonS4, respectively. We discuss the “Membrane at the End of the Universe” idea and its relation to the corresponding superconformal singleton theories that live on the boundary of the AdS4, AdS5 and AdS7 vacua. The massless sectors of these compactifications are described by the maximally supersymmetric D =4,D=5andD= 7 gauged supergravities. We construct the non-linear Kaluza-Klein ans¨atze describing the embeddings of the U(1)4, U(1)3 and U(1)2 truncations of these supergravities, which admit 4-charge AdS4,3-chargeAdS5 and 2- charge AdS7 black hole solutions. These enable us to embed the black hole solutions back in ten and eleven dimensions and reinterpret them as M2, D3andM5 branes spinning in the transverse dimensions with the black hole charges given by the angular momenta of the branes. A comprehensive Appendix lists the field equations, symmetries and transformation rules of D =11supergravity,TypeIIB supergravity, and the M2, D3andM5branes. 1Based on talks delivered at the Theoretical Advanced Study Institute, Boulder, Colorado, June 1999 and the Banff Summer School, Alberta, Canada, June-July 1999. Research supported in part by NSF Grant PHY-9722090. 2mduff@umich.edu Contents 1INTRODUCTION 2 1.1 Supergravity, supermembranes and M-theory................. 2 1.2TheKaluza-Kleinidea.............................. 3 1.3Thefieldcontent................................. 8 1.4TheAdS/CFTcorrespondence......................... 12 1.5Planofthelectures................................ 13 1.6Problems1 .................................... 15 2 ELEVEN DIMENSIONAL SUPERGRAVITY 15 2.1Bosonicfieldequations.............................. 15 2.2 AdS S7 ..................................... 16 4× 2.3Consistenttruncationtothemasslessmodes.................. 19 2.4Thesupermembranesolution.......................... 21 2.5 AdS S4 ..................................... 24 7× 2.6Thesuperfivebranesolution........................... 25 2.7Problems2 .................................... 27 3 TYPE IIB SUPERGRAVITY 27 3.1Bosonicfieldequations.............................. 27 3.2 AdS S5 ..................................... 28 5× 3.3Theself-dualsuperthreebranesolution..................... 29 3.4Problems3 .................................... 31 4 THE M2-BRANE, D3-BRANE AND M5-BRANE 31 4.1TheM2-brane................................... 31 4.2TheM5-brane................................... 32 4.3TheD3-brane................................... 35 4.4Problems4 .................................... 35 5 ADS/CFT : THE MEMBRANE AT THE END OF THE UNIVERSE 35 5.1 Singletons live on the boundary . ...................... 35 5.2 The membrane as a singleton: the membrane/supergravity bootstrap . 37 5.3Doubletonsandtripletonsrevisited....................... 40 5.4Themembraneattheendoftheuniverse................... 42 1 5.5Nearhorizongeometryandp-branearistocracy................ 45 5.6Supermembraneswithfewersupersymmetries.Skew-whiffing......... 47 5.7TheMaldacenaconjecture............................ 49 5.8Problems5 .................................... 50 6 ANTI-DE SITTER BLACK HOLES 51 6.1Introduction.................................... 51 6.2 S5 reductionofTypeIIBsupergravity..................... 54 6.3D=5AdSblackholes............................... 56 6.4RotatingD3-brane................................ 56 6.5 S7 reductionofD=11supergravity....................... 58 6.6D=4AdSblackholes............................... 62 6.7RotatingM2-brane................................ 62 6.8 S4 reductionofD=11supergravity....................... 64 6.9D=7AdSblackholes............................... 66 6.10RotatingM5-brane................................ 66 6.11Chargeasangularmomentum.......................... 67 6.12Magneticblackholes............................... 68 6.13Kaluza-Kleinstatesasblackholes....................... 69 6.14 Recent Developments . ............................. 71 6.15Problems6 .................................... 71 7 SOLUTIONS TO PROBLEMS 72 7.1Solutions1 .................................... 72 7.2Solutions2 .................................... 74 7.3Solutions3 .................................... 77 7.4Solutions4 .................................... 80 7.5Solutions5 .................................... 82 7.6Solutions6 .................................... 83 8 ACKNOWLEDGEMENTS 87 9 APPENDICES 87 A The Lagrangian, symmetries and transformation rules of D=11 super- gravity 87 2 B The field equations, symmetries and transformation rules of Type IIB supergravity 89 C The Lagrangian, symmetries and transformation rules of the M2-brane 92 D The field equations, symmetries and transformation rules of the M5- brane 93 E The Lagrangian, symmetries and transformation rules of the D3-brane 97 F D=4, N=2 gauged supergravity 99 3 1 INTRODUCTION 1.1 Supergravity, supermembranes and M-theory A vital ingredient in the quest for a unified theory embracing all physical phenomena is supersymmetry, a symmetry which (a) unites bosons and fermions, (b) requires the exis- tence of gravity and (c) places an upper limit of eleven on the dimension of spacetime. For these reasons, in the early 1980s many physicists looked to eleven-dimensional supergravity in the hope that it might provide that elusive superunified theory. Then in 1984 superuni- fication underwent a major paradigm-shift: eleven-dimensional supergravity was knocked off its pedestal by ten-dimensional superstrings , one-dimensional objects whose vibrational modes represent the elementary particles. Unlike eleven-dimensional supergravity, super- strings provided a perturbatively finite theory of gravity which, after compactification to four spacetime dimensions, seemed in principle capable of explaining the Standard Model of the strong, weak and electromagnetic forces including the required chiral representations of quarks and leptons. Despite these major successes, however, nagging doubts persisted about superstrings. First, many of the most important questions in string theory — How do strings break supersymmetry? How do they choose the right vacuum state? How do they explain the smallness of the cosmological constant? How do they resolve the apparent paradoxes of quantum black holes? — seemed incapable of being answered within the framework of a weak coupling perturbation expansion. They seemed to call for some new, non-perturbative, physics. Secondly, why did there appear to be five different mathematically consistent superstring theories: the E E heterotic string, the SO(32) heterotic string, the SO(32) 8 × 8 Type I string, the Type IIA and Type IIB strings? If one is looking for a unique Theory of Everything, this seems like an embarrassment of riches! Thirdly, if supersymmetry permits eleven dimensions, why do superstrings stop at ten? This question became more acute with the discoveries of the elementary supermembrane in 1987 and its dual partner, the solitonic superfivebrane, in 1992. These are supersymmetric extended objects with respectively two and five dimensions moving in an eleven-dimensional spacetime. Finally, therefore, if we are going to generalize zero-dimensional point particles to one-dimensional strings, why stop there? Why not two-dimensional membranes or more generally p-dimensional objects (inevitably dubbed p-branes)? Although this latter possibility was pursued by a small but dedicated group of theorists, starting in about 1986, it was largely ignored by the mainstream physics community. 4 Well, the year 1995 witnessed a new paradigm-shift: perturbative ten-dimensional su- perstrings have in their turn been superseded by a new non-perturbative theory called M- theory, which describes, amongst other things, supermembranes and superfivebranes, which subsumes the above five consistent strings theories, and which has as its low-energy limit, eleven-dimensional supergravity! According to Fields Medalist Edward Witten “M stands for magical, mystery or membrane, according to taste”. New evidence in favor of this the- ory is appearing daily on the internet and represents the most exciting development in the subject since 1984 when the superstring revolution first burst on the scene. E E heterotic string 8 × 8 SO(32) heterotic string 9 > > SO(32) Type I string > M theory (1.1) > > T ype IIA string = > Type IIB string > > > > Thus this new framework now provides the starting;> point for understanding a wealth of new non-perturbative phenomena, including string/string duality, Seiberg-Witten theory, quark confinement, QCD, particle physics phenomenology, quantum black holes, cosmology and, ultimately, their complete synthesis in a final theory of physics. 1.2 The Kaluza-Klein idea Cast your minds back to 1919. Maxwell’s theory of electromagnetism was well established and Einstein had recently formulated his General Theory of Relativity. By contrast, the strong and weak interactions were not well understood. In searching for a unified theory of the fundamental forces, therefore, it was natural to attempt to merge gravity with elec-
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