Gauge/Gravity Duality: Foundations and Applications

548 Pages·2015·6.333 MB·English

by  Martin Ammon, Johanna Erdmenger

Gauge/GravityDuality
FoundationsandApplications
Gauge/gravitydualitycreatesnewlinksbetweenquantumtheoryandgravity.Ithasledto
newconceptsinmathematicsandphysics,andprovidesnewtoolsforsolvingproblemsin
many areas of theoretical physics. This book is the first comprehensive textbook on this
important topic, enabling graduate students and researchers in string theory and particle,
nuclearandcondensedmatterphysicstobecomeacquaintedwiththesubject.
Focusing on the fundamental aspects as well as on applications, this textbook guides
readers through a thorough explanation of the central concepts of gauge/gravity duality.
For the AdS/CFT correspondence, it explains in detail how string theory provides the
conjectured map. Generalisations to less symmetric cases of gauge/gravity duality and
their applications are then presented, in particular to finite temperature and density,
hydrodynamics, QCD-like theories, the quark–gluon plasma and condensed matter sys-
tems.Thetextbookfeaturesalargenumberofexercises,withsolutionsavailableonlineat
www.cambridge.org/9781107010345.
JohannaErdmenger is a Research Group Leader at the Max Planck Institute for Physics
(Werner Heisenberg Institute), Munich, Germany, and Honorary Professor at Ludwig
Maximilian University, Munich. She is one of the pioneers of applying gauge/gravity
dualitytoelementaryparticle,nuclearandcondensedmatterphysics.
MartinAmmonisaJuniorProfessoratFriedrichSchillerUniversity,Jena,Germany,leading
a research group on gauge/gravity duality. He was awarded the prestigious Otto Hahn
Medal of the Max Planck Society for his Ph.D. thesis on applying gauge/gravity duality
tocondensedmatterphysics.
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Gauge/Gravity Duality
Foundations and Applications
MARTIN AMMON
FriedrichSchillerUniversity,Jena
JOHANNA ERDMENGER
MaxPlanckInstituteforPhysics,Munich
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Cambridge Books Online © Cambridge University Press, 2015
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©M.AmmonandJ.Erdmenger2015
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Firstpublished2015
PrintedintheUnitedKingdombyTJInternationalLtd.PadstowCornwall
AcataloguerecordforthispublicationisavailablefromtheBritishLibrary
Ammon,Martin,1981-
Gauge-gravityduality:foundationsandapplications/MartinAmmon,
FriedrichSchillerUniversity,Jena,JohannaErdmenger,
MaxPlanckInstituteforPhysics,Munich.
pages cm.
Includesbibliographicalreferencesandindex.
ISBN978-1-107-01034-5(hardback:alk.paper)
1. Gaugefields(Physics). 2. Gravity. 3. Holography.
4. Mathematicalphysics. I. Erdmenger,Johanna. II. Title.
QC793.3.G38A562015
530.14(cid:2)35–dc23 2015003328
ISBN978-1-107-01034-5Hardback
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Contents
Preface pageix
Acknowledgements xiii
PartI Prerequisites 1
1 Elementsoffieldtheory 3
1.1 Classicalscalarfieldtheory 3
1.2 Symmetriesandconservedcurrents 6
1.3 Quantisation 9
1.4 Wickrotationandstatisticalmechanics 18
1.5 Regularisationandrenormalisation 19
1.6 Diracfermions 28
1.7 Gaugetheory 30
1.8 Symmetries,Wardidentitiesandanomalies 45
1.9 Furtherreading 48
References 48
2 Elementsofgravity 50
2.1 Differentialgeometry 50
2.2 Einstein’sfieldequations 65
2.3 Maximallysymmetricspacetimes 66
2.4 Blackholes 76
2.5 Energyconditions 88
2.6 Furtherreading 89
References 89
3 Symmetriesinquantumfieldtheory 91
3.1 LorentzandPoincarésymmetry 91
3.2 Conformalsymmetry 102
3.3 Supersymmetry 118
3.4 Superconformalsymmetry 139
3.5 Furtherreading 143
References 143
v
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vi Contents
4 Introductiontosuperstringtheory 145
4.1 Bosonicstringtheory 145
4.2 Superstringtheory 156
4.3 Webofdualities 164
4.4 D-branesandothernon-perturbativeobjects 168
4.5 Furtherreading 174
References 174
PartII Gauge/GravityDuality 177
5 TheAdS/CFTcorrespondence 179
5.1 TheAdS/CFTcorrespondence:afirstglance 180
5.2 D3-branesandtheirtwofaces 182
5.3 Field-operatormap 189
5.4 Correlationfunctions 196
5.5 Holographicrenormalisation 204
5.6 WilsonloopsinN =4SuperYang–Millstheory 211
5.7 Furtherreading 216
References 216
6 TestsoftheAdS/CFTcorrespondence 219
6.1 Correlationfunctionof1/2BPSoperators 220
6.2 Four-pointfunctions 229
6.3 Theconformalanomaly 233
6.4 Furtherreading 237
References 238
7 Integrabilityandscatteringamplitudes 240
7.1 Integrablestructuresonthegaugetheoryside 241
7.2 Integrabilityonthegravity(stringtheory)side 248
7.3 BMNlimitandclassicalstringconfigurations 254
7.4 Dualsuperconformalsymmetry 258
7.5 Furtherreading 271
References 271
8 FurtherexamplesoftheAdS/CFTcorrespondence 273
8.1 D3-branesatsingularities 273
8.2 M2-branes:AdS /CFT 278
4 3
8.3 Gravitydualsofconformalfieldtheories:furtherexamples 287
8.4 Towardsnon-conformalfieldtheories 289
8.5 Furtherreading 294
References 294
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vii Contents
9 Holographicrenormalisationgroupflows 296
9.1 Renormalisationgroupflowsinquantumfieldtheory 296
9.2 Holographicrenormalisationgroupflows 302
9.3 ∗SupersymmetricflowswithinIIBSupergravityinD=10 308
9.4 Furtherreading 323
References 324
10 DualitywithD-branesinsupergravity 326
10.1 Branesasflavourdegreesoffreedom 326
10.2 AdS/CFTcorrespondencewithprobebranes 331
10.3 D7-branefluctuationsandmesonsinN =2theory 335
∗
10.4 D3/D5-branesystem 340
10.5 Furtherreading 342
References 343
11 Finitetemperatureanddensity 344
11.1 Finitetemperaturefieldtheory 344
11.2 Gravitydualthermodynamics 352
11.3 Finitedensityandchemicalpotential 362
11.4 Furtherreading 365
References 365
PartIII Applications 367
12 Linearresponseandhydrodynamics 369
12.1 Linearresponse 370
12.2 Hydrodynamics 380
12.3 Transportcoefficientsfromlinearresponse 384
12.4 Fluid/gravitycorrespondence 389
12.5 Furtherreading 396
References 396
13 QCDandholography:confinementandchiralsymmetrybreaking 399
13.1 ReviewofQCD 399
13.2 Gauge/gravitydualitydescriptionofconfinement 405
13.3 ChiralsymmetrybreakingfromD7-braneprobes 413
13.4 Non-Abelianchiralsymmetries:theSakai–Sugimotomodel 421
13.5 AdS/QCDcorrespondence 426
13.6 Furtherreading 430
References 431
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viii Contents
14 QCDandholography:finitetemperatureanddensity 435
14.1 QCDatfinitetemperatureanddensity 435
14.2 Gauge/gravityapproachtothequark–gluonplasma 438
14.3 Holographicflavouratfinitetemperatureanddensity 445
14.4 Sakai–Sugimotomodelatfinitetemperature 454
14.5 Holographicpredictionsforthequark–gluonplasma 455
14.6 Furtherreading 456
References 457
15 Stronglycoupledcondensedmattersystems 460
15.1 Quantumphasetransitions 461
15.2 Chargesandfinitedensity 463
15.3 Holographicsuperfluidsandsuperconductors 477
15.4 Fermions 485
15.5 Towardsnon-relativisticsystemsandhyperscalingviolation 489
15.6 Entanglemententropy 496
15.7 Furtherreading 501
References 502
AppendixA Grassmannnumbers 505
AppendixB Liealgebrasandsuperalgebras 508
AppendixC Conventions 526
Index 527
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Preface
Gauge/gravity duality is a major new development within theoretical physics. It brings
together string theory, quantum field theory and general relativity, and has applications
toelementaryparticle,nuclearandcondensedmatterphysics.Gauge/gravitydualityisof
fundamentalimportancesinceitprovidesnewlinksbetweenquantumtheoryandgravity
which are based on string theory. It has led to both new insights about the structure of
string theory and quantum gravity, and new methods and applications in many areas of
physics. In a particular case, the duality maps strongly coupled quantum field theories,
whicharegenericallyhardtodescribe,tomoretractableclassicalgravitytheories.Inthis
way, it provides a wealth of applications to strongly coupled systems. Examples include
theories similar to low-energy quantum chromodynamics (QCD), the theory of strong
interactions in elementary particle physics, and models for quantum phase transitions
relevantincondensedmattersystems.
Gauge/gravity duality realises the holographic principle and is therefore referred to
as holography. The holographic principle states that the entire information content of a
quantum gravity theory in a given volume can be encoded in an effective theory at the
boundarysurfaceofthisvolume.Thetheorydescribingtheboundarydegreesoffreedom
thus encodes all information about the bulk degrees of freedom and their dynamics, and
viceversa.Theholographicprincipleisofverygeneralnatureandisexpectedtoberealised
in many examples. In many of these cases, however, the precise form of the boundary
theoryisunknown,sothatitcannotbeusedtodescribethebulkdynamics.
Stringtheory,however,givesrisetoapreciserealisationoftheholographicprinciple,in
whichbothbulkandboundarytheoryareknown:thisisgauge/gravityduality.Inthiscase,
aquantumfieldtheoryattheboundary,whichinvolvesagaugesymmetry,isconjectured
tobeequivalenttoatheoryinvolvinggravityinthebulk.Moreover,stringtheoryprovides
many examples of dualities: a physical theory may generically have different equivalent
formulations which are referred to as being dual to each other. Two formulations are
equivalentifthereisaone-to-onemapbetweenthestatesineachofthem,andthedynamics
arethesame.Dualityisparticularlyusefulifphysicalprocessesarehardtocalculateinone
formulation, but easy to obtain in another. An example of a duality of this type is a map
betweentwoequivalentformulationsindifferentcouplingconstantregimes.Forinstance,
in a particular limit gauge/gravity duality maps a strongly coupled gauge theory, which
generallyishardtodescribe,toaweaklycoupledgravitytheory,inwhichitismuchmore
straightforwardtoperformexplicitcalculations.
The most prominent and best understood example of gauge/gravity duality is the
AdS/CFT correspondence, the celebrated proposal by Maldacena. The AdS/CFT corre-
spondence is characterised by a very high degree of symmetry. Here, ‘AdS’ stands for
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