Large N Field Theories, String Theory and Gravity - O. Aharony, S.S. Gubser, J. Maldacena, H. Ooguri

Large N Field Theories, String Theory and Gravity - O. Aharony, S.S. Gubser, J. Maldacena, H. Ooguri

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arXiv:hep-th/9905111 v3 1 Oct 1999 December 10, 2001 CERN-TH/99-122 hep-th/9905111 HUTP-99/A027 LBNL-43113 RU-99-18 UCB-PTH-99/16 Large N Field Theories, String Theory and Gravity Ofer Aharony, 1 Steven S. Gubser, 2 Juan Maldacena, 2 , 3 Hirosi Ooguri, 4 , 5 and Yaron Oz 6 1 Department of Physics and Astronomy, Rutgers University, Piscataway, NJ 08855-0849, USA 2 Lyman Laboratory of Physics, Harvard University, Cambridge, MA 02138, USA 3 School of Natural Sciences, Institute for Advanced Study, Princeton, NJ 08540 4 Department of Physics, University of California, Berkeley, CA 94720-7300, USA 5 Lawrence Berkeley National Laboratory, MS 50A-5101, Berkeley, CA 94720, USA 6 Theory Division, CERN, CH-1211, Geneva 23, Switzerland [email protected], [email protected], [email protected], [email protected], [email protected] Abstract We review the holographic correspondence between field theories and string/M theory, focusing on the relation between compactifications of string/M theory on Anti-de Sitter spaces and conformal field theories. We review the background for this correspondence and discuss its motivations and the evidence for its correctness. We describe the main results that have been derived from the correspondence in the regime that the field theory is approximated by classical or semiclassical gravity. We focus on the case of the N = 4 supersymmetric gauge theory in four dimensions, but we discuss also field theories in other dimensions, conformal and non-conformal, with or without supersym- metry, and in particular the relation to QCD. We also discuss some implications for black hole physics. ( To be published in Physics Reports )
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Contents 1 Introduction 4 1.1 General Introduction and Overview . . . . . . . . . . . . . . . . . . . . 4 1.2 Large N Gauge Theories as String Theories . . . . . . . . . . . . . . . 10 1.3 Black p -Branes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.3.1 Classical Solutions . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.3.2 D-Branes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.3.3 Greybody Factors and Black Holes . . . . . . . . . . . . . . . . 21 2 Conformal Field Theories and AdS Spaces 30 2.1 Conformal Field Theories . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.1.1 The Conformal Group and Algebra . . . . . . . . . . . . . . . . 31 2.1.2 Primary Fields, Correlation Functions, and Operator Product Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.1.3 Superconformal Algebras and Field Theories . . . . . . . . . . . 34 2.2 Anti-de Sitter Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.2.1 Geometry of Anti-de Sitter Space . . . . . . . . . . . . . . . . . 36 2.2.2 Particles and Fields in Anti-de Sitter Space . . . . . . . . . . . 45 2.2.3 Supersymmetry in Anti-de Sitter Space . . . . . . . . . . . . . . 47 2.2.4 Gauged Supergravities and Kaluza-Klein Compactifications . . . 48 2.2.5 Consistent Truncation of Kaluza-Klein Compactifications . . . . 52 3 AdS/CFT Correspondence 55 3.1 The Correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.1.1 Brane Probes and Multicenter Solutions . . . . . . . . . . . . . 61 3.1.2 The Field Operator Correspondence . . . . . . . . . . . . . . 62 3.1.3 Holography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.2 Tests of the AdS/CFT Correspondence . . . . . . . . . . . . . . . . . . 68 1
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3.2.1 The Spectrum of Chiral Primary Operators . . . . . . . . . . . 70 3.2.2 Matching of Correlation Functions and Anomalies . . . . . . . . 78 3.3 Correlation Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 3.3.1 Two-point Functions . . . . . . . . . . . . . . . . . . . . . . . . 82 3.3.2 Three-point Functions . . . . . . . . . . . . . . . . . . . . . . . 85 3.3.3 Four-point Functions . . . . . . . . . . . . . . . . . . . . . . . . 89 3.4 Isomorphism of Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . 90 3.4.1 Hilbert Space of String Theory . . . . . . . . . . . . . . . . . . 91 3.4.2 Hilbert Space of Conformal Field Theory . . . . . . . . . . . . . 96 3.5 Wilson Loops . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 3.5.1 Wilson Loops and Minimum Surfaces . . . . . . . . . . . . . . . 98 3.5.2 Other Branes Ending on the Boundary . . . . . . . . . . . . . . 103 3.6 Theories at Finite Temperature . . . . . . . . . . . . . . . . . . . . . . 104 3.6.1 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 3.6.2 Thermal Phase Transition . . . . . . . . . . . . . . . . . . . . . 107
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