WarpedWarped ConifoldsConifolds andand TheirTheir ApplicationsApplications toto CosmologyCosmology

Igor Klebanov

Department of Physics

PiTP Seminar IAS, July 19, 2007 FromFrom DD --branesbranes toto AdSAdS /CFT/CFT

•• AA stackstack ofof NN DirichletDirichlet 33--branesbranes realizesrealizes NN=4=4 supersymmetricsupersymmetric SU(N)SU(N) gaugegauge theorytheory inin 44 dimensions.dimensions. ItIt alsoalso createscreates aa curvedcurved 1010 --dd backgroundbackground ofof closedclosed typetype IIBIIB superstringsuperstring theorytheory (artwork by E.Imeroni )

whichwhich forfor smallsmall rr approachesapproaches TheThe AdSAdS /CFT/CFT dualityduality Maldacena ; Gubser , IK, Polyakov ; Witten •• RelatesRelates conformalconformal gaugegauge theorytheory inin 44 dimensionsdimensions toto stringstring theorytheory onon 55 --dd AntiAnti --dede SitterSitter spacespace timestimes aa 55 --dd compactcompact space.space. ForFor thethe NN=4=4 SYMSYM theorytheory thisthis compactcompact spacespace isis aa 55 --dd sphere.sphere.

•• TheThe geometricalgeometrical symmetrysymmetry ofof thethe AdSAdS 5 spacespace realizesrealizes thethe conformalconformal symmetrysymmetry ofof thethe gaugegauge theory.theory.

•• TheThe AdSAdS d spacespace isis aa hyperboloidhyperboloid

•• ItsIts metricmetric isis ConeCone --BraneBrane DualitiesDualities •• To reduce the number of supersymmetries in AdS /CFT, we may place the stack of N D3 -branes at the tip of a 6 -d Ricci -flat cone X whose base is a 5 -d Einstein space Y:

•• Taking the near -horizon limit of the background created by the N D3 -branes, we find the space AdS 5 x Y, with N units of RR 5 -form flux, whose radius is given by •• This type IIB background is conjectured to be dual to the IR limit of the gauge theory on N D3 -branes at the tip of the cone X. Kachru , Silverstein; … D3D3 --branesbranes onon thethe ConifoldConifold •• TheThe conifoldconifold isis aa CalabiCalabi --YauYau 33--foldfold conecone XX describeddescribed byby thethe constraintconstraint onon 44 complexcomplex variables.variables. •• ItsIts basebase isis calledcalled TT 1,1 ;; itit hashas symmetrysymmetry

SO(4)~SU(2)SO(4)~SU(2) AxSU(2)xSU(2) B thatthat rotatesrotates thethe zz’’ss,, andand alsoalso U(1)U(1) R :: •• TheThe SasakiSasaki --EinsteinEinstein metricmetric onon TT 1,1 isis

wherewhere •• TheThe topologytopology ofof TT 1,1 isis SS 2 xx SS 3.. •• ToTo `solve`solve ’’ thethe conifoldconifold constraintconstraint detdet ZZ == 00 wewe introduceintroduce anotheranother setset ofof convenientconvenient coordinates:coordinates:

•• TheThe actionaction ofof globalglobal symmetriessymmetries isis

•• ThereThere isis aa redundancyredundancy underunder

whichwhich isis partlypartly fixedfixed byby imposingimposing •• ItIt remainsremains toto quotientquotient thethe spacespace byby thethe phasephase rotationrotation (in(in thethe gaugegauge theory,theory, thisthis willwill havehave thethe meaningmeaning ofof thethe U(1)U(1) baryonbaryon numbernumber symmetry).symmetry). •• InIn thethe IRIR gaugegauge theorytheory onon D3D3 --branesbranes atat thethe apexapex ofof thethe conifoldconifold ,, thethe coordinatescoordinates

aa1,, aa 2,, bb 1,, bb 2 areare replacedreplaced byby chiralchiral superfieldssuperfields .. ForFor aa singlesingle D3D3 --branebrane itit isis necessarynecessary toto introduceintroduce gaugegauge groupgroup U(1)U(1) xx U(1).U(1). •• TheThe NN=1=1 SCFTSCFT onon NN D3D3 --branesbranes atat thethe apexapex ofof thethe conifoldconifold hashas gaugegauge groupgroup SU(N)xSU(NSU(N)xSU(N ))

coupledcoupled toto bifundamentalbifundamental chiralchiral superfieldssuperfields AA1,, AA2,, inin ,, andand BB 1,, BB 2 inin .. IK, Witten •• TheThe RR --chargecharge ofof eacheach fieldfield isis ½½.. ThisThis insuresinsures U(1)U(1) R anomalyanomaly cancellation.cancellation.

•• TheThe uniqueunique SU(2)SU(2) AxSU(2)xSU(2) B invariant,invariant, exactlyexactly marginalmarginal quarticquartic superpotentialsuperpotential isis added:added:

•• ThisThis theorytheory alsoalso hashas aa baryonicbaryonic U(1)U(1) ia -ia symmetrysymmetry underunder whichwhich AAk -->> ee AAk;; BBl -->> ee BBl ,,

andand aa ZZ 2 symmetrysymmetry whichwhich interchangesinterchanges thethe AA ’’ss withwith thethe BB ’’ss andand implementsimplements chargecharge conjugation.conjugation. ResolutionResolution andand DeformationDeformation

•• ThereThere areare twotwo wellwell --knownknown CalabiCalabi --YauYau blowblow --upsups ofof thethe conifoldconifold singularity.singularity. •• TheThe `deformation`deformation ’’ replacesreplaces thethe constraintconstraint onon thethe zz -- coordinatescoordinates byby 2 2 2 2 2 zz1 +z+z 2 +z+z 3 +z+z 4 ==ee •• ThisThis replacesreplaces thethe singularitysingularity byby aa finitefinite 33 --sphere.sphere. •• InIn thethe `small`small resolutionresolution ’’ thethe singularitysingularity isis replacedreplaced byby aa finitefinite 22 --sphere.sphere. ThisThis isis implementedimplemented byby modifyingmodifying thethe constraintconstraint onon thethe aa andand bb variablesvariables WarpedWarped ResolvedResolved ConifoldConifold •• InIn thethe gaugegauge theorytheory thethe resolutionresolution isis achievedachieved byby givinggiving VEVVEV ’’ss toto thethe chiralchiral superfieldssuperfields .. IK, Witten •• ForFor example,example, wewe maymay givegive aa VEVVEV toto onlyonly oneone ofof thethe fourfour superfieldssuperfields :: •• TheThe dualdual ofof suchsuch aa gaugegauge theorytheory isis aa resolvedresolved conifoldconifold ,, whichwhich isis warpedwarped byby aa stackstack ofof NN D3D3 --branesbranes placedplaced atat thethe northnorth polepole ofof thethe blownblown upup 22 --sphere.sphere. •• TheThe explicitexplicit CYCY metricmetric onon thethe resolvedresolved conifoldconifold isis Pando Zayas , Tseytlin

•• TheThe warpwarp factorfactor isis thethe GreenGreen ’’ss functionfunction onon thisthis spacespace IK, Murugan

•• TheThe radialradial functionsfunctions areare hyperhyper --geometric:geometric: •• WeWe getget anan explicitexplicit `localized`localized ’’ solutionsolution whichwhich describesdescribes SU(2)xU(1)xU(1)SU(2)xU(1)xU(1) symmetricsymmetric holographicholographic RGRG flowflow toto thethe NN=4=4 SU(N)SU(N) SYM.SYM.

•• AA previouslypreviously knownknown `smeared`smeared ’’ solutionsolution correspondscorresponds toto takingtaking justjust thethe l=0l=0 harmonic.harmonic. ThisThis solutionsolution isis singularsingular Pando Zayas , Tseytlin StringString TheoreticTheoretic ApproachesApproaches toto ConfinementConfinement •• It is possible to generalize the AdS /CFT correspondence so that the quark -antiquark potential is linear at large distance and nearly logarithmic at small. •• A “cartoon ’’ of the necessary metric is

•• The space ends at a maximum value of z where the warp factor is finite. Then the confining string tension is WarpedWarped DeformedDeformed ConifoldConifold •• AA usefuluseful tooltool isis toto addadd toto thethe NN D3D3 --branesbranes MM D5D5 --branesbranes wrappedwrapped overover thethe SS 2 atat thethe tiptip ofof thethe conifoldconifold .. •• TheThe 1010 --dd geometrygeometry dualdual toto thethe gaugegauge theorytheory onon thesethese branesbranes isis thethe warpedwarped deformeddeformed conifoldconifold (IK, Strassler )

•• isis thethe metricmetric ofof thethe deformeddeformed conifoldconifold ,, aa simplesimple CalabiCalabi --YauYau spacespace defineddefined byby thethe followingfollowing constraintconstraint onon 44 complexcomplex variables:variables: •• TheThe warpwarp factorfactor isis finitefinite atat thethe `end`end ofof spacespace ’’ t=0,t=0, asas requiredrequired forfor thethe confinement:confinement: h(th(t )=)= 22 -8/3 γγ I(tI(t ))

•• TheThe standardstandard warpwarp factorfactor aa 2 ,, whichwhich measuresmeasures thethe stringstring tension,tension, isis identifiedidentified withwith h(th(t )) -1/2 andand isis minimizedminimized atat t=0.t=0. ItIt blowsblows upup atat largelarge tt (near(near thethe boundary).boundary). •• TheThe dilatondilaton isis exactlyexactly constantconstant duedue toto thethe selfself --dualityduality ofof thethe 33 --formform backgroundbackground •• TheThe radiusradius --squaredsquared ofof thethe SS 3 atat t=0t=0 isis

~~ggsMM inin stringstring units.units.

•• WhenWhen ggsMM isis large,large, thethe curvaturescurvatures areare smallsmall everywhere,everywhere, andand thethe SUGRASUGRA solutionsolution isis reliablereliable inin `solving`solving ’’ thisthis confiningconfining gaugegauge theory.theory.

•• EvenEven whenwhen ggsMM isis small,small, thethe curvaturecurvature getsgets smallsmall atat largelarge tt (in(in thethe UV).UV). TheThe SUGRASUGRA descriptiondescription ofof thethe dualityduality cascadecascade isis robust.robust. LogLog runningrunning ofof couplingscouplings inin UVUV •• TheThe largelarge radiusradius asymptoticasymptotic solutionsolution isis characterizedcharacterized byby logarithmiclogarithmic deviationsdeviations 1,1 fromfrom AdSAdS 5 xx TT IK, Tseytlin •• TheThe nearnear --AdSAdS radialradial coordinatecoordinate isis •• TheThe NSNS --NSNS andand RR --RR 22 --formform potentials:potentials: •• ThisThis translatestranslates intointo loglog runningrunning ofof thethe gaugegauge couplingscouplings throughthrough

•• ThisThis agreesagrees withwith thethe bb--functionsfunctions inin thethe gaugegauge theorytheory

•• InIn thethe UVUV thethe anomalousanomalous dimensiondimension ofof operatorsoperators isis gg ~~ --1/21/2 •• TheThe warpwarp factorfactor deviatesdeviates fromfrom thethe M=0M=0 solutionsolution logarithmically.logarithmically.

•• Remarkably,Remarkably, thethe 55 --formform flux,flux, dualdual toto thethe numbernumber ofof colors,colors, alsoalso changeschanges logarithmicallylogarithmically withwith thethe RGRG scale.scale. •• WhatWhat isis thethe explanationexplanation inin thethe dualdual SU(kM)xSU((kSU(kM)xSU((k --1)M)1)M) SYMSYM theorytheory coupledcoupled toto bifundamentalbifundamental chiralchiral superfieldssuperfields AA1,, AA 2,, BB1,, BB 2 ?? AA novelnovel phenomenon,phenomenon, calledcalled aa dualityduality cascadecascade ,, takestakes place:place: kk repeatedlyrepeatedly changeschanges byby 11 asas aa resultresult ofof thethe SeibergSeiberg dualityduality IK, Strassler (diagram of RG flows from a review by M. Strassler ) •• ThereThere isis aa scalescale wherewhere thethe SU(kMSU(kM )) couplingcoupling becomesbecomes veryvery strong.strong. ThisThis gaugegauge

groupgroup hashas NNf=2(k=2(k --1)M1)M flavors.flavors. •• ToTo understandunderstand furtherfurther RGRG flow,flow, performperform

SeibergSeiberg dualityduality toto SU(NSU(N f--NNc)=SU((k)=SU((k --2)M).2)M). •• TheThe resultingresulting SU((kSU((k --1)M)xSU((k1)M)xSU((k --2)M)2)M) theorytheory hashas thethe samesame structurestructure asas thethe originaloriginal one,one, butbut kk isis reducedreduced byby 1.1. •• TheThe flowflow proceedsproceeds inin aa quasiquasi --periodicperiodic fashionfashion untiluntil kk becomesbecomes O(1)O(1) inin thethe IR.IR. •• ComparisonComparison ofof warpwarp factorsfactors inin thethe AdSAdS ,, warpedwarped conifoldconifold ,, andand warpedwarped deformeddeformed conifoldconifold cases.cases. TheThe warpedwarped conifoldconifold (KT)(KT) solution,solution, whichwhich hashas aa nakednaked singularity,singularity, shouldshould bebe interpretedinterpreted asas asymptoticasymptotic (UV)(UV) approximationapproximation toto thethe correctcorrect solution.solution. •• TheThe graphgraph ofof quarkquark antianti -- quarkquark potentialpotential isis qualitativelyqualitatively similarsimilar toto thatthat foundfound inin numericalnumerical simulationssimulations ofof QCD.QCD. TheThe upperupper graph,graph, fromfrom thethe recentrecent SeniorSenior ThesisThesis ofof V.V. CvicekCvicek showsshows thethe stringstring theorytheory resultresult forfor thethe warpedwarped deformeddeformed conifoldconifold .. •• TheThe lowerlower graphgraph showsshows latticelattice QCDQCD resultsresults byby G.G. BaliBali etet alal withwith rr 0 ~~ 0.50.5 fm.fm. •• AllAll ofof thisthis providesprovides usus withwith anan exactexact solutionsolution ofof aa 44 --dd largelarge NN confiningconfining supersymmetricsupersymmetric gaugegauge theory.theory. •• ThisThis shouldshould bebe aa goodgood playgroundplayground forfor testingtesting variousvarious ideasideas aboutabout stronglystrongly coupledcoupled gaugegauge theory.theory. •• SomeSome resultsresults onon glueballglueball spectraspectra areare alreadyalready available,available, andand furtherfurther calculationscalculations areare ongoing.ongoing. Krasnitz ; Caceres, Hernandez; Dymarsky , Melnikov ; Berg,

Haack , Muck •• CouldCould therethere bebe applicationsapplications ofof thesethese modelsmodels toto newnew physics?physics? • The Inflationary Universe ( Guth ; DD--BraneBrane InflationInflation Linde ; Albrecht, Steinhardt) is a very promising idea for generating the CMB anisotropy spectrum observed by the WMAP. • Finding slow -roll models has proven to be difficult. Recent string theory constructions use moving D -branes . Dvali , Tye , …

• In the KKLT/KKLMMT model, the warped deformed conifold is embedded into a string compactification . An anti -D3 -brane is added at the bottom to break SUSY and generate a potential. A D3 -brane rolls in the throat. Its radial coordinate plays the role of an inflaton . Kachru , Kallosh , Linde , Maldacena , McAllister, Trivedi CosmicCosmic StringsStrings •• BraneBrane antianti --branebrane annihilationannihilation producesproduces variousvarious kindskinds ofof longlong stringsstrings atat thethe bottombottom ofof thethe throat:throat: fundamental,fundamental, DD --stringsstrings andand theirtheir (( p,qp,q )) boundbound states.states. Sarangi , Tye ; Copeland, Myers, Polchinski •• TheirTheir tensiontension mm isis `warped`warped downdown ’’ byby thethe factorfactor -1/2 hh0 ~~ expexp (( --44pp K/3K/3 ggs M)M) ifif KK cascadecascade stepssteps taketake placeplace inin thethe throat.throat. ItIt isis notnot hardhard toto attainattain GG mm << 1010 -7 dictateddictated byby presentpresent experimentalexperimental bounds.bounds. •• TheThe FF --stringsstrings areare dualdual toto confiningconfining stringsstrings inin thethe gaugegauge theory;theory; DD --stringsstrings toto certaincertain solitonicsolitonic strings.strings. Gubser , IK, Herzog SlowSlow rollroll DD --branebrane inflation?inflation?

•• EffectsEffects ofof D7D7 --branesbranes andand ofof compactificationcompactification genericallygenerically spoilspoil thethe flatnessflatness ofof thethe potential.potential. NonNon --perturbativeperturbative effectseffects introduceintroduce thethe KKLTKKLT -- typetype superpotentialsuperpotential wherewhere XX denotesdenotes thethe D3D3 -- branebrane position.position. InIn anyany warpedwarped throatthroat DD --branebrane inflationinflation model,model, itit isis importantimportant toto calculatecalculate A(X).A(X). •• TheThe gaugegauge theorytheory onon D7D7 --branesbranes wrappingwrapping aa 44 -- cyclecycle ΣΣ4 hashas couplingcoupling •• TheThe nonnon --perturbativeperturbative superpotentialsuperpotential dependsdepends onon thethe D3D3 --branebrane locationlocation throughthrough thethe warpedwarped volumevolume •• InIn thethe throatthroat approximation,approximation, thethe warpwarp factorfactor cancan bebe calculatedcalculated andand integratedintegrated overover aa 44 -- cyclecycle explicitly.explicitly. Baumann, Dymarsky , IK, Maldacena , McAllister, Murugan . •• IfIf thethe D7D7 --branebrane emeddingemedding isis specifiedspecified byby •• TheThe FF --termterm potentialpotential inin NN=1=1 SUGRASUGRA isis

•• UsingUsing thethe DeWolfeDeWolfe --GiddingsGiddings KaehlerKaehler potentialpotential forfor thethe volumevolume modulusmodulus rr andand thethe threethree D3D3 --branebrane coordinatescoordinates zza onon thethe conifoldconifold thethe FF --termterm potentialpotential isis foundfound toto bebe Burgess, Cline, Dasgupta , Firouzjahi ; Baumann, Dymarsky , IK, McAllister, Steinhardt •• ThisThis generallygenerally givesgives HubbleHubble --scalescale correctionscorrections toto thethe inflatoninflaton potential,potential, soso finefine --tuningtuning isis needed.needed. •• TheThe `uplifting`uplifting ’’ isis accomplishedaccomplished byby thethe D3D3 antianti --D3D3 potentialpotential

•• ForFor thethe KKLMMTKKLMMT modelmodel withwith antianti --D3D3 branebrane ,, wherewhere hh 0 isis thethe warpwarp factorfactor atat thethe bottombottom ofof thethe throat.throat. •• WeWe havehave studiedstudied aa simplesimple andand symmetricsymmetric KupersteinKuperstein embeddingembedding •• TheThe stablestable trajectorytrajectory forfor positivepositive mm isis •• TheThe effectiveeffective potentialpotential forfor thethe inflatoninflaton genericallygenerically hashas aa locallocal maximummaximum andand miniumminium .. ItIt cancan bebe finefine --tunedtuned toto havehave anan inflectioninflection point.point. •• MotionMotion nearnear thethe inflectioninflection pointpoint cancan produceproduce enoughenough ee--foldsfolds ofof inflation.inflation. •• ButBut cosmologicalcosmological predictionspredictions areare veryvery sensitive,sensitive, e.g.e.g. •• TheThe signsign ofof dependsdepends onon itsits positionposition relativerelative toto inflectioninflection point.point. ThisThis isis aa `Delicate`Delicate Universe.Universe. ’’ Baumann, Dymarsky , IK, McAllister, Steinhardt ConclusionsConclusions

•• PlacingPlacing D3D3 --branesbranes atat thethe tiptip ofof aa CYCY cone,cone, suchsuch asas thethe conifoldconifold ,, leadsleads toto AdSAdS /CFT/CFT dualitiesdualities withwith NN=1=1 SUSY.SUSY. SymmetrySymmetry breakingbreaking inin thethe gaugegauge theorytheory producesproduces warpedwarped resolvedresolved conifoldsconifolds .. •• AddingAdding wrappedwrapped D5D5 --branesbranes atat thethe apexapex producesproduces aa cascadingcascading confiningconfining gaugegauge theorytheory whosewhose dualdual isis thethe warpedwarped deformeddeformed conifoldconifold .. •• ThisThis exampleexample ofof gauge/stringgauge/string dualityduality givesgives aa newnew geometricalgeometrical viewview ofof suchsuch importantimportant phenomenaphenomena asas dimensionaldimensional transmutation,transmutation, chiralchiral symmetrysymmetry breaking,breaking, andand quantumquantum deformationdeformation ofof modulimoduli space.space. •• EmbeddingEmbedding gauge/stringgauge/string dualitiesdualities intointo stringstring compactificationscompactifications offersoffers newnew possibilitiespossibilities forfor modelingmodeling inflationinflation andand cosmiccosmic strings.strings. •• CalculationCalculation ofof nonnon --perturbativeperturbative correctionscorrections toto thethe inflatoninflaton potentialpotential isis importantimportant forfor determiningdetermining ifif thesethese modelsmodels cancan bebe finefine -- tunedtuned toto produceproduce slowslow --rollroll inflation.inflation.