Non-Abelian Finite Gauge Theories

arXiv:hep-th/9811183v315Apr1999MIT-CTP-2803hep-th/9811183Non-AbelianFiniteGaugeTheoriesAmihayHananyandYang-HuiHehanany,[e-mail]hysics,MassachusettsInstituteofTechnologyCambridge,Massachusetts02139,U.S.A.November,1998AbstractWestudyorbifoldsofN=4U(n)super-Yang-Millstheorygivenbydiscretesub-groupsofSU(2)andSU(3).Wehavereachedmanyinterestingobservationsthathavegraph-theoreticinterpretations.ForthesubgroupsofSU(2),wehaveshownhowthemattercontentagreeswithcurrentquivertheoriesandhaveofferedapossibleexpla-nation.InthecaseofSU(3)wehaveconstructedacatalogueofcandidatesforfinite(chiral)N=1theories,givingthegaugegroupandmattercontent.Finally,wecon-jectureaMcKay-typecorrespondenceforGorensteinsingularitiesindimension3withmodularinvariantsofWZWconformalmodels.ThisimpliesaconnectionbetweenaclassoffiniteN=1supersymmetricgaugetheoriesinfourdimensionsandtheclassificationofaffineSU(3)modularinvariantpartitionfunctionsintwodimensions.1IntroductionRecentadvancesonfinitefourdimensionalgaugetheoriesfromstringtheoryconstruc-tionshavebeendichotomous:eitherfromthegeometricalperspectiveofstudyingalgebro-geometricsingularitiessuchasorbifolds[4][5][6],orfromtheintuitiveperspectiveofstudy-ingvariousconfigurationsofbranessuchastheso-calledbrane-boxmodels[7].(See[8]andreferencesthereinforadetaileddescriptionofthesemodels.Arecentpaperdiscussesthebendingofnon-finitemodelsinthiscontext[9].)Thetwoapproachesleadtotherealisationoffinite,possiblychiral,N=1supersymmetricgaugetheories,suchasthosediscussedin[10].Ourultimatedreamisofcoursetohavetheflexbilityoftheequivalenceandcomple-tionoftheseapproaches,allowingustocomputesay,thedualitygroupactingonthemodulispaceofmarginalgaugecouplings[11].(ThedualitygroupsfortheN=2supersymmetrictheorieswerediscussedinthecontextofthesetwoapproachesin[12]and[13].)Thebrane-boxmethodhasmetgreatsuccessinprovidingtheintuitivepicturefororbifoldsbyAbeliangroups:theellipticmodelconsistingofk×k′branesconvenientlyreproducesthetheoriesonorbifoldsbyZZk×ZZk′[8].OrbifoldsbyZZksubgroupsofSU(3)aregivenbyBraneBoxMod-elswithnon-trivialidentificationonthetorus[11][8].SincebythestructuretheoremthatallfiniteAbeliangroupsaredirectsumsofcyclicones,thisprocedurecanbepresumablyextendedtoallAbelianquotientsingularities.Thenon-Abeliangroupshowever,presentdifficulties.Byaddingorientifoldplanes,thedihedralgroupshavealsobeensuccessfullyattackedfortheorieswithN=2supersymmetry[14].Thequestionstillremainsastowhatcouldbedoneforthemyriadoffinitegroups,andthustogeneralGorensteinsingularities.InthispaperweshallpresentacatalogueoftheseGorensteinsingularitiesindimensions2and3,i.e.,orbifoldsconstructedfromdiscretesubgroupsofSU(2)andSU(3)whoseclassificationarecomplete.Inparticularweshallconcentrateonthegaugegroup,thefermionicandbosonicmattercontentresultingfromtheorbifoldingofanN=4U(n)super-Yang-Millstheory.InSection2,wepresentthegeneralargumentsthatdictatethemattercontentforarbitraryfinitegroupΓ.TheninSection3,westudythecaseofΓ⊂SU(2)wherewenoticeinterestinggraph-theoreticdescriptionsofthemattermatrices.Weanalogouslyanalysecasebycase,thediscretesubgroupsofSU(3)inSection4,followedbyabriefdigressionofpossiblemathematicalinterestinSection5.ThisleadstoaMckay-typeconnectionbetweentheclassificationoftwodimensionalSU(3)kmodularinvariantpartitionfunctionsandtheclassoffiniteN=1supersymmetricgaugetheoriescalculatedinthispaper.FinallywetabulatepossiblechiraltheoriesobtainablebysuchorbifoldingtechniquesfortheseSU(3)subgroups.2TheOrbifoldingTechniquePromptedbyworksbyDouglas,Greene,MooreandMorrisonongaugetheorieswhicharisebyplacingD3branesonorbifoldsingularities[1][2],[3],KachruandSilverstein[4]andsubsequentlyLawrence,NekrasovandVafa[5]notedthatanorbifoldtheoryinvolvingtheprojectionofasupersymmetricN=4gaugetheoryonsomediscretesubgroupΓ⊂SU(4)leadstoaconformalfieldtheorywithN≤4supersymmetry.Weshallfirstbrieflysummarisetheirresultshere.WebeginwithaU(n)N=4super-Yang-MillstheorywhichhasanR-symmetryofSpin(6)≃SU(4).TherearegaugebosonsAIJ(I,J=1,...,n)beingsingletsofSpin(6),alongwithadjointWeylfermionsΨ4IJinthefundamental4ofSU(4)andadjointscalarsΦ6IJintheantisymmetric6ofSU(4).Thenwechooseadiscrete(finite)subgroupΓ⊂SU(4)withthesetofirreduciblerepresentations{ri}actingonthegaugegroupbybreakingtheI-indicesupaccordingto{ri},i.e.,byLiri=LiCNirisuchthatCNiaccountsforthemultiplicityofeachriandn=Pi=1Nidim(ri).Inthestringtheorypicture,thisdecompositionofthegaugegroupcorrespondstopermutingnD3-branesandhencetheirChan-Patonfactors2whichcontaintheIJindices,onorbifoldsofIR6.SubsequentlybytheMaldecenalargeNconjecture[15],wehaveanorbifoldtheoryonAdS5×S5,withtheR-symmetrymanifestingastheSO(6)symmetrygroupofS5inwhichthebranesnowlive[4].Thestringperturbativecalculationinthiscontext,especiallywithrespecttovanishingtheoremsforβ-functions,hasbeenperformed[6].Havingdecomposedthegaugegroup,wemustlikewisedosoforthematterfields:sinceanorbifoldisinvariantundertheΓ-action,weperformtheso-calledprojectiononthefieldsbykeepingonlytheΓ-invariantfieldsinthetheory.Subsequentlywearriveata(supercon-formal)fieldt