Correspondence

February 1, 2008
hep-th/9905111

CERN-TH/99-122
HUTP-99/A027
LBNL-43113
RU-99-18
UCB-PTH-99/16

Large N Field Theories,

arXiv:hep-th/9905111v3 1 Oct 1999

String Theory and Gravity
Ofer Aharony,1 Steven S. Gubser,2 Juan Maldacena,2,3
Hirosi Ooguri,4,5 and Yaron Oz6
1

2

Lyman Laboratory of Physics, Harvard University, Cambridge, MA 02138, USA

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Department ofPhysics and Astronomy, Rutgers University,
Piscataway, NJ 08855-0849, USA

School of Natural Sciences, Institute for Advanced Study, Princeton, NJ 08540
Department of Physics, University of California, Berkeley, CA 94720-7300, USA

Lawrence Berkeley National Laboratory, MS 50A-5101, Berkeley, CA 94720, USA
6

Theory Division, CERN, CH-1211, Geneva 23, Switzerlandoferah@physics.rutgers.edu, ssgubser@bohr.harvard.edu,
malda@pauli.harvard.edu, hooguri@lbl.gov, yaron.oz@cern.ch
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 theevidence 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 supersymmetry, and inparticular the relation to QCD. We also discuss some implications for
black hole physics.
(To be published in Physics Reports)

Contents
1 Introduction
1.1 General Introduction and Overview . . . . .
1.2 Large N Gauge Theories as String Theories
1.3 Black p-Branes . . . . . . . . . . . . . . . .
1.3.1 Classical Solutions . . . . . . . . . .
1.3.2 D-Branes . . . . . . . . . . . . . . .1.3.3 Greybody Factors and Black Holes .

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2 Conformal Field Theoriesand AdS Spaces
2.1 Conformal Field Theories . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 The Conformal Group and Algebra . . . . . . . . . . . . . . . .
2.1.2 Primary Fields, Correlation Functions, and Operator Product
Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.3 Superconformal Algebras and Field Theories . . . . . . . . . . .
2.2 Anti-de Sitter Space .. . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Geometry of Anti-de Sitter Space . . . . . . . . . . . . . . . . .
2.2.2 Particles and Fields in Anti-de Sitter Space . . . . . . . . . . .
2.2.3 Supersymmetry in Anti-de Sitter Space . . . . . . . . . . . . . .
2.2.4 Gauged Supergravities and Kaluza-Klein Compactifications . . .
2.2.5 Consistent Truncation of Kaluza-KleinCompactifications . . . .

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3 AdS/CFT Correspondence
3.1 The Correspondence . . . . . . . . . . . . . . .
3.1.1 Brane Probes and Multicenter Solutions
3.1.2 The Field ↔ Operator Correspondence .
3.1.3 Holography . . . . . . . . . . . . . . . .
3.2 Tests of the AdS/CFT Correspondence . . . . .

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3.3

3.4

3.5

3.6

3.2.1 The Spectrum of Chiral Primary Operators . . .
3.2.2 Matching of Correlation Functions and Anomalies
Correlation Functions . . . . . . . . . . . . . . . . . . . .
3.3.1...