From Weak to Strong Coupling in N=4 SYM Theory從弱到N =4 SYM理論強(qiáng)耦合(-48)

FromWeaktoStrongCouplinginN=4SYMTheory

IgorKlebanovDepartmentofPhysicsTalkatRutgersUniversityMarch6,2007LargeNGaugeTheoriesConnectionofgaugetheorywithstringtheoryismostapparentin`tHooft’sgeneralizationfrom3colors(SU(3)gaugegroup)toNcolors(SU(N)gaugegroup).MakeNlarge,whilekeepingthe`tHooftcouplingfixed:Theprobabilityofsnappingafluxtubebyquark-antiquarkcreation(mesondecay)is1/N.Thestringcouplingis1/N.InthelargeNlimitonlyplanardiagramscontribute,but4-dgaugetheoryisstillverydifficult.BreakingtheIceDirichletbranes(Polchinski)ledstringtheorybacktogaugetheoryinthemid-90’s.AstackofNDirichlet3-branesrealizesN=4supersymmetricSU(N)gaugetheoryin4dimensions.Italsocreatesacurvedbackgroundof10-dtheoryofclosedsuperstrings(artworkby)

whichforsmallrapproachesSuccessfulmatchingofgravitonabsorptionbyD3-branes,relatedto2-pointfunctionofstress-energytensorintheSYMtheory,withagravitycalculationinthe3-branemetric(IK;Gubser,IK,Tseytlin)wasaprecursoroftheAdS/CFTcorrespondence.Super-ConformalInvarianceIntheN=4SYMtheorythereare6scalarfields(itisusefultocombinetheminto3complexscalars:Z,W,V)and4gluinosinteractingwiththegluons.AllthefieldsareintheadjointrepresentationoftheSU(N)gaugegroup.TheAsymptoticFreedomiscanceledbytheextrafields;thebetafunctionisexactlyzeroforanycomplexcoupling.Thetheoryisinvariantunderscaletransformationsxm->axm.Itisalsoinvariantunderspace-timeinversions.Thefullsuper-conformalgroupisSU(2,2|4).Itisinterestingtostudyobservablesoftheplanartheoryasfunctionsofthe`tHooftcouplingl.EntropyofthermalsupersymmetricSU(N)theoryThermalCFTisdescribedbyanear-extremal3-branebackgroundwhosenear-horizonformisablackholeinAdS5TheCFTtemperatureisidentifiedwiththeHawkingTofthehorizonlocatedatzhAnyeventhorizoncontainsBekenstein-HawkingentropyAbriefcalculationgivestheentropydensityGubser,IK,PeetThisisinterpretedasthestrongcouplinglimitofTheweak`tHooftcouplingbehavioroftheinterpolatingfunctionisdeterminedbyFeynmangraphcalculationsintheN=4SYMtheoryWededucefromAdS/CFTdualitythatTheentropydensityismultipliedonlyby?asthecouplingchangesfromzerotoinfinity.Gubser,IK,TseytlinCorrectionstotheinterpolatingfunctionatstrongcouplingcomefromthehigher-derivativetermsinthetypeIIBeffectiveaction:Gubser,IK,TseytlinTheinterpolatingfunctionisusuallyassumedtohaveasmoothmonotonicform,butsofarwedonotknowitsformattheintermediatecoupling.Asimilarreductionofentropybystrong-couplingeffectsisobservedinlatticenon-supersymmetricgaugetheoriesforN=3:thearrowsshowfreefieldvalues.Karsch(hep-lat/0106019).N-dependenceinthepuregluetheoryenterslargelythroughtheoverallnormalization.

BringoltzandTeper(hep-lat/0506034)

TheAdS/CFTduality

Maldacena;Gubser,IK,Polyakov;WittenRelatesconformalgaugetheoryin4dimensionstostringtheoryon5-dAnti-deSitterspacetimesa5-dcompactspace.FortheN=4SYMtheorythiscompactspaceisa5-dsphere.Whenagaugetheoryisstronglycoupled,theradiusofcurvatureofthedualAdS5andofthe5-dcompactspacebecomeslarge:Stringtheoryinsuchaweaklycurvedbackgroundcanbestudiedintheeffective(super)-gravityapproximation,whichallowsforahostofexplicitcalculations.CorrectionstoitproceedinpowersofFeynmangraphsinsteaddevelopaweakcouplingexpansioninpowersofl.Atweakcouplingthedualstringtheorybecomesdifficult.GaugeinvariantoperatorsintheCFT4areinone-to-onecorrespondencewithfields(orextendedobjects)inAdS5Operatordimensionisdeterminedbythemassofthedualfield;e.g.forscalaroperatorsGKPWTheBPSprotectedoperatorsaredualtoSUGRAfieldsofm~1/L.Theirdimensionsareindependentofl.Theunprotectedoperators(Konishioperatoristhesimplest)aredualtomassivestringstates.AdS/CFTpredictsthatatstrongcouplingtheirdimensionsgrowasl1/4.Untilrecently,theonly(probably)exactnon-BPSinterpolatingfunctionwastheexpectationvalueofacircularWilsonloopobtainedfromrainbowgraphsummationErickson,Semenoff,ZaremboHeretheradiusofconvergenceoflnWisrelatedtothezerooftheBesselfunctionJ1.Thereisaninfinitenumberofbranchpointsatnegativel.ThestrongcouplinglimitisinagreementwiththeAdScalculation.ExactIntegrabilityPerturbativecalculationsofanomalousdimensionsaremappedtointegrablespinchains,suggestingexactintegrabilityoftheN=4SYMtheory.Minahan,Zarembo;Beisert,StaudacherForexample,forthe`SU(2)sector’operators

Tr(ZZZWZW…ZW),whereZandWaretwocomplexscalars,theHeisenbergspinchainemergesat1loop.HigherloopscorrecttheHamiltonianbutseemtopreserveitsintegrability.ThismeshesnicelywithearlierfindingsofintegrabilityincertainsubsectorsofQCD.Lipatov;Faddeev,Korchemsky;Braun,Derkachov,ManashovThedualstringtheoryapproachindicatesthatintheSYMtheorytheexactintegrabilityispresentatverystrongcoupling(Bena,Polchinski,Roiban).Henceitislikelytoexistforallvaluesofthecoupling.SpinningStringsvs.HighlyChargedOperatorsVibratingclosedstringswithlargeangularmomentumonthe5-spherearedualtoSYMoperatorswithlargeR-charge(thenumberoffieldsZ)Berenstein,Maldacena,NastaseGenerally,semi-classicalspinningstringsaredualtolongoperators,e.g.thedualofahigh-spinoperatorisafoldedstringspinningaroundthecenterofAdS5.Gubser,IK,PolyakovThestructureofdimensionsofhigh-spinoperatorsisTheuniversalfunctionf(g)isindependentofthetwist;itisuniversalintheplanarlimit.ItalsoentersthecuspanomalyofWilsonloopsinMinkowskispace.ThiscanbecalculatedusingAdS/CFT.KruczenskiAtstrongcoupling,theAdS/CFTcorrespondspredictsviathespinningstringenergycalculationsGubser,IK,Polyakov;Frolov,TseytlinAtweakcouplingtheexpansionoftheuniversalfunctionf(g)upto3loopsisThecoefficientsinf(g)arerelatedtothecorrespondingcoefficientsinQCDthroughselectingateachorderthetermwiththehighesttranscendentality.Kotikov,Lipatov,Onishchenko,VelizhaninRecently,greatprogresshasbeenachievedonfindingf(g)at4loopsandbeyond!Usingtheofthespinchainsymmetries,theBetheansatzequationswererestrictedtotheformStaudacher,Beisert

ThemagnondispersionrelationisThecomplexx-variablesencodethemomentumpandenergyC:Ofparticularimportanceisthecrossingsymmetry(Janik)TheDressingPhaseThe`dressingphase’inthemagnonS-matrixappearsfirstat4-looporderinperturbationtheory:whereqr(u)areeigenvaluesofthemagnoncharges.Itwasdeterminedfirstinthelargegexpansion(Beisert,Hernandez,Lopez),andthenviaappropriateresummationinthesmallgexpansion(Beisert,Eden,Staudacher).f(g)isdeterminedthroughsolvinganintegralequation,whichcorrectsanearliersimilarequationderivedbyEdenandStaudacher(whoassumedthatthephasevanishes)TheBESkernelisThefirsttermistheESkernelwhilethesecondoneisduetothedressingphaseinthemagnonS-matrixPerturbativeorder-by-ordersolutionoftheBESequationgivesthe4-loopterminf(g)

(itdiffersbyrelativesignfromtheESpredictionwhichdidnotincludethe`dressingphase’)Remarkably,anindependent4-loopcalculationbyBern,Dixon,Czakon,KosowerandSmirnovyieldedanumericalvaluethatpromptedthemtoconjectureexactlythesameanalyticalresult.Thishasledthetwogroupstothesameconjectureforthecompletestructureoftheperturbativeexpansionoff(g):itistheoneyieldedbytheBESintegralequation.ThisapproachyieldsanalyticpredictionsforallplanarperturbativecoefficientsSofarthe4-loopanswerisonlyknownnumerically.Recently,anewmethodyieldedimprovednumericalprecisionandagreeswiththeanalyticalpredictiontoaround0.001%.

Cachazo,Spradlin,VolovichThealternationoftheseriesandthegeometricbehaviorofthecoefficientsremoveallsingularitiesfromtherealaxis,allowingsmoothextrapolationtoinfinitecoupling.Theradiusofconvergenceis?.Theclosestsingularitiesaresquare-rootbranchpointsatTheanalyticstructureoff(g)isnotcompletelyclear,butthereappeartobeaninfinitenumberofbranchcutsalongtheimaginaryaxis.Thereisanessentialsingularityatinfinity.ThesequalitativefeaturesalsoappearforthecircularWilsonloop.Tosolvetheequationatfinitecoupling,weuseabasisoflinearlyindependentfunctionsDeterminationofistantamounttoinvertinganinfinitematrix.Truncationtofinitematricesconvergesveryrapidly.Benna,Benvenuti,IK,ScardicchioThebluelinereferstotheBESequation,redlinetotheES,greenlinetotheequationwherethedressingkernelisdividedby2.ThefirsttwotermsoftheBESlargegasymptoticsareinverypreciseagreementwiththeAdS/CFTspinningstringpredictions.Recently,theexactanalyticsolutionwasobtainedfortheleadingorderBESequation.Alday,Arutyunov,Benna,Eden,IK;Kostov,Serban,VolinExpandingatstrongcoupling,ThesolutionisSinces1=1/2,thisprovesthatf(g)=4g+O(1)

QuarkAnti-QuarkPotentialThez-directionisdualtotheenergyscaleofthegaugetheory:smallzistheUV;largezistheIR.Inapleasantsurprise,becauseofthe5-thdimensionz,thestringpictureapplieseventotheoriesthatareconformal(notconfining!).Thequarkandanti-quarkareplacedattheboundaryofAnti-deSitterspace(z=0),butthestringconnectingthembendsintotheinterior(z>0).DuetothescalingsymmetryoftheAdSspace,thisgivesCoulombpotential(Maldacena;Rey,Yee)ExcitationsoftheGluonicString

Callan,Guijosa;Brower,Tan,Thorn;IK,Maldacena,ThornThespectrumofsmalloscillationsofthestringwithDirichletboundaryconditionsatz=0isknown:Byconformalinvariance,allenergiesscaleinverselywithquarkanti-quarkdistanceL.Theirspectrumisknown,andtheycorrespondtogluonicexcitationsatstrong`tHooftcoupling.Butatweakcouplingtherearenosuchexcitations:onlythegroundstateofenergywhereAsimplifiedmodelwhereonlyladdergraphsaresummedindicatesthatthecouplingwhereaninfinitenumberofexcitationsappearisTheirappearanceisrelatedtothefall-to-thecenterinstabilityintheboundstateequationThenear-thresholdboundstatesareinexactagreementwiththespectrumofahighlyexcitedstringcontainingasingleverylongfold:Issucha`falltothecenter’transitionpossibleinQCD?Apparentlynot,sincetheasymptoticfreedommakesthecouplingweakwhentheendsofthestringapproacheachother.Arecentlatticecalculationofthestringexcitationspectrumshowsthisexplicitly.Onlyonelevelexistswithenergymuchlowerthanothersatshortseparation.Juge,Kuti,MorningstarWhyarethefluxtubeexcitationsstableinaCFT?ForacombinationofenergeticandlargeNreasons!Afteragluonisemittedtheresultingstatewouldbeinacoloradjoint,andwouldhavepositiveenergy.Hence,energyconservationforbidsone-gluonemissionfromanexcitedboundstate.Theenergyconservationdoesnotforbidglueball(closedstring)emission.ButitissuppressedatlargeN.ConclusionsTheAdS/CFTcorrespondencemakesamultitudeofdynamicalpredictionsaboutstronglycoupledconformalgaugetheories.Theyalwaysappeartomakesense,butaredifficulttocheckquantitatively(e.g.,the?intheentropy).Fornon-BPSquantitiesinN=4SYM,non-trivialinterpolatingfunctionsappear.Recently,theconjecturedintegrabilityhasledtodeterminationofthecuspanomalyfunction.ThisprovidesstrongnewevidenceforthevalidityoftheAdS/CFTduality.Therecentdevelopmentsraisehopeoffurtherexactresults,andperhapsaneventualproofofAdS/CFTduality.TheplanarN=4SYMisarichandcomplicatedtheory,andsomepropertiesofitsspectrumcandependdramaticallyonthevalueofthecoupling.Thez-directionofAdSisdualtotheenergyscaleofthegaugetheory:smallzistheUV;largezistheIR.Inapleasantsurprise,becauseofthe5-thdimensionz,thestringpictureapplieseventotheoriesthatareconformal(notconfining!).Thequarkandanti-quarkareplacedattheboundaryofAnti-deSitterspace(z=0),butthestringconnectingthembendsintotheinterior(z>0).DuetothescalingsymmetryoftheAdSspace,thisgivesCoulombpotential(Maldacena;Rey,Yee)StringTheoreticApproachestoConfinementItispossibletogeneralizetheAdS/CFTcorrespondenceinsuchawaythatthequark-antiquarkpotentialislinearatlargedistance.The5-dmetricshouldhaveawarpedform(Polyakov):Thespaceendsatamaximumvalueofzwherethewarpfactorisfinite.ThentheconfiningstringtensionisSemi-classicalNambuStringSemi-classicalquantizationaroundlongstraightNambustringpredictsthequark-antiquarkpotentialThecoefficientoftheuniversalLueschertermdependsonlyonthespace-timedimensiondandisproportionaltotheReggeintercept:ComparisonwiththeLatticeData

Luescher,Weisz(2002)Latticecalculationsoftheforcevs.distanceproducegoodagreementwiththesemi-classicalNambustringforr>0.7fm:EmbeddinginFluxCompactificationsAlongwarpedthroatembeddedintoacompactificationwithNS-NSandR-RfluxesleadstoasmallratiobetweentheIRscaleatthebottomofthethroatandthestringscale.

Randall,Sundrum;Verlinde;IK,Strassler;Giddings,Kachru,Polchinski;KKLT;KKLMMTInthedualcascadinggaugetheorytheIRscaleistheconfinementscale:confinementstabilizesthehierarchybetweenthePlanckscaleandtheSMortheinflationaryscale.Cascadinggaugetheoriesdualto“standardmodelthroats”mayofferinterestingpossibilitiesfornewphysicsbeyondthestandardmodel.Cascales,Franco,Hanany,Saad,Uranga,…ExactIntegrabilityPerturbativecalculationsofanomalousdimensionsaremappedtointegrablespinchains,suggestingexactintegrabilityoftheN=4SYMtheory.Minahan,Zarembo;Beisert,StaudacherForexample,fortheSU(2)sectoroperators

Tr(ZZZWZW…ZW),whereZandWaretwocomplexscalars,theHeisenbergspinchainemergesat1loop.HigherloopscorrecttheHamiltonianbutseemtopreserveitsintegrability.ThismeshesnicelywithearlierfindingsofintegrabilityincertainsubsectorsofQCD.Lipatov;Faddeev,Korchemsky;Braun,Derkachov,ManashovThedualstringtheoryapproachindicatesthatintheSYMtheorytheexactintegrabilityispresentatverystrongcoupling(Bena,Polchinski,Roiban).Henceitislikelytoexistforallvaluesofthecoupling.ConnectionwithCosmicstrings

Copeland,Myers,PolchinskiAfundamentalstringatthebottomofthewarpeddeformedconifoldisdualtoaconfiningstring.AD-stringisdualtoasolitonicstringwhichcouplestotheGoldstonemode.Uponembeddingofthewarpedthroatintoafluxcompactification,theseobjectscanbeusedtomodelcosmicstrings.Thisthroatisnotthe“standardmodelthroat’’butanotherthroat,the“inflationarythroat,”dualtosomehiddensectorcascadinggaugetheory.SpeedofSoundInaCFT,thepressureisrelatedtotheenergydensity,p=3e.Hence,Cs2=dp/de=1/3.TheplotofCs2inpureglueSU(3)latticegaugetheory

Gavai,Gupta,MukherjeeForT/Tc=3,thecouplingisstillquitestrong,l~7.ThissuggeststhatAdS/CFTmethodsmaybeuseful.FoldedStringsonS5andMagnonsFoldedstringspinningaroundthenorthpolehas,forlargeR-chargeJ:Gubser,IK,PolyakovRecentlythisstringwasidentifiedbyHofmanandMaldacenawitha2-magnonoperatorwhereeachmagnoncarriesp=p.TheexactenergyofafreemagnonisSantambrogio,ZanonBeisertItsdualisanopenstringspinningonS5:Hencethe2-magnonoperatordimensionis

inexactagreementwiththefoldedstringresultforlargel.ThegaugetheoryonD7-braneswrappinga4-cycleS4hascouplingThenon-perturbativesuperpotentialdependsontheD3-branelocationthroughthewarpedvolumeInthethroatapproximation,thewarpfactorcanbecalculatedandintegratedovera4-cycleexplicitly.Baumann,Dymarsky,Klebanov,Maldacena,McAllister,Murugan.

ForaclassofconifoldembeddingsArean,Crooks,Ramallo

(w1=z1+iz2,etc.)

theresultisThisformulaappliesbothtonwrappedD7-branes,andtoawrappedEuclideanD3(n=1).Forthelattercase,GanorshowedthatAhasasimplezerowhentheD3-braneapproachesthe4-cycle.Ourresultagreeswiththis.Wehavealsocarriedoutsuchcalculationsfor4-cycleswithintheCalabi-YauconesoverYp,qwithanalogousresul