Noncommutative Gauge Theory on Fuzzy Sphere from M

arXiv:hep-th/0101102v422Mar2001January,2001KEK-TH-737TIT/HEP-462NoncommutativeGaugeTheoryonFuzzySpherefromMatrixModelSATOSHIISOa)c)1,YUSUKEKIMURAa)b)2,KANJITANAKAa)c)3andKAZUNORIWAKATSUKIa)d)4a)HighEnergyAcceleratorResearchOrganization(KEK),Tsukuba,Ibaraki305-0801,Japanb)DepartmentofPhysics,TokyoInstituteofTechnology,Oh-okayama,Meguro-ku,Tokyo152,Japanc)DepartmentofParticleandNuclearPhysics,TheGraduateUniversityforAdvancedStudies,Tsukuba,Ibaraki305-0801,Japand)DepartmentofPhysics,TokyoMetropolitanUniversity,Hachiohji,Tokyo192-0397,JapanAbstractWederiveanoncommutativeU(1)andU(n)gaugetheoryonthefuzzyspherefromathreedimensionalmatrixmodelbyexpandingthemodelaroundaclassicalsolutionofthefuzzysphere.Chern-Simonstermisaddedinthematrixmodeltomakethefuzzysphereasaclassicalsolutionofthemodel.Majoranamasstermisalsoaddedtomakeitsupersymmetric.WeconsidertwolargeNlimits,onecorrespondingtoagaugetheoryonacommutativesphereandtheothertothatonanoncommutativeplane.Wealsoinvestigatestabilityofthefuzzyspherebycalculatingone-loopeffectiveactionaroundclassicalsolutions.Inthefinalpartofthispaper,weconsideranothermatrixmodelwhichgivesasupersymmetricgaugetheoryonthefuzzysphere.Inthismatrixmodel,onlyChern-Simonstermisaddedandsupersymmetrytransformationismodified.1e-mailaddress:[e-mail]2e-mailaddress:[e-mail]3e-mailaddress:[e-mail]4e-mailaddress:[e-mail]11IntroductionToformulatenonperturbativeaspectsofstringtheoryorMtheory,severalkindsofmatrixmodelshavebeenproposed[1,2].TheseproposalsarebasedonthedevelopmentsofD-branephysics[3,4].IIBmatrixmodelisoneoftheseproposals[1].ItisalargeNreducedmodel[5]often-dimensionalsupersymmetricYang-MillstheoryandtheactionhasamatrixregularizedformoftheGreen-SchwarzactionofIIBsuperstring.ItispostulatedthatitgivesaconstructivedefinitionoftypeIIBsuperstringtheory.Inthematrixmodel,matterandevenspacetimearedynamicallyemergedoutofmatrices[6,7].Spacetimecoordinatesarerepresentedbymatricesandthereforenoncommu-tativegeometryappearsnaturally.Theideaofthenoncommutativegeometryistomodifythemicroscopicstructureofthespacetime.Thismodificationisimplementedbyreplacingfieldsonthespacetimebymatrices.Itwasshown[8,9,10,11]thatnoncommutativeYang-Millstheoriesinaflatbackgroundareobtainedbyexpandingthematrixmodelaroundaflatnoncommutativebackground.ThenoncommutativebackgroundisaD-brane-likeback-groundwhichisasolutionoftheequationofmotionandpreservesapartofsupersymmetry.VariouspropertiesofnoncommutativeYang-Millshavebeenstudiedfromthematrixmodelpointofview[12].Instringtheory,itisdiscussedthattheworldvolumetheoryonD-braneswithNS-NStwo-formbackgroundisdescribedbynoncommutativeYang-Millstheory[13].Adifferentkindofnoncommutativebackgrounds,anoncommutativesphere,orafuzzysphereisalsostudiedinmanycontexts.In[14],itisdiscussedintheframeworkofmatrixregularizationofamembrane.Inthelight-conegauge,theygaveamapbetweenfunctionsonsphericalmembraneandhermitianmatrices.InBFSSmatrixmodel[2],membranesofsphericaltopologyareconsideredin[23,24,25].Anoncommutativegaugetheoryonafuzzysphereinstringtheorycontextisdiscussedin[26,27].Theapproachtoconstructagaugetheoryonthefuzzyspherewerepursuedin[15,16,17,18,19,20,21,22].Thefuzzysphere[15]canbeconstructedbyintroducingacutoffparameterNforangularmomentumofthesphericalharmonics.ThenumberofindependentfunctionsisPNl=0(2l+1)=(N+1)2.Therefore,wecanreplacethefunctionsby(N+1)×(N+1)hermitianmatricesonthefuzzysphere.Thus,thealgebraonthefuzzyspherebecomesnoncommutative.Inthispaper,toconstructanoncommutativegaugetheoryonthefuzzysphere,wefirstconsiderathreedimensionalsupersymmetricreducedmodel.WeaddChern-SimonstermandMajoranaspinormasstermtotheactionoforiginalsupersymmetricmatrixmodel.AnactionwithbothofYang-MillsandChern-Simonstermsarealsoconsideredin[27].Althoughanordinarymatrixmodelhasonlyaflatbackgroundasaclassicalsolution,ourmatrixmodelcandescribeacurvedbackgroundowingtotheseterms.WealsostudyanothertypeofactioncontainingChern-SimonstermbutnotMajoranamassterm.Althoughthisactionisnotinvariantundertheoriginalsupersymmetry,itis2invariantunderamodifiedsupersymmetry.ItisalsoinvariantunderaconstantshiftofthefermionandhenceithasN=2supersymmetry.Thispaperisorganizedasfollows.Insection2,westudyamatrixmodelwhichgivesanoncommutativegaugetheoryonafuzzysphere.Wederiveanoncommutativegaugetheorybyexpandingthemodelaroundaclassicalsolutionwhichrepresentsafuzzysphere.Itisshownthatinacommutativelimit,itgivesastandardtheoryonacommutativesphere.Inthelatterpartofthissection,anotherlargeNlimitcorrespondingtoaspherewithalargeradiusbutthenoncommutativityfixedisconsidered.Bytakingthelimit,wecanobtainanoncommutativegaugetheoryinaflatnoncommutativebackground,whichisconsideredin[8].Insection3,somepropertiesofDiracoperatorandchiralityoperatorinthismodelareconsidered.Theseoperatorsbecomethecorrectoperatorsonacommutativesphereinacommutativelimit.Insection4,stabilityoftwoclsssicalsolutions,diagonalmatricesandthefuzzysphere,isexamined.One-loopeffectiveactioniscalculatedfortheclassicalsolutions.Fordiagonalmatrices,allsupersymmetryi