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arXiv:0807.3381v2[hep-th]6Aug2008July,2008OnN=2supergravityandprojectivesuperspace:DualformulationsSergeiM.Kuzenko1SchoolofPhysicsM013,TheUniversityofWesternAustralia35StirlingHighway,CrawleyW.A.6009,AustraliaDedicatedtoProfessorI.L.BuchbinderOntheOccasionofHis60thBirthdayAbstractThesuperspaceformulationforfour-dimensionalN=2matter-coupledsuper-gravityrecentlydevelopedin[1]makesuseofanewtypeofconformalcompensatorwithinfinitelymanyoff-shelldegreesoffreedom:theso-calledcovariantweight-onepolarhypermultiplet.Inthepresentnoteweprovethedualityofthisformulationtotheknownminimal(40+40)off-shellrealizationforN=2Poincar´esupergrav-ityinvolvingtheimprovedtensorcompensator.Withinthelatterformulation,wepresentnewoff-shellmattercouplingsrealizedintermsofcovariantweight-zeropo-larhypermultiplets.WealsoelaborateupontheprojectivesuperspacedescriptionofvectormultipletsinN=2conformalsupergravity.Analternativesuperspacerep-resentationforlocallysupersymmetricchiralactionsisgiven.Wepresentamodelformassiveimprovedtensormultipletwithboth“electric”and“magnetic”typesofmassterms.1kuzenko@cyllene.uwa.edu.au1IntroductionRecently,wehavedevelopedthesuperspaceformulationforfour-dimensionalN=2matter-coupledsupergravity[1],extendingtheearlierconstructionfor5DN=1super-gravity[2,3].Fromthepurelygeometricalpointofview,thisapproachmakesuseofGrimm’scurvedsuperspacegeometry[4],whichisperfectlysuitabletodescribeN=2conformalsupergravityandhasasimplerelationtoHowe’ssuperspaceformulation[5].Kinematically,matterfieldsin[1]aredescribedintermsofcovariantprojectivesuper-multipletswhicharecurved-spaceversionsofthesuperconformalprojectivemultiplets[6]livinginrigidprojectivesuperspace[7,8].InadditiontothelocalN=2superspacecoordinateszM=(xm,θμi,¯θi˙μ),wherem=0,1,···,3,μ=1,2,˙μ=1,2andi=1,2,suchasupermultiplet,Q(n)(z,u+),dependsonauxiliaryisotwistorvariablesu+i∈C2\{0},withrespecttowhichQ(n)isholomorphicandhomogeneous,Q(n)(cu+)=cnQ(n)(u+),onanopendomainofC2\{0}(theintegerparameterniscalledtheweightofQ(n)).Inotherwords,suchsuperfieldsareintrinsicallydefinedinCP1.Thecovariantprojectivesupermultipletsarerequiredtobeannihilatedbyhalfofthesupercharges1D+αQ(n)=¯D+˙αQ(n)=0,D+α:=u+iDiα,¯D+˙α:=u+i¯Di˙α,(1.1)withDA=(Da,Diα,¯D˙αi)thecovariantsuperspacederivatives.Intheapproachof[1],thedynamicsofsupergravity-mattersystemsisdescribedbyalocallysupersymmetricactionoftheform:S=12πI(u+du+)Zd4xd4θd4¯θEW¯WL++(Σ++)2,E−1=Ber(EAM),(1.2)whereΣ++:=14(D+)2+4S++W=14(¯D+)2+4eS++¯W=Σiju+iu+j.(1.3)HeretheLagrangianL++(z,u+)isacovariantrealprojectivemultipletofweighttwo,W(z)isthecovariantlychiralfieldstrengthofanAbelianvectormultiplet(i.e.thefirstsuperconformalcompensator),suchthatthebodyofW(z)iseverywherenon-vanishing,S++(z,u+)=Sij(z)u+iu+jandeS++(z,u+)=¯Sij(z)u+iu+jarespecialdimension-1compo-nentsofthesupertorsion;see[1]formoredetail.Theaction(1.2)canbeshowntobeinvariantunderthesupergravitygaugetransformations,andisalsomanifestlysuper-Weylinvariant.ItcanalsoberewrittenintheequivalentformS=12πI(u+du+)Zd4xd4θd4¯θEL++S++eS++(1.4)1Intherigidsupersymmetriccase,suchconstraintsinisotwistorsuperspaceR4|8×CP1wereintroducedfirstbyRosly[9],andlaterbytheharmonic[10,11]andprojective[7,8]superspacepractitioners.1inwhich,however,thesuper-Weylinvarianceisnotmanifest.In[1],wepresentedafamilyofsupergravity-mattersystemsinwhichthematterhyper-multipletsaredescribedbycovariantweight-zeropolarmultiplets2,andthesecondsuper-conformalcompensatorisidentifiedwithacovariantweight-onepolarmultiplet.Thisisanewtypeofsupergravitycompensator,althoughitisrelatedtotheq+-hypermultipletcompensatorwhichemergeswithintheharmonicsuperspaceapproach[13,11]to4DN=2supergravity,e.g.,inthesenseof[14].Inthepresentpaper,wewishtostudydu-alityofthesupergravityformulationgivenin[1]to(oneof)theminimal(40+40)off-shellformulationsforN=2Poincar´esupergravityconstructedinthe1980s.Theseformu-lationsareobtainedbycouplingtheminimalfieldrepresentationwith32+32off-shelldegreesoffreedom[15](thatistheWeylmultiplet[16,17,18]coupledtoanAbelianvec-tormultiplet,thelatterbeingthefirstsuperconformalcompensator)tovariousoff-shellversionsforthesecondcompensatorwith(8+8)degreesoffreedom.Theseinclude:(i)the“standard”minimalrealizationwithanonlinearmultiplet[19,16];(ii)thealternativeformulationinvolvinganoff-shellhypermultipletwithintrinsiccentralcharge[20];(iii)thenewminimalrealizationwithanimprovedtensormultiplet[21].Superspacerealizationsforthesesupergravityformulationshavebeenstudied,e.g.,in[22,5,11].OuranalysiswillbespecificallyconcernedwiththedualitybetweenthethirdminimalPoincar´esuper-gravity[21]andthesupergravityformulationgivenin[1].ThepointisthattheformerisknowntobeanalogoustothenewminimalN=1Poincar´esupergravity[23].Wearegoingtodemonstratebelowthattheprojective-superspaceformulation[1]isanalogoustotheoldminimalN=1Poincar´esupergravity[24,25].3Thispaperisorganizedasfollows.Insection2,weelaborateupontheprojective-superspacedescriptionofAbelianvectormultipletsinN=2conformalsupergravity,andpresentanalternativesuperspacerepresentationforlocallysupersymmetricchiralactio
本文标题:On N = 2 supergravity and projective superspace Du
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