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8.821F2008Lecture06: SupersymmetricLagrangiansandBasic

Checks

of

AdS/CFT

Lecturer:

McGreevy

September24,2008

Weareonourwaytotalkingaboutreallyawesomethingsaboutsupercoolstuff. Beforeweget

there,though,weneedtodevelopsomeverypowerfultechnology.Tothatend,todaywewilltalkabout

1. SUSYLagrangiansandawhirlwindtourofthebeautiesofsuperspace.

2.moreonN = 4 SYM.

3. BacktotheBigPicture:SomebasicchecksofAdS/CFT

Lookingpastthislecture,wewillbetalkingaboutstringsfromgaugetheorynext.

N = 4SYMandOtherSupersymmetricLagrangians

RecallthatthefieldcontentofN = 4SYMisavectorA, gauginiI=1...4,andsixscalarsXi,all

in theadjointofthe gauge group.The Lagrangian density (which iscompletelydetermined bytheamountof SUSY,upto two parameters, (gY M, )), is

L

= 2

1

tr F

2

+ (DX

i

)

2

/+iDgY M

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[Xi,Xj ]2 [X, ] + [X, ] ])i


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[Q,X] =

{Q,} = F+ + [X, X]

{Q, } = DX

[Q,A] = (2)

ANote: Thereareobviouslyindicesandgamma/sigmamatricessuppressedALLovertheplace(Lorentzvector,Lorentzspinorand SO(6) vector/spinor,supersymmetry).Ifyouwanttoputtheindicesin,eitherleavethatasafunexercise,orcheckoutWeinberg,volume3. Asanexample,F+ F.

1.1

A

Superspace

Detour

TheN

= 4 SYM Lagrangian isanexampleofa (highly)supersymmetric Lagrangian. So far, Ijust told youwhat it wasand that it was SUSY invariant (something youcouldsit down in theprivacyofyourofficeandcheck,ifyouwanted). Itdbenice,though,ifthereweresomesortofamachinethatonecouldcranktogeneratesupersymmetriclagrangians.Thatcrankablemachineissuperspace.

To understand whysuperspace is useful, we should thinkaboutwhy fields are useful for representingtranslationallyinvariantLagrangiansinordinaryQFT.Onereasonisthattherepresentationsofthetranslationgrouponthefieldsareparticularlysimple

(x) =eiPx

(0) (3)

Wed like to introduceasuperfield (thatcomeswith itsownsupercapitalization)(x, ),whichisnowafunctionofspacecoordinatesx andsuperspacecoordinates,thatwellrepresentstransla-tionalinvarianceANDsupersymmetry

(x, ) =eiPx+iQ(0, 0) (4)

where theQs aretheoperatorsthatgeneratesupersymmetrytransformations.

Now, in QFT,onecanautomatically get translationally invariantactions

S = ddx L(,) (5)

aslongasxL= 0.Similarlythe (hereunproven)claim is that

ddxd N s L((x, )) (6)

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issupersymmetricaslongasL=0. Heres denotesthesmallest (real)dimensionofthespinorrepresentation ind dimensionsandN,is,asusualthenumberofsupersymmetries.ThenN s is

justthenumberofrealsupercharges. ForexampleforN = 1,d = 4,N s =4,becauseeitheraWeylor Majoranaspinor (theminimum in four dimensions)has fourrealcomponents.

1.1.1

BPS

or

Chiral

Multiplets

In lecture 5wemade a big dealabout specialrepresentationsofsupersymmetrywhichare killed bysomeofthesupercharges. Suchmultipletshavecorrespondinglyspecialpropertiesinsuperspace.Considerafieldwhichsatisfies

[Q,] = 0

[Q,] 0= (7)

These multiplets,whichare BPS (halfofthe supersymmetries annihilate them)are generallycalledchiralmultiplets.Sometimes theyareactuallychiral (in thesenseofthe Lorentz group), butoftentimestheyarenot. Thisfollowsa longtraditioninphysicsofcallingthingsotherthingswhichtheyarenot.

Thesemultipletsarefunctionsofonlyhalfofsuperspaceas

(x,, ) =ei(Q+Q)(x, 0, 0)= (x,, 0) (8)

Ok,wellthisequationisnotexactlycorrect(asSenthilpointedout;reallytheRHSshouldbe(y

x +i correct: thesarefunctionsofhalfofsuperspace. Because,)),butitismorallyofthis,itispossibletoaddtermstotheLagrangiandensitythatareintegratedoveronlyhalfofsuperspaceandmaintainsupersymmetry:

d2 W (L= d2 W() + ) (9)

Here,weareexplicitlyworkingind = 4,N =1superspace. Thesetermsaresupersymmetric,aslongasW isaholomorphicfunctionof,W = 0.Withthisconstraint,W isafunctionwearefreetochoose,andisknownasthesuperpotential.

Twoexamplesofasuperpotentialare

The second line ofequation (1). Here,werefer to the fact that this linecan be written ind = 4,N = 1superspace (wherewe pickouta particularN =1subgroupfromtheN=4).Inacertainsense,oneofthesandF canbethoughtofascomprisingoneN =1chiralmultiplet,whiletheremainingthreesandsixscanbethoughtofasanotherthreechiralmultiplets.The second line of equation (1) is a superpotential for these three chiralmultiplets.Onpset2youwillhaveachancetothinkaboutthismoreprecisely.

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The gauge kinetic termsandthird lineofequation (1)can be thoughtofascomingfrom theN = 1superpotential

d2 tr() (10)

where the appearingisthesuperpartnerofF ( isaspinorindex),and isacomplexifiedcouplingconstant

g42

i +2. Here shouldbethoughtofasasuperfieldwhoselowest

Y M

component is the gaugino.The superfieldexpansioncontainsa =. . .F term.Multiplyingtwotogether,onegets

1F2 d2 2F2 d2 tr(

) (11)2gY M

Therestofthegaugekinetictermsandthetaangletermcanbeunderstoodsimilarly.

1.2

Holomorphy

and

Non-Renormalization

(aka

Seibergology)

Whatsthebigdealwiththisnewfangledsuperpotential?

Well,aswesaidbefore,thesuperpotentialhastobeholomorphicinorderforsupersymmetrytobepreserved. Wecantakethislineofreasoningonestepfurtherwecanthinkofpromotingthecouplings to dynamicalsuperfields (whose lowestcomponentvevsare just theconstantcouplings).Then,thesuperpotentialmustbeholomorphic,also,inthecouplings.

Thisstatement isuncomfortably powerful.Forexample, it implies that if SUSY isnot broken, theformofradiativelygeneratedcorrectionstothesuperpotentialareseverelyconstrainedtheymust

be holmorphic in thefieldsandthecouplings.Forexample,onecouldnever generatea term in thesuperpotentialthatwasafunctionofboth and,W =W().

Thisleadstomanynonrenormalizationtheoremsinsupersymmetricfieldtheories,oneexampleofwhich is for the functionofsupersymmetric gauge theories.Thissays that

Y M = 1 loop + nonperturbative (12)

Thismakessensebecauseweknowthethetaanglecannotappearinthebetafunctionperturba-tively. However,sincetheeffectivegaugecouplingfunctionmustbeholomorphicin,theonly

perturbativecontributiontothebetafunctioncanbeO(0),i.e.,theoneloop contribution.

ForN = 4 SYM,onecan go homeandcalculate (for pset 2)that theone loopcontribution to thebetafunctionisidenticallyzero. Inthistheory,thenonperturbativetermscanbeunderstoodascomingfrominstantons. However,asitturnsout,theydonotcorrectthebetafunction,butdoaddhigherderivativetermselsewhereintheaction1. Hence,inthistheorythebetafunctioniszero,evennonperturbatively,andthetheoryreallyisscaleinvariant.

1Thiscorrectsawrongstatementfrom lecture4.

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FormoreonSeibergology,seetheexcellentnotesofArgyres:http://www.physics.uc.edu/argyres/661/index.html

SomeMoreCommentsonN = 4SYM

1. IthasaCoulombBranchofVacua:

ThescalarpotentialintheN = 4action (1)is:

V tr[Xi, Xj ]2 (13)i,j

Ingeneral,supersymmetryisunbrokeniffV = 0,andsothereisamodulispaceofsupersymmetricvacua,labeledbyvevsofX whichsatisfy

M

={X|[Xi,Xj] = 0, i, j}/{gaugesymmetry}

(14)

Wedivideoutbythegaugesymmetry,sincevacuarelatedbygaugerotationarephysicallyequivalent.We canfix this gauge symmetryandsatisfythe modulispace condition by pickingabasissuchthat

Xi=1,...,6= diag(xi1, . . . xi ) (15)N

whereN is thenumberofcolors.Hence, themodulispace is justan 6N dimensionalspacegiven bythesexs.

Atagenericpointinthismodulispacexim =xin,andsothegaugebosonsthatcommute

witheachXi

areall just proportionalto theXisthemselves,andso,atagenericpoint,the gauge group is broken fromU(N) totheU(1)N subgroupgeneratedbythese. Sincewehave broken toawhole bunchofcopiesofelectromagnetism (withsomechargedscalarsandspinors),thismodulispaceisknownasaCoulombbranch.

Sincethevevsofthe Xsgiveamassscale,notonlydoesagenericpoint inthismodulispacebreakthegaugesymmetry,butitalsospontaneouslybreaksthedilation,andhence,thesuperconformalsymmetry. ThehiggsingfromU(N)toU(1)N givesWbosonswithamassderivedfromthismassscale

ab ZabmW =|xa xb|= (16)

TheW bosonsareBPSobjectswithrespecttothestillunbrokenN

=4supersymmetry.Thismustbetruesincewesaidinlecture5thattheN = 4 BPSmassivemultiplet (whatwehavecreatedbyhiggsing)hasthesamenumberofdegreesoffreedomastheN = 4masslessmultiplet (whatwe had before higgsing).The masslessN = 4 gaugemultiplet carries its foodaroundwithit.TheZ aboveare just thecentralchargesofthese BPSobjects.

Recallalso that this story has a Dbrane interpretation:N = 4 SYMwith gauge groupU(N)istheworldvolumetheoryof N (coincident)D3branes. Wecan thinkaboutseparatingthese parallelN D3branes. Sincetheyareparallel,andsitinatendimensionalspacetime,

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therearesixcoordinateswhichlabeltheirpositions:xia wherei runsoverthesixtransversedimensionsanda runsoverN,thenumberofDbranes. Actually,tomakesurethemassdimensionsofxa

i ,matchwiththoseabove,wewritetheseparationbetweenbranea andb as

i i iyab =|xa xb| (17)

Recall, however, thatwhen theDbraneswerecoincident, we could interpret the lowest(massless) string stateswhich startedononeDbraneandendedonanotherasmasslessgauge bosons for theU(N) gaugetheory.Asweseparatethebranes,itisnolongerthecasethatstringsstretchedbetweenbranesaremassless:theyarestretched.Therefore,thegaugebosonsareacquiringamass!In thiscontext,the Higgsmechanism is just pullingastackofcoincidentbranesapartwhichiskindofsuperawesome.OnecancalculatethemassoftheseWbosons,andnotsurprisinglytheymatchwiththepictureabove

1abmW = (string tension)x (length)= yab =|xa xb| (18)

The BPS propertymeans that this relation must be true for anyvalue ofthe stringcoupling.2. Sduality (a.k.a.MontonenOlive)

Sdualitysaysthefollowingabsurdthing

1[N = 4withgaugegroupG, coupling] = [N = 4withgaugegroupLG, coupling ]

(19)

whereLG is thedualgauge group, more on that in a second.As withmanysuch dualities,itsaysthatthesetwotheoriesarecompletelyequal,thoughitmightbehardtofigureouthowtomaptheobservablesofoneontoobservablesoftheother. ThedualgroupisdefinedsuchthattheweightlatticeofLG isequaltothedualoftheweightlatticeofG

w(LG) = ( w(G))

(20)

Forexample,

LSU(N) = SU(N)/ZNLSO(2N) = Sp(N)

LU(N) = U(N) (21)

TheLisforLanglands.

Really,Sdualityisasubsetofalargerdualitythatthetheoryenjoys: itisinvariantundershiftsofthethetaangle, + 2.ThesetwotransformationsareusuallycalledS andT:

1622S ( = 0):gY M 2 (22)gY M

T : + 2 (23)

whichtogether generate the groupSL(2,Z):

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2.Perturbations

ThefirstclaimeachlinearizedsupergravityperturbationcorrespondstosomeN = 4 gaugeinvariantoperator. Fornowwefocusonlocaloperators,combinationsoffieldstakenatthesamespacetime point.Since theoperatorsare gauge invariant, theyinvolvea trace,or prod-

uctoftracesoverSU(N) indices.Well focus onsingle traceoperators (thereasons for whichwillbeclearinafuturelecture).

Wecanorganizeoperator representationsof thesuperconformalalgebrabystartingwithoperatorsthataresocalledsuperconformalprimaries.Theseoperators OareannihilatedbybothK andS

[K, O] = [S, O] = 0 (26)

Wecan then getotheroperators in therepresentation byactingwiththeoperatorsQ andP.Since, forasuperconformal primary [S, O] = 0, it cannot be written as [Q, anotheroperator].Glancingat (2),thissuggeststhatallsuperconformal primariesare builtoutof theXs. Infact,since commutators appear on the RHSof (2)onlysymmetric combinations of theXswillbesuperconformalprimaries.Thuswearelookingatoperatorsoftheform

Oi1...il Tr(X{i1 . . . X il}) (27)

On the supergravityside,fields (perturbations)onAdS5 S5 canbeexpandedinspherical

harmonicsontheS5. Forexample,usingx ascoordinatesonAdS5 andy ontheS5 (with

y2 = 1),anyfield(x, y) canbeexpandedas

(x, y) = l(x)Yl(y) (28)

l

andthesphericalharmonicscanbewrittenas

Yl(y) =Ti1...ilyi1

. . . y il|Py2=1 (29)

The correspondence says thatwe should identifythe spherical harmonics withsuperconformalprimariesas

Ti1...ilyi1

. . . y il Ti1...ilTr(X{i1

. . . X il}) (30)

inawaythatwillbemadepreciseinnottoofuturelectures. Thisisenoughtoorganizethewholespectrumofsupergravityperturbations. Sinceweknowwhichsupergravityfields

correspondtosuperconformalprimaries,wecan getdescendantoperatorsbyactingwithQ andP. Similarly,wecanget the supergravityperturbationsthatcorrespondtotheseoperatorsbyactingwiththecorrespondingsymmetriesinAdS5 S5.

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