Introduction to the AdS/CFT Correspondence

457 Pages·2015·8.69 MB·English

by  Horaţiu Năstase

IntroductiontotheAdS/CFTCorrespondence
Providing a pedagogical introduction to the rapidly developing field of AdS/CFT cor-
respondence, this is one of the first texts to provide an accessible introduction to all
the necessary concepts needed to engage with the methods, tools, and applications of
AdS/CFT. Without assuming anything beyond an introductory course in quantum field
theory,itbeginsbyguidingthereaderthroughthebasicconceptsoffieldtheoryandgauge
theory,general relativity,supersymmetry, supergravity,stringtheory,andconformal field
theory,beforemovingontogiveaclearandrigorousaccountofAdS/CFTcorrespondence.
Thefinalsectiondiscussesthemorespecializedapplications,includingQCD,quark–gluon
plasma,andcondensedmatter.Thisbookisself-containedandlearner-focused,featuring
numerousexercisesandexamples.Itisessentialreadingforbothstudentsandresearchers
acrossthefieldsofparticle,nuclear,andcondensedmatterphysics.
Horat¸iuNa˘staseisaResearcherattheInstituteforTheoreticalPhysicsattheStateUniversity
ofSãoPaulo,Brazil.Todate,hiscareerhasspannedfourcontinents:asanundergraduate
hestudiedattheUniversityofBucharestandatCopenhagenUniversity.Helatercompleted
hisPh.D.attheStateUniversityofNewYorkatStonyBrook,beforemovingtotheInstitute
for Advanced Study, Princeton, where his collaboration with David Berenstein and Juan
Maldacena defined the pp wave correspondence. He has also held research and teaching
positionsatBrownUniversityandtheTokyoInstituteofTechnology.
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Introduction to the AdS/CFT
Correspondence
˘
HORAT¸IU NASTASE
InstituteforTheoreticalPhysicsattheStateUniversityofSãoPaulo
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Firstpublished2015
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Na˘stase,Hora¸tiu,1972–author.
IntroductiontotheAdS/CFTcorrespondence/Hora¸tiuNa˘stase,Institutefor
TheoreticalPhysicsattheStateUniversityofSãoPaulo.
pages cm
ISBN978-1-107-08585-5
1. Gaugefields(Physics) 2. Supergravity. 3. Supersymmetry. 4. String
models. I. Title.
QC793.3.G38N38 2015
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2015005313
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Tothememoryofmymother,
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Contents
Preface pagexv
Acknowledgments xvi
Introduction xvii
PartI Background 1
1 Elementsofquantumfieldtheoryandgaugetheory 3
1.1 Noteonconventions 3
1.2 TheFeynmanpathintegralandFeynmandiagrams 3
1.3 Smatricesvs.correlationfunctions 8
1.4 Electromagnetism,Yang–Millsfieldsandgaugegroups 10
1.5 Couplingtofermionsandotherfieldsandgaugingasymmetry;the
Noethertheorem 12
1.6 Symmetrycurrentsandthecurrentanomaly 16
Exercises 22
2 Basicsofgeneralrelativity;Anti-deSitterspace 23
2.1 Curvedspacetimeandgeometry;theequivalenceprinciple 23
2.2 Kinematics:Christoffelsymbolsandtensors 27
2.3 Dynamics:Einstein’sequations 31
2.4 Globalstructure:Penrosediagrams 34
2.5 Anti-deSitterspace:definition,metrics,Penrosediagram 35
Exercises 43
3 Basicsofsupersymmetry 44
3.1 Liealgebras;theColeman–Mandulatheorem 44
3.2 Supersymmetry:asymmetrybetweenbosonsandfermions 44
3.3 Spinorsinvariousdimensions 47
3.4 The2-dimensionalWess–Zuminomodel:on-shellsupersymmetry 50
3.5 The2-dimensionalWess–Zuminomodel:off-shellsupersymmetry 53
3.6 The4-dimensionalWess–Zuminomodel 54
3.7 Two-component notation, extended supersymmetry algebra, and
multiplets;R-symmetry 55
3.8 N =1superspaceinfourdimensions 60
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viii Contents
3.9 The N =1SuperYang–Mills(SYM)action 64
3.10 The N =4SuperYang–Mills(SYM)action 67
Exercises 70
4 Basicsofsupergravity 72
4.1 Thevielbeinandspinconnectionformulationofgeneralrelativity 72
4.2 Countingdegreesoffreedomofon-shellandoff-shellfields 75
4.3 Localsupersymmetry:supergravity 78
4.4 N =1on-shellsupergravityinfourdimensions 79
4.5 Genericfeaturesofsupergravitytheories 82
Exercises 85
5 Kaluza–Kleindimensionalreduction 87
5.1 TheKKbackground,KKexpansion,andKKreduction 87
5.2 TheKKdimensionalreduction 89
5.3 Theexpansionofvariousfieldsontori 90
5.4 Consistenttruncationandnonlinearansatz 91
5.5 Example:originalKaluza–Kleinreduction 93
5.6 Generalproperties;symmetries 94
Exercises 96
6 Blackholesandp-branes 98
6.1 TheSchwarzschildsolution:metric,horizon,blackholes 98
6.2 Continuationinsidethehorizon;globalstructure 100
6.3 Solutionswithcharge;solutionsinsideAdSspace 102
6.4 Blackholesinhigherdimensions 105
6.5 Blackholesextendedinpspatialdimensions:“p-branesolutions” 109
Exercises 113
7 Stringtheoryactionsandspectra 114
7.1 Worldlineparticleactionandworldlineconstructionof
Feynmandiagrams 114
7.2 Firstorderparticleaction 117
7.3 Arelativistictensionfulstring:theNambu–Gotoaction 118
7.4 ThePolyakovaction 120
7.5 Equationsofmotion,constraints,andquantizationincovariantgauge 121
7.6 Quantizationinlight-conegauge;thebosonicstringspectrum 126
7.7 Stringsinbackgroundfields 129
7.8 Supersymmetricstrings 131
7.9 Supergravitiesintheα(cid:3) →0limitandthedualityweb 135
7.10 ConstructingS-matrices 139
Exercises 141
8 Elementsofconformalfieldtheory 142
8.1 Conformaltransformationsandtheconformalgroup 142
8.2 ConformalfieldsintwoEuclideandimensions 145
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ix Contents
8.3 Conformalfieldsandcorrelatorsind >2 149
8.4 N =4SuperYang–Millsasaconformalfieldtheory 151
Exercises 152
9 D-branes 154
9.1 DirichletboundaryconditionsandD-branes 154
9.2 D-branefluctuations:fields,action,andtension 155
9.3 Chan–PattonfactorsandquantizationofopenstringsonD-branes 159
9.4 TheactionofmultipleD3-branesandtheN =4SYMlimit 162
Exercises 164
PartII BasicsofAdS/CFTforN = 4SYMvs.AdS5 ×S5 167
10 TheAdS/CFTcorrespondence:motivation,definition,andspectra 169
10.1 D-branes=extremalp-branes 171
10.2 Motivation:near-horizonlimit,Hawkingradiation,andthetwopoints
ofview 172
10.3 Definition:limit,validity,operatormap 174
10.4 Spectraand“experimental”evidence 179
10.5 GlobalAdS/CFT;dimensionalreductiononS3 180
Exercises 184
11 Wittenprescriptionand3-pointcorrelatorcalculations 185
11.1 Wittenprescriptionforcorrelationfunctions 185
11.2 Set-up:the2-pointfunctionofscalarsinxspace 187
11.3 Set-up:2-pointfunctionofgaugefieldsinxspaceandmomentum
space 189
11.4 3-pointfunctions;example:R-currentanomaly 193
11.5 Calculationoffull3-pointfunctionofR-currents 197
Exercises 199
12 HolographyinLorentziansignature:Poincaréandglobalcoordinates 201
12.1 ModeandpropagatorcalculationsinLorentziansignatureforPoincaré
coordinates 202
12.2 PrescriptionforholographyinLorentziansignaturefor
Poincarécoordinates 203
12.3 ModeandpropagatorcalculationsinglobalLorentzian
coordinates 204
12.4 HolographyinglobalLorentziancoordinates:interpretation 207
Exercises 208
13 SolitonicobjectsinAdS/CFT 209
13.1 Instantonsvs.D-instantons 209
13.2 BaryonsingaugetheoriesandviaAdS/CFT 212
13.3 TheD3–D5systemandthefuzzy2-sphereviaAdS/CFT 214
Exercises 219
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