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

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

Department of Physics and Astronomy, Rutgers University,

Piscataway, NJ 08855-0849, USA

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

Theory Division, CERN, CH-1211, Geneva 23, Switzerland

oferah@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 eld theories and string/M theory,

focusing on the relation between compactications of string/M theory on Anti-de Sitter

spaces and conformal eld 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 eld

theory is approximated by classical or semiclassical gravity. We focus on the case of

theN= 4 supersymmetric gauge theory in four dimensions, but we discuss also eld

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)

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 Compactications . . . 48

2.2.5 Consistent Truncation of Kaluza-Klein Compactications . . . . 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

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

4 More on the Correspondence 111

4.1 Other AdS 5 Backgrounds . . . . . . . . . . . . . . . . . . . . . . . . . . 111

4.1.1 Orbifolds of AdS 5 × S 5 . . . . . . . . . . . . . . . . . . . . . . . 113

4.1.2 Orientifolds of AdS 5 × S 5 . . . . . . . . . . . . . . . . . . . . . 118

4.1.3 Conifold theories . . . . . . . . . . . . . . . . . . . . . . . . . . 121

4.2 D-Branes in AdS, Baryons and Instantons . . . . . . . . . . . . . . . . 129

4.3 Deformations of the Conformal Field Theory . . . . . . . . . . . . . . . 134

4.3.1 Deformations in the AdS/CFT Correspondence . . . . . . . . . 135

4.3.2 A c-theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

4.3.3 Deformations of theN= 4 SU ( N ) SYM Theory . . . . . . . . 138

4.3.4 Deformations of String Theory on AdS 5 × S 5 . . . . . . . . . . . 144

5 AdS 3 150

5.1 The Virasoro Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

5.2 The BTZ Black Hole . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

5.3 Type IIB String Theory on AdS 3 × S 3 × M 4 . . . . . . . . . . . . . . . 155

5.3.1 The Conformal Field Theory . . . . . . . . . . . . . . . . . . . . 155

5.3.2 Black Holes Revisited . . . . . . . . . . . . . . . . . . . . . . . . 159

5.3.3 Matching of Chiral-Chiral Primaries . . . . . . . . . . . . . . . 162

5.3.4 Calculation of the Elliptic Genus in Supergravity . . . . . . . . 167

5.4 Other AdS 3 Compactications . . . . . . . . . . . . . . . . . . . . . . . 168

5.5 Pure Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

5.6 Greybody Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

5.7 Black Holes in Five Dimensions . . . . . . . . . . . . . . . . . . . . . . 178

6 Other AdS Spaces and Non-Conformal Theories 180

6.1 Other Branes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

6.1.1 M5 Branes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

6.1.2 M2 Branes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

6.1.3 Dp Branes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

6.1.4 NS5 Branes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

6.2 QCD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

6.2.1 QCD 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

6.2.2 QCD 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204

6.2.3 Other Directions . . . . . . . . . . . . . . . . . . . . . . . . . . 218

7 Summary and Discussion

223

Chapter 1

Introduction

1.1 General Introduction and Overview

The microscopic description of nature as presently understood and veried by experi-

ment involves quantum eld theories. All particles are excitations of some eld. These

particles are pointlike and they interact locally with other particles. Even though

quantum eld theories describe nature at the distance scales we observe, there are

strong indications that new elements will be involved at very short distances (or very

high energies), distances of the order of the Planck scale. The reason is that at those

distances (or energies) quantum gravity eects become important. It has not been

possible to quantize gravity following the usual perturbative methods. Nevertheless,

one can incorporate quantum gravity in a consistent quantum theory by giving up the

notion that particles are pointlike and assuming that the fundamental objects in the

theory are strings, namely one-dimensional extended objects [1, 2]. These strings can

oscillate, and there is a spectrum of energies, or masses, for these oscillating strings.

The oscillating strings look like localized, particle-like excitations to a low energy ob-

server. So, a single oscillating string can eectively give rise to many types of particles,

depending on its state of oscillation. All string theories include a particle with zero

mass and spin two. Strings can interact by splitting and joining interactions. The only

consistent interaction for massless spin two particles is that of gravity. Therefore, any

string theory will contain gravity. The structure of string theory is highly constrained.

String theories do not make sense in an arbitrary number of dimensions or on any

arbitrary geometry. Flat space string theory exists (at least in perturbation theory)

only in ten dimensions. Actually, 10-dimensional string theory is described by a string

which also has fermionic excitations and gives rise to a supersymmetric theory. 1 String

theory is then a candidate for a quantum theory of gravity. One can get down to four

1 One could consider a string with no fermionic excitations, the so called “bosonic” string. It lives

in 26 dimensions and contains tachyons, signaling an instability of the theory.

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

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