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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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