hep-th9712074 PUPT-1748 Restrictions imposed by Superconform

We derive unitarity restrictions on the scaling dimensions of primary quantum operators in a superconformal quantum field theory, in d =3, 4, 5,

hep-th/9712074PUPT-1748

RestrictionsimposedbySuperconformalInvarianceOnQuantumFieldTheories

ShirazMinwalla1

TheDepartmentofPhysics

PrincetonUniversity

Princeton,NJ08540,USA

Wederiveunitarityrestrictionsonthescalingdimensionsofprimaryquan-

tumoperatorsinasuperconformalquantum eldtheory,ind=3,4,5,

6.12/97

1minwalla@princeton.edu

We derive unitarity restrictions on the scaling dimensions of primary quantum operators in a superconformal quantum field theory, in d =3, 4, 5,

1.Introduction

Whereasclassicalmassless eldtheoriesareoftenconformallyinvariant,thesameisrarelytrueoftheirquantumcounterparts.Tode neaquantum eldtheory,youneedtospecifybothaLagrangianandacuto .Thecuto introducesahiddendimensionalscaleinthede nitionofthetheory.Ascaletransformationrelatesaquantumtheorywithagivencuto toanotherquantumtheorywiththescaledcuto .ThereforeaQFTwithnontrivialcuto dependence(i.e.,nonzerobetafunction)isneverscaleinvariant,andsoneverconformallyinvariant.

Sincequantum eldtheoriesgenericallyhavenonzerobetafunction,astudyoftheimplicationsofexactconformalinvarianceonaquantum eldtheorymightseemtobeoflimitedinterest.Thatis,however,notthecase.AQFTisindeednotscaleinvariantatarbitrarycuto ,butasyoulowerthecuto inyourtheory,atrulygaplesstheory ows,intheWilsoniansense,toa xedpointoftherenormalizationgroup-i.e.,apointwithzerobetafunction.Longwavelengthphysicsinsuchatheoryisthusgovernedbyane ectivequantumtheorythatisexactlyscaleinvariant.Inmanyexamplesthise ectivequantum eldtheoryisafreemasslesstheory.Insomecasesofinteresthowever,thelowenergye ectivetheoryisaninteractingscaleinvariant,andgenericallyconformallyinvariant[1]quantum eldtheory.Lowenergye ectivetheoriescompletelyspecifythevacuumstructureofatheory,andthenatureofitsmasslessexcitations,andhenceareofinterest.

Supersymmetric eldtheorieshaveprovedtobeparticularlyamenabletoexactanal-ysisintheinfrared.Interactinggaplesstheorieswithunbrokensupersymmetrytypicallyexhibitbothconformalandsupersymmetricinvarianceintheinfrared.Thefullsymmetryofthe(lowenergye ective)theorymustthenbegeneratedbyasuperalgebrathatcontainsinit,assubsuperalgebras,boththeconformalalgebraandthesupersymmetryalgebra.Suchalgebrasareveryconstrained-infactnoneexistind≥6,andtheirpossibleformsareknownind≤6.Thesealgebrasarecalledsuperconformalalgebras,andthesymmetrytheygenerateiscalledsuperconformalsymmetry.

Threeexamplesofinteractingsuperconformaltheoriesofcurrentinterestarethein-trinsictheorieslivingontheworldvolumesofcoincidentM2,M5,D3branes.AfourthexampleisthetheorythatgovernsthelowenergybehaviourofN=1,d=4superQCDwithNfquarksandNccolourswith3Nc<Nf<3Nc.

Inthispaperwewillderiveunitarityconstraintsonquantum eldtheoriesind=3,4,5,6thatexactlyimplementthesuperconformalalgebra.Theconstraintswewillderive

1

We derive unitarity restrictions on the scaling dimensions of primary quantum operators in a superconformal quantum field theory, in d =3, 4, 5,

willtaketheformofinequalitiesthatscalingdimensionsofprimarylocaloperatorsinsuchatheorywillbeforcedtoobey.

Thispaperisarrangedasfollows.

InSection2wewilldetermineconstraintsthatconformalinvarianceimposesonaquantumtheoryinarbitrarydimension.

Insection3wewillconstructthesuperconformalalgebrasthatexist,i.e.listallcommutationrelationsexplicitly.Wewillalsodeterminesomeconstraintsonscalingdi-mension,imposedbydemandingtheunitarityoftheserepresentations.

Insection4wewillreviewwellknownresultsinthetheoryofLieSuperAlgebras,followingKac[2].Inparticularwewillreviewtheclassi cationofsuchalgebras,andde neexplicitlyallalgebrasthatwewilluseintherestofthepaper.Wewillexplicitly,following

[3],identifythesuperalgebrascorrespondingtothesuperconformalalgebrasind≤6,andverifythattherearenosuperconformalalgebrasind>6.WewillproceedtouseamethoddevelopedbyDobrevandPetkovain[4],[5],[6],(usingthetheorysetupin[7],)toalmostcompletelydeterminetheconstraintsimposedbyunitarityonrepresentationsofthesealgebras.TheseconstraintswilldeterminetherequiredinequalitiesonthescalingdimensionsoflocaloperatorsinaunitaryQFT.

Insection5weexaminehowourresults tintowhatisknownaboutsystemswithsuperconformalinvariance.WeexaminesomeprimaryoperatorsintheworldvolumetheoryoftheM2,D3,M5brane,andidentifytherepresentationsthattheseoperatorslabelfromthelistofallowedrepresentationsgeneratedinprevioussection.Wethengoontoreviewtheoscillatorconstruction[8]ofthed=6,n=2algebra,andexplicitlyconstructtheshortrepresentationsofthisalgebrawhoseexistencecouldnotbeconclusivelyestablishedinsection4.

Insection6wepresentadetailedlistingofallourresults.Thereaderinterestedonlyinourresultsmayjumpdirectlytosection6.

2.TheConformalGroup.

Inthissectionwewilldiscusstheunitaryrepresentationsoftheconformalgroupinarbitrarydimension,asrelevanttoQuantumFieldTheory.Ourchiefresultwillbethefollowing.Primary eldsinaconformallyinvariantQFTappearinmultipletsthatform

2

We derive unitarity restrictions on the scaling dimensions of primary quantum operators in a superconformal quantum field theory, in d =3, 4, 5,

representationsofSO(d).Theyalsohaveascalingdimension, 0.Unitarityforces 0toobeyaninequalityoftheform

0≥f(representationoffield)(2.1)

wherefisafunctionwedeterminebelow.WewillalsoobtainrestrictionsontheSO(d)contentoffreeoperatorsinaconformalQFT.

2.1.De nitionoftheConformalGroup

Theconformalgroupinddimensionsisgeneratedbyd(d 1)

LorentzgeneratorsMµν,

dmomentaPµ,dspecialconformalgeneratorsKµandadilatationD.(Throughthissectiongreekindicesrunfrom0tod 1).Thesequantitiesobeycommutationrelations

[Mµν,Mαβ]=( i)[ηµβMνα+ηναMµβ ηµαMνβ ηνβMµα]

[Mµν,Pα]=( i)[ηναP
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