Instantons and Conformal Holography

arXiv:hep-th/0606276v316Feb2007AEI-2006-046InstantonsandConformalHolographySebastiandeHaro⋆andAnastasiosC.Petkou†⋆Max-Planck-Institutf¨urGravitationsphysikAlbert-Einstein-Institut14476Golm,[e-mail]†UniversityofCrete,DepartmentofPhysics71003Heraklion,[e-mail]toroftheAdS4/CFT3correspondencewhereaclassofsolutionsinthebulkandontheboundarycanbeexplicitlycompared.Thebulkgravitationaltheorycontainsaconformallycoupledscalarfieldwithaφ4potential,andisholographicallyrelatedtoamasslessscalarwithaφ6interactioninthreedimensions.Weconsiderthescalarsectorofthebulktheoryandmatchbulkandboundaryclassicalsolutionsoftheequationsofmotion.Ofparticularinterestisthematchingofthebulkandtheboundaryinstantonsolutionswhichunderliestherelationshipbetweenbulkandboundaryvacuawithbrokenconformalinvariance.Usingaformofradialquantizationweshowthatquantumstatesinthebulkcorrespondtomultiply-occupiedsingleparticlequantumstatesintheboundarytheory.Thisallowsustoexplicitlyidentifytheboundarycompositeoperatorwhichisdualtothebulkscalar,atthefreetheorylevelaswellasintheinstantonvacuum.Weconcludewithadiscussionofpossibleimplicationsofourresults.1Contents1IntroductionandMotivation32ClassicalHolography52.1TheModel....................................62.2InstantonsandtheVacuumStructureoftheBoundaryCFTs........82.3TheProposalfortheDualboundaryCFT..................103ConformalHolography:FreeCase123.1GeometricSet-upforRadialQuantization..................123.2ClassicalCorrespondence............................133.3RadialQuantizationintheBulk........................163.4MappingoftheHilbertSpaces.........................184InstantonsandConformalHolography214.1FluctuationsAroundtheBulkInstanton...................224.2FluctuationsAroundtheBoundaryInstanton................244.3TheBulk-BoundaryCorrespondence.....................244.4Quantization..................................274.5QuantizationoftheSpecialModes......................295InstantonDecayanddeSitterSpace316DiscussionandOutlook32Acknowledgements34AConventionsandCoordinateSystems35BHypersphericalHarmonicsandHolographyinCylinderCoordinates35CSolutionsintheInteractingCase37DFluctuationEquationinPoincareCoordinates40EProofofformula(97)(byTomKoornwinder)4121IntroductionandMotivationThedeeprelationshipbetweenquantumfieldtheoryandgravityviaholographyhaspro-videdaverystrongdriveinthecurrentresearchinstringtheory.Inparticular,classicalfielddynamicsinAdSspaceshasenjoyedalargeamountofattentionoverthelastfewyearsandanenormousamountofworkonclassicalsupergravitiesinvariousdimensionshasfoundimportantapplicationsinclassicalgravity,StandardModelphenomenology,andCosmology.Progress,however,hasbeenlimitiedbywhatisalsooneofthevirtuesofAdS/CFT[1,2,3],namelythefactthatitisaweak/strongcouplingduality.Inpracticethismeansthattheboundaryfieldtheories,beingstronglycoupled,areusuallynotundercontrol.Thereareveryfewexampleswheretheholographyofcontrolablequantumfieldtheoriescanbestudiedinperturbationtheory.Higher-Spingaugetheories1mightbesuchanexampleasithasbeenarguedthattheyprovidetheholographicdualoffree-fieldtheories.Ontheotherhand,thereareexampleswheretheboundaryfieldtheoryisbasicallyunknown,suchastheholographicdualoftheM2brane.2BothexamplesaboveinvolvetheAdS4/CFT3correspondence,whichinmanyrespectsseemsratherspecial.Thecrucialobservationthathassparkedourinvestigationistheexistenceofclassicalinstantonsolutionstotheequationsofmotionbothforthebulkandfortheboundarytheoriesandthatthereisanapparentrelationshipbetweenthem:bulkinstantonsrestrictedtotheboundaryarethesquareofboundaryinstantons.Remarkably,thistoymodeltouchesuponbothHigher-Spingaugetheories,whichpossessaconformallycoupledscalarwithaquarticself-interaction,aswellasthelow-energyactionoftheM2brane[6].Wehopethatourresultsmighthelpsheddingmorelightontheabovetwoimportantproblems.Oneoftheaimsinthisworkistopresentasimpletoy-modeloftheAdS4/CFT3cor-respondencewhereboththebulkandboundarytheoriesareundercontrol.Thismodel,alreadystudiedin[6],containsaconformallycoupledscalarwithφ4interactioninthebulkofAdS4,whichisholographicallyrelatedtoamasslessscalarwithaφ6interactioninthreedimensions.Despitethedrasticsimplificationoftheboundarytheorytoasinglescalarfield,wefindthatitreproducesthebulkresultsremarkablywellforalargeclassofsolutions.Inparticular,therearebulkinstantonsolutionsthatpreserveAdSboundaryconditionsandcorrespondtotheexpectationvalueofacompositeoperatorofdimension1intheboundarytheory,whichisreproducedbyaclassicalcomputationinthemodelCFT.Sincewewillbeinterestedinthissingleboundaryoperator,werestricttothescalarsectorinthebulk.Wefurthersimplifythemodelbyneglectingtheback-reactionwhich1See[4]forarecentreviewandfurtherreferences.2See[5]forarecentdiscussion.3ispartlyjustifiedinourlinearizedapproximationsinceweexpandaroundexactsolutionsoftheequationsofmotionwhoseimprovedstress-energytensorisidenticallyzero[23].Afullholographicanalysis,forexamplethediscussionoftheboundarytwo-pointfunctions,doesrequiretheinclusionofback-reactionofthefluctuationsinthe