DD--BranesBranes onon ConesCones andand Gauge/StringGauge/String DualitiesDualities

Igor Klebanov

Department of Physics Princeton University

Lectures at PiTP 2006 IAS, Princeton QCDQCD andand StringString TheoryTheory •• AtAt shortshort distances,distances, mustmust smallersmaller thanthan 11 fermifermi,, thethe quarkquark-- antiquarkantiquark potentialpotential isis approximatelyapproximately CoulombicCoulombic,, duedue toto thethe AsymptoticAsymptotic Freedom.Freedom. •• AtAt largelarge distancesdistances thethe potentialpotential shouldshould bebe linearlinear (Wilson)(Wilson) duedue toto formationformation ofof confiningconfining fluxflux tubes.tubes. FluxFlux TubesTubes inin QCDQCD •• TheseThese objectsobjects maymay bebe approximatelyapproximately describeddescribed byby thethe NambuNambu stringsstrings (animation from lattice work by D. Leinweber et al, Univ. of Adelaide) •• TheThe tubestubes areare widelywidely used,used, forfor example,example, inin jetjet hadronizationhadronization algorithmsalgorithms (the(the LundLund StringString Model)Model) wherewhere theythey snapsnap throughthrough quarkquark--antiquarkantiquark creation.creation. LargeLarge NN GaugeGauge TheoriesTheories •• ConnectionConnection ofof gaugegauge theorytheory withwith stringstring theorytheory isis strengthenedstrengthened inin `t`t HooftHooft’’ss generalizationgeneralization fromfrom 33 colorscolors (SU(3)(SU(3) gaugegauge group)group) toto NN colorscolors (SU(N)(SU(N) gaugegauge group).group). •• MakeMake NN large,large, whilewhile keepingkeeping thethe `t`t HooftHooft couplingcoupling fixed.fixed.

•• TheThe probabilityprobability ofof snappingsnapping aa fluxflux tubetube byby quarkquark-- antiquarkantiquark creationcreation (meson(meson decay)decay) isis 1/N.1/N. TheThe stringstring couplingcoupling isis 1/N.1/N. •• Yet,Yet, thethe planarplanar diagramsdiagrams neededneeded inin thethe largelarge NN limitlimit areare veryvery difficultdifficult toto sumsum explicitly.explicitly. DD--BranesBranes vs.vs. GeometryGeometry •• Dirichlet branes (Polchinski) led string theory back to gauge theory in the mid-90’s. •• A stack of N Dirichlet 3-branes realizes N=4 supersymmetric SU(N) gauge theory in 4 dimensions. It also creates a curved background of 10-d theory of closed superstrings (artwork by E.Imeroni)

which for small r approaches

•• Successful matching of graviton absorption by D3- branes, related to 2-point function of stress-energy tensor in the SYM theory, with a gravity calculation in the 3-brane metric (IK; Gubser, IK, Tseytlin) was a precursor of the AdS/CFT correspondence. ConformalConformal InvarianceInvariance •• InIn thethe NN=4=4 SU(N)SU(N) SYMSYM theorytheory theorytheory therethere areare 33 adjointadjoint chiralchiral superfieldssuperfields ZZi coupledcoupled toto thethe NN=1=1 SU(N)SU(N) SYMSYM theorytheory withwith superpotentialsuperpotential TrTr ZZ1 [Z[Z2,Z,Z3].]. •• TheThe AsymptoticAsymptotic FreedomFreedom isis canceledcanceled byby thethe extraextra fields;fields; thethe betabeta functionfunction isis exactlyexactly zero!zero! Hence,Hence, thethe theorytheory isis invariantinvariant underunder scalescale transformationstransformations xxμ -->> λλ xxμ .. ItIt isis alsoalso invariantinvariant underunder spacespace--timetime inversions.inversions. •• SuchSuch aa theorytheory isis calledcalled aa ConformalConformal FieldField TheoryTheory (CFT).(CFT). •• TheThe NN=4=4 SU(N)SU(N) SYMSYM isis alsoalso invariantinvariant underunder thethe SU(4)SU(4) RR--symmetry.symmetry. ItsIts fullfull supersuper--conformalconformal symmetrysymmetry isis SU(2,2|4).SU(2,2|4). 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--spheresphere realizingrealizing thethe SU(4)SU(4) RR-- symmetry.symmetry.

•• TheThe SO(2,4)SO(2,4) geometricalgeometrical symmetrysymmetry ofof thethe AdSAdS5 spacespace realizesrealizes thethe conformalconformal symmetrysymmetry ofof thethe gaugegauge theory.theory. •• TheThe dd--dimensionaldimensional AdSAdS spacespace isis aa hyperboloidhyperboloid

•• ItsIts metricmetric isis •• WhenWhen aa gaugegauge theorytheory isis stronglystrongly coupled,coupled, thethe radiusradius ofof curvaturecurvature ofof thethe dualdual AdSAdS5 andand ofof thethe 55--dd compactcompact spacespace becomesbecomes large:large:

•• StringString theorytheory inin suchsuch aa weaklyweakly curvedcurved backgroundbackground cancan bebe studiedstudied inin thethe effectiveeffective (super)(super)--gravitygravity approximation,approximation, whichwhich allowsallows forfor aa hosthost ofof explicitexplicit calculations.calculations. CorrectionsCorrections toto itit proceedproceed inin powerspowers ofof

•• FeynmanFeynman graphsgraphs insteadinstead developdevelop aa weakweak couplingcoupling expansionexpansion inin powerspowers ofof λ.λ. AtAt weakweak couplingcoupling thethe dualdual stringstring theorytheory becomesbecomes difficult.difficult. •• GaugeGauge invariantinvariant operatorsoperators inin thethe CFTCFT4 areare inin oneone--toto--oneone correspondencecorrespondence withwith fieldsfields (or(or

extendedextended objects)objects) inin AdSAdS5 •• OperatorOperator dimensiondimension isis determineddetermined byby thethe massmass ofof thethe dualdual field;field; e.g.e.g. forfor scalarscalar operatorsoperators

•• CorrelationCorrelation functionsfunctions areare calculatedcalculated fromfrom thethe dependencedependence ofof stringstring theorytheory pathpath integralintegral onon

boundaryboundary conditionsconditions φφ0 inin AdSAdS5,, imposedimposed nearnear z=0:z=0:

•• InIn thethe largelarge NN limitlimit thethe pathpath integralintegral isis foundfound fromfrom thethe classicalclassical stringstring action:action: ConebraneConebrane 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 AdS5 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. TraceTrace AnomalyAnomaly •• InIn aa 44--dd CFTCFT therethere areare twotwo tracetrace anomalyanomaly coefficients,coefficients, aa andand c:c:

•• CalculationsCalculations onon AdSAdS5 xx YY givegive theirtheir leadingleading largelarge NN valuesvalues Henningson, Skenderis; Gubser •• InIn supersuper--conformalconformal theoriestheories thethe anomaliesanomalies areare relatedrelated toto thethe spectrumspectrum ofof RR--chargescharges ofof thethe chiralchiral fermions:fermions: Anselmi, Freedman, Grisaru, Johansen •• ThisThis providesprovides basicbasic checkschecks ofof thethe dualities.dualities. •• ForFor thethe NN=4=4 SYMSYM theorytheory thethe gauginosgauginos havehave R=1,R=1, whilewhile thethe fermionfermion fieldsfields fromfrom thethe ZZi chiralchiral multipletsmultiplets havehave R=R=--1/3.1/3. •• SinceSince TrTr R=0,R=0, wewe findfind a=c,a=c, andand

•• OnOn thethe gravitygravity side,side, thethe volumevolume ofof SS5 isis ππ3 (the(the radiusradius ofof YY isis fixedfixed soso thatthat )) •• ForFor largelarge NN thethe twotwo calculationscalculations ofof anomalyanomaly coefficientscoefficients agree.agree. OrbifoldOrbifold ConesCones Kachru, Silverstein; Lawrence, Nekrasov, Vafa

•• The simplest set of examples is provided by cones that are orbifolds R6/Γ , where Γ is a subgroup of the rotation group SU(4). •• For abelian orbifolds, all group elements can be brought to the form

•• For Zk orbifolds, the n-th group element is specified by three integers mi defined mod k: xi=nmi/k . •• If none of the eigenvalues of the generator = 1, then all SUSY is broken; if one of the eigenvalues = 1, then N=1 SUSY is preserved; if two of the eigenvalues = 1, then N=2 SUSY is preserved. •• TThehe actionaction inin thethe nn--thth twistedtwisted sectorsector onon 33 complexcomplex coordinatescoordinates ofof CC3,,

andand theirtheir complexcomplex conjugates,conjugates, isis

wherewhere •• IfIf nonenone ofof thesethese phasesphases == 1,1, thenthen thethe orbifoldorbifold actsacts freelyfreely onon SS5//Γ.Γ. ((TThehe tiptip ofof thethe conecone isis aa fixedfixed pointpoint thatthat isis removedremoved inin thethe basicbasic nearnear--horizonhorizon limit.)limit.)

•• AA wellwell--knownknown exampleexample ofof aa freelyfreely--actingacting orbifoldorbifold isis ZZ3 withwith mmi=1.=1. SinceSince oneone ofof thethe eigenvalueseigenvalues ofof thethe generatorgenerator == 1,1, i.e.i.e. ,, thisthis orbifoldorbifold preservespreserves NN=1=1 SUSY.SUSY. ConstructionConstruction ofof thethe quiverquiver gaugegauge theoriestheories Douglas, Moore •• GaugeGauge theorytheory onon NN D3D3--branesbranes atat thethe tiptip ofof RR6//ΓΓ isis foundfound byby applyingapplying projectionsprojections toto thethe U(NkU(Nk)) gaugegauge theorytheory onon thethe coveringcovering space.space. RetainRetain onlyonly thethe fieldsfields invariantinvariant underunder thethe orbifoldorbifold actionaction combinedcombined withwith conjugationconjugation byby aa U(NkU(Nk)) matrixmatrix γγ actingacting onon thethe gaugegauge indices:indices: 3 •• In the supersymmetric examples, such as C /Z3 or the conifold, the conebrane dualities have been tested almost as thoroughly as in the maximally supersymmetric case. •• But when all SUSY is broken, problems may arise. •• When the orbifold Γ breaks all SUSY and is not freely 5 acting, then the weakly curved background AdS5 x S /Γ is unstable due to the presence of tachyons that have (mL)2<-4, and therefore violate the BF bound. •• But for freely acting orbifolds, the negative zero-point energy is compensated by the large stretching of the twisted sector closed strings in the compact space. Hence, at large radius, there are no `bad tachyons.’ This makes freely acting orbifolds particularly interesting from AdS/CFT point of view. •• Yet, before the formal decoupling limit, the non-SUSY freely acting orbifolds have closed-string tachyons localized at the tip of the cone. ClosedClosed StringString ZeroZero--PointPoint EnergyEnergy •• InIn lightlight--ConeCone GreenGreen--Schwarz,Schwarz, therethere areare 44 complexcomplex worldworld sheetsheet bosonsbosons andand fermions.fermions. •• InIn thethe nn--thth twistedtwisted sectorsector thethe boundaryboundary conditionsconditions onon thethe fermionsfermions areare andand onon thethe bosonsbosons

•• TheThe totaltotal zerozero--pointpoint energyenergy ofof thesethese modesmodes isis WeakWeak couplingcoupling analysisanalysis ofof nonnon--SUSYSUSY quiversquivers •• My recent work with Dymarsky and Roiban (hep- th/0505099 and 0509132) reconsiders quiver gauge theory on a stack of D3-branes at the tip of a cone R6/Γ where the orbifold group Γ breaks all the supersymmetry. •• At first sight, the gauge theory seems conformal because the planar beta functions for all single-trace operators 5 vanish. The candidate string dual is AdS5 x S /Γ. Kachru, Silverstein; Lawrence, Nekrasov, Vafa; Bershadsky, Johanson •• However, dimension 4 double-trace operators made out of twisted single-trace ones, f On O-n, are induced at one- loop. Their planar beta-functions have the form 2 2 βf = a λ + 2 γ f λ + f βλ = 0 AA NoteNote onon NormalizationsNormalizations

•• TheThe VEVVEV ofof aa singlesingle tracetrace operatoroperator isis ofof orderorder N.N. •• TheThe standardstandard YangYang--MillsMills actionaction isis ofof orderorder NN2 inin thethe `t`t HooftHooft limit.limit.

•• TheThe doubledouble--tracetrace operatorsoperators ff OOn OO-n makemake contributionscontributions ofof thethe samesame orderorder (for(for thethe couplingcoupling constantconstant ofof orderorder 1).1). TheyThey cannotcannot bebe ignoredignored inin thethe leadingleading largelarge NN limit.limit. •• InIn fact,fact, thethe treetree--levellevel potentialpotential ofof thethe SU(N)SU(N)k quiverquiver theoriestheories (with(with thethe interactinginteracting U(1)U(1)’’ss decoupled)decoupled) containscontains suchsuch doubledouble--tracetrace terms.terms.

• If D=γ2 - a < 0, then there is no real fixed point for f.

• A class of Zk orbifolds with global SU(3) symmetry, that are freely acting on the 5- sphere, has the group action in the fundamental of SU(4)

• Here is a plot of a one-loop SU(N)k gauge theory discriminant, D, and of the ground state closed string m2 on the cone without the D-branes. n=1, …, k-1 labels the twisted sector, and x=n/k. • The simplest freely acting non-susy example is Z5 where there are four induced double- trace couplings

• For example, the SU(3) adjoints are

(α=2π/5) •• ForFor moremore complicatedcomplicated orbifoldsorbifolds,, crossingcrossing ofof eigenvalueseigenvalues ofof thethe discriminantdiscriminant matrixmatrix becomesbecomes important.important. TheThe agreementagreement withwith closedclosed stringsstrings continuescontinues toto hold.hold. •• Generally,Generally, therethere areare threethree twisttwist anglesangles xxi thatthat definedefine aa cube.cube. TheThe stability/instabilitystability/instability regionsregions agreeagree betweenbetween oneone--looploop gaugegauge theorytheory andand stringstring theory.theory. •• AnyAny nonnon--SUSYSUSY abelianabelian orbifoldorbifold containscontains unstableunstable operators.operators. ThisThis appearsappears toto removeremove allall suchsuch orbifoldorbifold quiversquivers fromfrom aa listlist ofof largelarge NN perturbativelyperturbatively conformalconformal gaugegauge theories.theories. •• TheThe oneone--looploop betabeta functionsfunctions destroydestroy thethe conformalconformal invarianceinvariance preciselyprecisely inin thosethose twistedtwisted sectorssectors wherewhere therethere existexist closedclosed--stringstring tachyonstachyons localizedlocalized atat thethe tiptip ofof RR6//Γ.Γ. Thus,Thus, aa veryvery simplesimple correspondencecorrespondence emergesemerges betweenbetween perturbativeperturbative gaugegauge theorytheory andand freefree closedclosed stringstring onon anan orbifoldorbifold.. Why?Why? Perhaps,Perhaps, inin thethe presencepresence ofof tachyons,tachyons, thethe standardstandard AdSAdS/CFT/CFT decouplingdecoupling argumentargument maymay fail.fail. 5 •• TheThe AdSAdS5 xx SS //ΓΓ backgroundbackground isis tachyontachyon--freefree atat largelarge radius.radius. CouldCould itit havehave somesome instabilities?instabilities? IfIf not,not, thenthen therethere isis aa transitiontransition fromfrom instabilityinstability toto stabilitystability asas λλ isis increased.increased. •• WhatWhat isis thethe endend--pointpoint ofof thethe RGRG flow?flow? •• CondensationCondensation ofof localizedlocalized tachyontachyon smoothessmoothes outout thethe tiptip ofof thethe cone.cone. Adams, Polchinski, Silverstein •• TheThe gaugegauge theorytheory onon D3D3--branesbranes atat aa smoothsmooth pointpoint isis NN=4=4 SYM.SYM. Hence,Hence, aa naturalnatural conjectureconjecture isis thatthat thethe gaugegauge theorytheory flowsflows fromfrom thethe nonnon--SUSYSUSY SU(N)SU(N)k quiverquiver gaugegauge theorytheory toto thethe NN=4=4 SU(N)SU(N) SYM.SYM. Dymarsky, Franco, Roiban, IK (work in progress) 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 YY isis aa cosetcoset TT1,1 whichwhich hashas

symmetrysymmetry SU(2)SU(2)AxSU(2)xSU(2)B thatthat rotatesrotates thethe zz’’ss,, andand alsoalso U(1)U(1)R :: •• TheThe SasakiSasaki--EinsteinEinstein metricmetric onon TT1,1 isis

wherewhere •• TheThe topologytopology ofof TT1,1 isis SS2 xx SS3.. •• TheThe NN=1=1 SCFTSCFT onon thethe 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 BB1,, BB2 inin .. IK, Witten •• TheThe RR--chargecharge ofof eacheach fieldsfields 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 ZZ2 symmetrysymmetry whichwhich interchangesinterchanges thethe AA’’ss withwith thethe BB’’ss andand implementsimplements chargecharge conjugation.conjugation. ComparisonComparison withwith aa ZZ22 OrbifoldOrbifold QuiverQuiver •• TheThe simplestsimplest NN=2=2 SUSYSUSY quiverquiver hashas k=2;k=2;

mm1=m=m2=1,=1, mm3=0.=0. TheThe gaugegauge groupgroup isis againagain SU(N)xSU(NSU(N)xSU(N),), butbut inin additionaddition toto thethe

bifundamentalsbifundamentals AAi,, BBj,, therethere isis oneone adjointadjoint chiralchiral superfieldsuperfield forfor eacheach gaugegauge group,group, withwith superpotentialsuperpotential

•• AddingAdding aa ZZ2 oddodd massmass termterm andand integratingintegrating outout thethe adjointsadjoints,, wewe obtainobtain thethe superpotentialsuperpotential ofof thethe conifoldconifold theory,theory,

BreakingBreaking thethe ConformalConformal SymmetrySymmetry •• AA usefuluseful tooltool isis toto addadd toto thethe NN D3D3--branesbranes MM D5D5--branesbranes wrappedwrapped overover thethe SS2 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: StringString TheoreticTheoretic ApproachApproach toto ConfinementConfinement •• It is possible to generalize the AdS/CFT correspondence in such a way that the quark- antiquark potential is linear at large distance. •• 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 •• 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 aa2 ,, 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 3 •• TheThe radiusradius--squaredsquared ofof thethe SS 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. AnomalouslyAnomalously lightlight glueballsglueballs

•• TheThe confiningconfining stringstring tensiontension isis •• TheThe glueballsglueballs areare thethe normalizablenormalizable modesmodes localizedlocalized nearnear atat smallsmall t.t. InIn thethe supergravitysupergravity limitlimit (at(at largelarge ggs M)M) theirtheir massmass scalesscales areare

•• InIn orderorder toto eliminateeliminate thethe anomalouslyanomalously lightlight boundbound states,states, wewe needneed aa smallsmall ggs M,M, whichwhich requiresrequires aa departuredeparture fromfrom thethe SUGRASUGRA limit.limit.

•• EvenEven forfor smallsmall ggs M,M, SUGRASUGRA becomesbecomes reliablereliable inin thethe UVUV (at(at largelarge t).t). LogLog runningrunning ofof couplingscouplings •• TheThe largelarge radiusradius asymptoticasymptotic solutionsolution isis characterizedcharacterized byby logarithmiclogarithmic deviationsdeviations 1,1 fromfrom AdSAdS5 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 •• 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,, AA2,, BB1,, BB2 ?? 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) YYp,qp,q DualitiesDualities •• CascadingCascading behaviorbehavior isis notnot limitedlimited toto thethe conifoldconifold.. Recently,Recently, anan infiniteinfinite familyfamily ofof newnew CYCY conescones overover SasakiSasaki--EinsteinEinstein spacesspaces YYp,q ofof topologytopology SS2 xx SS3 havehave beenbeen constructedconstructed (p(p andand qq areare coco--primeprime integers).integers). Gauntlett, Martelli, Sparks, Waldram •• yy rangesranges betweenbetween twotwo smallersmaller rootsroots ofof v(yv(y):):

•• TheThe SU(N)SU(N)2p SCFTSCFT’’ss onon NN D3D3-- branesbranes atat thethe tiptip ofof thethe conescones havehave alsoalso beenbeen constructed.constructed. Benvenuti, Franco, Hanany, Martelli, Sparks •• ForFor example,example, herehere isis thethe quiverquiver diagramdiagram forfor thethe SCFTSCFT dualdual toto 4,3 AdSAdS5 xx YY RR--chargescharges fromfrom aa--maximizationmaximization •• TheThe conformalconformal invarianceinvariance conditionsconditions dodo notnot fullyfully determinedetermine thethe RR--charges.charges. LetLet RRZ=x,=x, RRY=y,=y, RRU=1=1--(x+y)/2,(x+y)/2, RRV=1+=1+ (x(x--y)/2y)/2 •• TheThe techniquetechnique ofof aa--maximizationmaximization Intriligator, Wecht givesgives

•• Remarkably,Remarkably, thisthis givesgives thethe tracetrace anomalyanomaly agreeingagreeing withwith thethe AdSAdS/CFT/CFT

Benvenuti et al; Bertolini, Bigazzi, Cotrone •• AdditionAddition ofof MM D5D5--branesbranes wrappedwrapped overover thethe SS2 modifiesmodifies thethe quiverquiver diagramdiagram toto •• PerformingPerforming thethe SeibergSeiberg dualityduality onon thethe biggestbiggest gaugegauge groupgroup givesgives thethe samesame quiverquiver withwith NN -->> NN--M.M. ThisThis factfact isis necessarynecessary forfor thethe existenceexistence ofof aa selfself--similarsimilar cascadingcascading RGRG flow.flow. •• TheThe gravitygravity dualsduals ofof thesethese cascadescascades includeinclude thethe ISDISD 33-- formform fieldfield strength.strength. Herzog, Ejaz, IK •• TheThe metricmetric andand 55--formform areare determineddetermined byby aa singlesingle warpwarp factor,factor, whichwhich howeverhowever dependsdepends onon 22 coordinates.coordinates. •• Luckily,Luckily, thethe PDEPDE forfor h(r,yh(r,y)) isis exactlyexactly solvablesolvable

•• TheThe loglog behaviorbehavior characteristiccharacteristic ofof thethe cascadecascade producesproduces aa nakednaked singularitysingularity inin thethe IR.IR. IsIs therethere aa smoothsmooth solutionsolution withwith thethe aboveabove largelarge rr behavior?behavior? IRIR BehaviorBehavior ofof thethe ConifoldConifold CascadeCascade

•• HereHere thethe dynamicaldynamical deformationdeformation ofof thethe conifoldconifold rendersrenders thethe solutionsolution smooth,smooth, andand explainsexplains thethe IRIR dynamicsdynamics ofof thethe gaugegauge theory.theory. •• DimensionalDimensional transmutationtransmutation inin thethe IR.IR. TheThe dynamicallydynamically generatedgenerated confinementconfinement scalescale isis

•• TheThe patternpattern ofof RR--symmetrysymmetry breakingbreaking isis thethe samesame asas inin thethe SU(M)SU(M) SYMSYM theory:theory: ZZ2M -->> ZZ2 •• Yet,Yet, thethe IRIR gaugegauge theorytheory isis somewhatsomewhat moremore complicated.complicated. . •• InIn thethe IRIR thethe gaugegauge theorytheory cascadescascades downdown toto SU(2M)SU(2M) xx SU(M).SU(M). TheThe SU(2M)SU(2M) gaugegauge groupgroup

effectivelyeffectively hashas NNf==NNc.. •• TheThe baryonbaryon andand antianti--baryonbaryon operatorsoperators Seiberg

acquireacquire expectationexpectation valuesvalues andand breakbreak thethe U(1)U(1) ia -ia symmetrysymmetry underunder whichwhich AAk -->> ee AAk;; BBl -->> ee BBl.. Hence,Hence, wewe observeobserve confinementconfinement withoutwithout aa massmass

gap:gap: duedue toto U(1)U(1)baryon chiralchiral symmetrysymmetry breakingbreaking therethere existexist aa GoldstoneGoldstone bosonboson andand itsits masslessmassless scalarscalar superpartnersuperpartner.. ThereThere existsexists aa baryonicbaryonic branchbranch ofof thethe modulimoduli spacespace •• TheThe KSKS solutionsolution isis partpart ofof aa modulimoduli spacespace ofof confiningconfining SUGRASUGRA backgrounds,backgrounds, resolvedresolved warpedwarped deformeddeformed conifoldsconifolds.. Gubser, Herzog, IK; Butti, Grana, Minasian, Petrini, Zaffaroni •• ToTo looklook forfor themthem wewe needneed toto useuse thethe PTPT ansatzansatz::

•• H,H, x,x, g,g, a,a, v,v, andand thethe dilatondilaton areare functionsfunctions ofof thethe radialradial variablevariable t.t. •• AdditionalAdditional radialradial functionsfunctions enterenter intointo thethe ansatzansatz forfor thethe 33--formform fieldfield strengths.strengths. TheThe PTPT ansatzansatz

preservespreserves thethe SO(4)SO(4) butbut breaksbreaks aa ZZ2 chargecharge conjugationconjugation symetrysymetry,, exceptexcept atat thethe KSKS point.point. •• BGMPZBGMPZ usedused thethe methodmethod ofof SU(3)SU(3) structuresstructures toto derivederive thethe completecomplete setset ofof coupledcoupled firstfirst--orderorder equations.equations. •• AA resultresult ofof theirtheir integrationintegration isis thatthat thethe warpwarp factorfactor andand thethe dilatondilaton areare related:related:

Dymarsky, IK, Seiberg •• TheThe integrationintegration constantconstant determinesdetermines thethe `modulus`modulus’’ U:U: wherewhere •• AtAt largelarge tt thethe solutionsolution approachesapproaches thethe KTKT `cascade`cascade asymptoticsasymptotics’’:: •• The resolution parameter U is proportional to the VEV of the operator

•• This family of resolved warped deformed conifolds is dual to the `baryonic branch’ in the gauge theory (the quantum deformed moduli space). •• At large U the IR part of the solution approaches that of the Maldacena-Nunez solution. But we always have the `cascade’ asymptotics at large t. •• Here are plots of the string tension (a fundamental string at the bottom of the throat is dual to an `emergent’ chromo- electric flux tube) and of the dilaton profiles as a function of the modulus U=ln |ζ|. Dymarsky, IK, Seiberg BPSBPS DomainDomain WallsWalls

•• A D5-brane wrapped over the 3- sphere at the bottom of the throat is the domain wall separating two adjacent vacua of the theory. •• Since it is BPS saturated, its tension cannot depend on the baryonic branch modulus. This is indeed the case. This fact provides a check on the choice of the UV boundary conditions, and on the numerical integration procedure necessary away from the KS point. •• Analytic proof? ApplicationsApplications toto DD--branebrane InflationInflation • The Slow-Roll Inflationary Universe (Linde; AAlbrecht,lbrecht, Steinhardt) is a very promising idea for generating the CMB anisotropy spectrum observed by the WMAP. • Finding models with very flat potentials 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, TrivedTrivedii AA relatedrelated suggestionsuggestion forfor DD--branebrane inflationinflation (A. Dymarsky, IK, N. Seiberg) •• In a flux compactification, the U(1)baryon is gauged. Turn on a Fayet-Iliopoulos parameter ξ . •• This makes the throat a resolved warped deformed conifold. •• The probe D3-brane potential on this space is asymptotically flat, if we ignore effects of compactification and D7-branes. The plots are for two different values of U~ξ. •• No anti-D3 needed: in presence of the D3-brane, SUSY is broken by the D-term ξ. Related to the `D-term Inflation’ Binetruy, Dvali; Halyo 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, Klebanov, Maldacena, McAllister, Murugan. •• ForFor aa classclass ofof conifoldconifold embeddingsembeddings

Arean, Crooks, Ramallo (w1=z1+iz2 , etc.) thethe resultresult isis •• ThisThis formulaformula appliesapplies bothboth toto nn wrappedwrapped D7D7-- branes,branes, andand toto aa wrappedwrapped EuclideanEuclidean D3D3 ((n=1n=1).). •• ForFor thethe latterlatter case,case, GanorGanor showedshowed thatthat AA hashas aa simplesimple zerozero whenwhen thethe D3D3--branebrane approachesapproaches thethe 44--cycle.cycle. OurOur resultresult agreesagrees withwith this.this. •• WeWe havehave alsoalso carriedcarried outout suchsuch calculationscalculations forfor 44-- cyclescycles withinwithin thethe CalabiCalabi--YauYau conescones overover YYp,q withwith analogousanalogous results:results: A(X)A(X) isis proportionalproportional toto thethe embeddingembedding equationequation raisedraised toto thethe powerpower 1/n1/n.. ThisThis appearsappears toto bebe aa generalgeneral rulerule forfor 44--cyclescycles inin thethe throat.throat. •• TheThe dependencedependence ofof thethe nonnon--perturbativeperturbative superpotentialsuperpotential onon D3D3--branebrane position,position, andand otherother compactificationcompactification effects,effects, givegive HubbleHubble--scalescale correctionscorrections toto thethe inflatoninflaton potential.potential. •• SomeSome `fine`fine--tuningtuning’’ isis generallygenerally neededneeded toto cancelcancel differentdifferent correctionscorrections toto thethe D3D3-- branebrane potential.potential. ThisThis isis currentlycurrently underunder investigationinvestigation withwith D.D. Baumann,Baumann, A.A. DymarskyDymarsky,, J.J. MaldacenaMaldacena,, L.L. McAllisterMcAllister andand P.P. Steinhardt.Steinhardt. ConclusionsConclusions

•• In the first part, we investigated non-SUSY orbifolds of AdS/CFT. At one loop, flow of double-trace couplings spoils conformal invariance even in the large N limit. There is a precise connection of this instability with presence of twisted sector closed string tachyons. •• Gauge/string dualities for confining gauge theories give a new geometrical view of such important phenomena as dimensional transmutation, chiral symmetry breaking, and quantum deformation of moduli space. We have also discussed new UV phenomena: the duality cascades. •• Embedding gauge/string dualities into string compactifications offers new possibilities for modeling inflation. In particular, D3-branes on resolved warped deformed conifolds may realize D-term inflation. •• Calculation of non-perturbative corrections to the inflaton potential is important for determining if these models can produce slow-roll inflation.