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