AdS/CFT Duality User Guide

299 Pages·2015·5.086 MB·English

by  Makoto Natsuume (auth.)

Lecture Notes in Physics 903
Makoto Natsuume
AdS/CFT
Duality
User Guide
Lecture Notes in Physics
Volume 903
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Makoto Natsuume
AdS/CFT Duality User
Guide
123
MakotoNatsuume
Theory Center
Instituteof Particle& Nuclear Studies
HighEnergy Accelerator Research
Organization (KEK)
Tsukuba
Japan
TYOGENRIRON NOOYObyMakotoNatsuume.
OriginalJapanese language edition publishedbySaiensu-sha Co.,Ltd.
1-3-25,Sendagaya, Shibuya-ku,Tokyo151-0051, Japan
Copyright©2012, Saiensu-sha Co.,Ltd.AllRightsreserved.
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Lecture Notesin Physics
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Preface
This book describes “real-world” applications of the AdS/CFT duality for begin-
ning graduate students in particle physics and for researchers in the other fields.
The AdS/CFT duality is a powerful tool for analyzing strongly coupled gauge
theories using classical gravitational theories. The duality originated from string
theory, so it has been actively investigated in particle physics. In recent years,
however,thedualityhasbeendiscussedbeyondtheoreticalparticlephysics.Infact,
theoriginalAdS/CFTpaperbyMaldacenahasbeencitedinallphysicsarXivs.This
is because the duality is becoming a powerful tool to analyze the “real world.”
For example, it turns out that one prediction of AdS/CFT is indeed close to the
experimental results of the real quark–gluon plasma. Since then, the duality has
been applied to various fields of physics; examples are QCD, nuclear physics,
nonequilibrium physics, and condensed matter physics.
In order to carry out such researches, one has to know many materials such as
string theory, general relativity, nuclear physics, nonequilibrium physics, and
condensed matter physics. The aim of this book is to provide these background
materials as well as some key applications of the AdS/CFT duality in a single
volume. The emphasis throughout the book is on a pedagogical and intuitive
approach focusing on the underlying physical concepts. Yet it also includes step-
by-step computations for important results which are useful for beginners. Most
of them are contained in the appendices.
Following conventions of many textbooks, I often do not refer to original
researchpapersandreferonlytotheothertextbooksandreviewsthatmaybemore
useful to readers. Also, the choice of references reflects my knowledge, and I
apologize in advance for possible omissions.
Initially, this project was begun for a book that was published in Japanese
(Saiensu-sha Co., Ltd, 2012), and this is the “translated” one. But I used this
opportunitytoimprovemanyexplanationsandtoaddmorematerialstotheJapanese
edition. So, this book isthe“second edition” in this sense.
Iwouldlike tothank many people who helped mewiththis book.This book is
based on review talks at various conferences and on courses I taught at various
v
vi Preface
graduateschools(TohokuUniversity,OchanomizuUniversity,GraduateUniversity
of Advanced Studies, and Rikkyo University). I thank the organizers and the par-
ticipantsoftheconferencesandthecourses.IalsoliketothankElenaCáceres,Koji
Hashimoto, Tetsuo Hatsuda, Gary Horowitz, and Joe Polchinski for encouraging
me to write this English edition. I also thank Tetsufumi Hirano, Akihiro Ishibashi,
andTakeshiMoritaforusefulcommentsanddiscussion.Iespeciallywouldliketo
thankTakashiOkamurawhoclarifiedmyunderstandingofthesubjectsinthisbook
through collaboration for many years. He also gave many suggestions for
improvement.IthankmyeditorHisakoNikoatSpringerJapanandthestaffofthe
LectureNotesinPhysics.Ofcourse,theresponsibilityforanyremainingmistakeis
solelymine.Iwillbehappytoreceivecommentsonthisbook.Pleasesendthemto
[email protected].
An updated list of corrections will be posted on my website. The tentative
address is http://research.kek.jp/people/natsuume/ads-real-world.html. Even if the
address changes in future, you can probably easily search the website because my
family name is rather rare.
I hope that this book will help readers to explore new applications of the AdS/
CFT duality.
Contents
1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Overview of AdS/CFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 AdS/Real-World. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.4 Notation and Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.5 Some Useful Textbooks. . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2 General Relativity and Black Holes. . . . . . . . . . . . . . . . . . . . . . . 9
2.1 Particle Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2 Einstein Equation and Schwarzschild Metric. . . . . . . . . . . . . . 14
2.3 Physics of the Schwarzschild Black Hole. . . . . . . . . . . . . . . . 15
2.3.1 Gravitational Redshift . . . . . . . . . . . . . . . . . . . . . . . 15
2.3.2 Particle Motion. . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.4 Kruskal Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
Appendix: Review of General Relativity. . . . . . . . . . . . . . . . . . . . . 21
3 Black Holes and Thermodynamics. . . . . . . . . . . . . . . . . . . . . . . . 25
3.1 Black Holes and Thermodynamics . . . . . . . . . . . . . . . . . . . . 25
3.1.1 Zeroth Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.1.2 Surface Gravity (cid:1) . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.1.3 First Law. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.2 From Analogy to Real Thermodynamics . . . . . . . . . . . . . . . . 29
3.2.1 Hawking Radiation. . . . . . . . . . . . . . . . . . . . . . . . . 29
3.2.2 Hawking Temperature and Euclidean Formalism. . . . . 31
3.2.3 On the Origin of Black Hole Entropy (cid:1). . . . . . . . . . . 33
3.3 Other Black Holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.3.1 Higher-Dimensional Schwarzschild Black Holes. . . . . 35
3.3.2 Black Branes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.3.3 Charged Black Holes. . . . . . . . . . . . . . . . . . . . . . . . 38
vii
viii Contents
3.4 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
Appendix: Black Holes and Thermodynamic Relations (cid:1). . . . . . . . . . 41
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4 Strong Interaction and Gauge Theories. . . . . . . . . . . . . . . . . . . . 43
4.1 Strong Interaction and QCD. . . . . . . . . . . . . . . . . . . . . . . . . 43
4.1.1 Overview of QCD . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.1.2 Phase Structure of QCD . . . . . . . . . . . . . . . . . . . . . 44
4.1.3 Heavy-Ion Experiments. . . . . . . . . . . . . . . . . . . . . . 45
4.1.4 “Unexpected Connection Between String
Theory and RHIC Collisions”. . . . . . . . . . . . . . . . . . 46
4.2 Large-Nc Gauge Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.3 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5 The Road to AdS/CFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
5.1 String Theory: Prehistory. . . . . . . . . . . . . . . . . . . . . . . . . . . 55
5.2 String Theory as the Unified Theory. . . . . . . . . . . . . . . . . . . 58
5.2.1 String Oscillations and Elementary Particles. . . . . . . . 58
5.2.2 D-brane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
5.2.3 Why Open and Closed Strings? . . . . . . . . . . . . . . . . 61
5.2.4 String Interactions. . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.2.5 Supergravity: Classical Gravity Approximation
of String Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.3 Reexamine String Theory as the Theory
of Strong Interaction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5.3.1 Comparison of Partition Functions . . . . . . . . . . . . . . 66
5.3.2 Scale Invariance and Its Consequences . . . . . . . . . . . 68
5.3.3 AdS/CFT, Finally. . . . . . . . . . . . . . . . . . . . . . . . . . 72
5.4 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
Appendix 1: Scale, Conformal, and Weyl Invariance (cid:1) . . . . . . . . . . . 75
Appendix 2: D-brane and AdS/CFT (cid:1). . . . . . . . . . . . . . . . . . . . . . . 77
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
6 The AdS Spacetime. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
6.1 Spacetimes with Constant Curvature . . . . . . . . . . . . . . . . . . . 85
6.1.1 Spaces with Constant Curvature . . . . . . . . . . . . . . . . 85
6.1.2 Spacetimes with Constant Curvature . . . . . . . . . . . . . 87
6.1.3 Relation with Constant Curvature Spaces. . . . . . . . . . 89
6.1.4 Various Coordinate Systems of AdS Spacetime . . . . . 89
6.1.5 Higher-Dimensional Cases. . . . . . . . . . . . . . . . . . . . 91
6.2 Particle Motion in AdS Spacetime (cid:1) . . . . . . . . . . . . . . . . . . . 94
6.3 Remarks on AdS/CFT Interpretations . . . . . . . . . . . . . . . . . . 98
6.4 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
Contents ix
7 AdS/CFT—Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
7.1 The AdS Black Hole. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
7.2 Thermodynamic Quantities of AdS Black Hole. . . . . . . . . . . . 102
7.2.1 Thermodynamic Quantities. . . . . . . . . . . . . . . . . . . . 102
7.2.2 Free Gas Computation. . . . . . . . . . . . . . . . . . . . . . . 106
7.3 The AdS Black Hole with Spherical Horizon. . . . . . . . . . . . . 108
7.4 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
Appendix: AdS Black Hole Partition Function (cid:1) . . . . . . . . . . . . . . . 110
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
8 AdS/CFT—Adding Probes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
8.1 Basics of Wilson Loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
8.2 Wilson Loops in AdS/CFT: Intuitive Approach . . . . . . . . . . . 120
8.3 String Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
8.4 Wilson Loops in AdS/CFT: Actual Computation . . . . . . . . . . 129
8.5 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
Appendix: A Simple Example of the Confining Phase . . . . . . . . . . . 132
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
9 Basics of Nonequilibrium Physics . . . . . . . . . . . . . . . . . . . . . . . . 135
9.1 Linear Response Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
9.1.1 Ensemble Average and Density Matrix . . . . . . . . . . . 135
9.1.2 Linear Response Theory . . . . . . . . . . . . . . . . . . . . . 137
9.1.3 Transport Coefficient: An Example. . . . . . . . . . . . . . 140
9.2 Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
9.3 Hydrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
9.3.1 Overview of Hydrodynamics . . . . . . . . . . . . . . . . . . 144
9.3.2 Example: Diffusion Problem . . . . . . . . . . . . . . . . . . 145
9.3.3 Perfect Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
9.3.4 Viscous Fluid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
9.3.5 When a Current Exists (cid:1) . . . . . . . . . . . . . . . . . . . . . 155
9.3.6 Kubo Formula for Viscosity. . . . . . . . . . . . . . . . . . . 157
9.3.7 Linearized Hydrodynamic Equations
and Their Poles . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
9.4 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
10 AdS/CFT—Non-equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
10.1 GKP-Witten Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
10.2 Example: Scalar Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
10.3 Other Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
10.3.1 Maxwell Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
10.3.2 Massive Scalar Field. . . . . . . . . . . . . . . . . . . . . . . . 174
10.3.3 Gravitational Field . . . . . . . . . . . . . . . . . . . . . . . . . 176