Baryons, Boundaries and Matrix Models

arXiv:hep-th/0211271v128Nov2002UCB-PTH-02/56UCLA-02-TEP-37Baryons,BoundariesandMatrixModels.IosifBena1,RaduRoiban2andRaduTatar31DepartmentofPhysics2DepartmentofPhysicsUniversityofCaliforniaUniversityofCaliforniaLosAngeles,CA90095SantaBarbara,[e-mail]@vulcan2.physics.ucsb.edu3DepartmentofPhysicsandTheoreticalPhysicsGroupLawrenceBerkeleyNationalLaboratoryUniversityofCaliforniaBerkeley,[e-mail]sionoftheDijkgraaf-Vafaproposalistoincludefieldsinthefundamentalrepresentationofthegaugegroup.Inthispaperweusefieldtheorytechniquestoanalyzegaugetheorieswhosetreelevelsuperpotentialisagenericpolynomialinmulti-traceopera-torsconstructedoutofsuchfields.Weshowthattheeffectivesuperpo-tentialisgeneratedbyplanardiagramswithatmostone(generalized)boundary.Thisjustifiestheproposalputforwardin[12].Wethenproceedtoextendthegaugetheory-matrixmodeldual-itytoincludebaryonicoperators.WeobtainthefullmodulispaceofvacuaforanU(N)theorywithNflavors.Wealsooutlineapro-gramleadingtoastringtheoryjustificationofthegaugetheory-matrixmodelcorrespondencewithfundamentalmatter.1IntroductionRecently,DijkgraafandVafahaveproposed[1,2,3]aperturbativemethodforcomputingtheeffectiveglueballsuperpotentialofseveralclassesofN=1theories.Theproposalinstructsonetocomputetheplanarfreeenergyofthematrixmodelwhosepotentialisthetree-levelsuperpotentialofthetheoryofinterest.Beingobtainedviaa“stringtheoryroute”,theoriginalproposalnatu-rallyincludestheorieswithfieldstransformingintheadjoint/bifundamentalrepresentationofthegaugegroup.Oneofthemostnaturalextensionsofthisdualityistotheorieswithfieldsinthefundamentalrepresentationofthegaugegroup.Itturnsoutthatthisgoalcanbereachedinarathersim-pleway:oneneedstocomputeonlythecontributiontothefreeenergyofthematrixmodelarisingfromFeynmandiagramswithoneboundary[9,12].Thus,thegaugetheoryeffectivesuperpotentialisconstructedasWeff(S,Λ)=NcS(1−lnSΛ3)+Nc∂Fχ=2∂S+NfFχ=1.(1)Suchaconstructionwassuccessfullyusedtocomparematrixmodelpre-dictionswithknowngaugetheoryresultsfortheorieswithmassiveandmass-lessflavors,withN=1andN=2supersymmetry[9,12,11,13,17,27].Otherinterestingrelatedworkhasappearedin[10,14,28,26,18,19,20,21,23,24,22].Ofamuchdeeperinteresthasbeentheinvestigationofthestructureunderlyingthisduality.In[4]itwasproventhatinagaugetheorywithadjointfields,thecomputationofthesuperpotentialisreproducedbythematrixcomputation,diagrambydiagram.Usinganomaly-basedarguments,in[15]itwasalsoshownthatthegaugetheoryandthematrixmodelresultsarerelatedasproposedbyDijkgraafandVafa.Inthispaperweproveusingfieldtheorytechniquesthattheeffectivesuperpotentialofatheorywithatreelevelsuperpotentialbeingagenericpolynomialinmulti-traceandbaryonicoperatorsisgeneratedbyplanardi-agramswithatmostone(generalized)boundary.Thisjustifiestheproposalputforwardin[12].Wethenproceedtoextendthegaugetheory-matrixmodeldualitytoincludebaryonicoperators1.Weobtainthefullmoduli1Aswewerepreparingthemanuscript,[25]appearedwhichhassomeoverlapwithourdiscussionofbaryons.1spaceofvacuaforanSU(N)theorywithNflavors.Wethenoutlineapro-gramforthestringtheoryjustificationoftheextensionsoftheDijkgraaf-Vafaduality.2Fieldtheoryanalysisoftheeffectivesuper-potentialInthissectionweanalyzeindetailtheeffectivesuperpotentialasafunctionoftheglueballsuperfieldS=Tr(W2)andthevariouscouplingconstantsthatexistinthetheory,andshowthatintheorieswithfieldstransforminginthefundamentalrepresentationofthegaugegroupthesuperpotentialisgeneratedentirelybyFeynmandiagramswithasingleboundary.Theanalysisissimilartotheonedescribedin[15].Wewillfirstconsidertreelevelsuperpotentialsbuiltoutoftracesofproductsofquarkbilinears.Thisanalysisprovestheproposalputforwardin[12],thatonlydiagramswithoneboundarycontributetotheeffectivesuperpotential.Wewillthenaddmulti-tracedeformationsaswellasbaryonicoperators.Webeginbyconsideringatheorywithafairlyarbitrary,polynomial,treelevelsuperpotential:WG=Xl≥1glTr[(Q˜Q)l],(2)wherethetraceisovertheflavorindices.Ifnotreelevelsuperpotentialispresent,theglobalsymmetryofthetheoryisSU(Nf)×SU(Nf)×QNfi−1U(1)i×QNfi−1eU(1)i×U(1)R.Theintroductionofthesuperpotentialabove,withscalarcouplings,breaksthissymmetrygrouptoSU(Nf)×U(1)i×eU(1)i×U(1)R.(3)Asinthecaseofatheorywithadjointfields[15],thesesymmetriesarenotsufficienttocompletelydeterminetheeffectivesuperpotential.Thisiseasytoseebyconsideringthefollowingratherunusualchargeassignmentforthevariousfieldsandcouplings:SU(Nf)U(1)gU(1)U(1)RΔQ:Nf1011˜Q:10111gl:1−l−l2−2l3−2lS:10023(4)2Thelastcolumninthetableaboverepresentstheengineeringdimensionofthecorrespondingfield.Asimplecountingargumentshowsthatallhigherloop2Feynmandia-gramswithinsertionsoftheglueballsuperfieldarefinite.Wearethuslook-ingforafunctionwhichdependsonlyonSandgi.Furthermore,sincethisfunctionisgeneratedperturbatively,neitheroneofitsargumentsisallowedtoappearatafractionalpower.ItisthennothardtoseethatthebasicinvariantcombinationsareglSl−1gl1(∀)l≥2(5)andtheyalsohavevanishingscalingdimension.Thus,themostgeneraleffectivesuperpotentialwhichcanbegeneratedisWeff=SF(glSl−1gl1)(6)wher