The Square of the Dirac and spin-Dirac Operators o

arXiv:math-ph/0703052v229Nov2007TheSquareoftheDiracandspin-DiracOperatorsonaRiemann-CartanSpace(time)∗E.A.Notte-CuelloDepartamentodeMatem´aticas,UniversidaddeLaSerena,Av.Cisterna1200,LaSerena-Chile.e-mail:[e-mail],StatisticsandScientificComputation,IMECC-UNICAMPCP6065,13083-859Campinas,SP,Brazil.e-mail:[e-mail]ics,StatisticsandScientificComputation,IMECC-UNICAMPCP6065,13083-859Campinas,SP,[e-mail]7,2008AbstractInthispaperweintroducetheDiracandspin-Diracoperatorsasso-ciatedtoaconnectiononRiemann-Cartanspace(time)andstandardDiracandspin-DiracoperatorsassociatedwithaLevi-CivitaconnectiononaRiemannian(Lorentzian)space(time)andcalculatethesquareoftheseoperators,whichplayanimportantroleinseveraltopicsofmodernMathematics,inparticularinthestudyofthegeometryofmodulispacesofaclassofblackholes,thegeometryofNS-5branesolutionsoftypeIIsupergravitytheoriesandBPSsolitonsinsomestringtheories.Weob-tainageneralizedLichnerowiczformula,decompositionsoftheDiracandspin-DiracoperatorsandtheirsquaresintermsofthestandardDiracandspin-Diracoperatorsandusingthefactthatspinorfields(sectionsofaspin-Cliffordbundle)haverepresentativesintheCliffordbundlewepresentalsoanoticeablere-lationinvolvingthespin-DiracandtheDiracoperators.Keywords:Spin-Cliffordbundles,DiracOperator,LichnerowiczFormula∗toappear:ReportsonMathematicalPhysics60(1),135-157(2007).11IntroductionRecently,inseveralapplicationsoftheoreticalphysicsanddifferentialgeometry,inawayoranothertheDiracoperatoranditssquareonaRiemann-Cartanspace(time)hasbeenused.In,e.g.,[12]RapoportproposedtogiveaCliffordbundleapproachtohistheoryofgeneralizedBrownianmotion;in[1]AgricolaandFriedrichinvestigatetheholonomygroupofalinearmetricconnectionwithskew-symmetrictorsionandin[2]theyintroducedalsoanelliptic,second-orderoperatoractingonaspinorfield,andinthecaseofanaturallyreductivespacetheycalculatedtheCasimiroperatoroftheisometrygroup.Thesquareofthespin-Diracoperatoralsoappearsnaturallyinthestudyofthegeometryofmodulispacesofaclassofblackholes,thegeometryofNS-5branesolutionsoftypeIIsupergravitytheoriesandBPSsolitonsinsomestringtheories([5])andmanyotherimportanttopicsofmodernmathematics(see[3,6]).Someoftheworksjustquotedpresentextremelysophisticatedandreallycomplicatedcalculationsandsometimesevenerroneousones.ThisbringstomindthatasimpletheoryofDiracoperatorsandtheirsquaresactingonsectionsoftheCliffordandSpin-CliffordbundlesonRiemann-Cartanspace(times)hasbeenpresentedin[15],andfurtherdevelopedin[14].Usingthattheory,inSection2wefirstintroducethestandardDiracoperator∂|(as-sociatedwithaLevi-CivitaconnectionDofametricfieldg)actingonsectionsoftheCliffordbundleofdifferentialformsCℓ(M,g)andnext,insection2.1weintroducetheDiracoperator∂(associatedwithanarbitraryconnection∇)andalsoactingonsectionsofCℓ(M,g).Next,wecalculateinSection2.2inasimpleanddirectwaythesquareoftheDiracoperatoronRiemann-Cartanspace,andthenspecializetheresultforthesimplestcaseofascalarfunctionf∈sec^0T∗M֒→secCℓ(M,g)inordertocompareourresultswiththeonespresentedin[12].Wegivetwocalculations,oneusingthedecompositionoftheDiracoperatorintothestandardDiracoperatorplusatermdependingonthetorsiontensor(seeEq.(16)below)andanotherone,whichfollowsdirectlyfromthedefinitionoftheDiracoperatorwithoutusingthestandardDiracop-erator.NextwepresentarelationbetweenthesquareoftheDiracandthestandardDiracoperators(actingonascalarfunction)intermsofthetorsiontensorandinvestigatealsoinSection2.3therelationbetweenthoseoperatorsinthecaseofanullstraintensor.InSection3,wepresentabriefsummaryofthetheoryoftheSpin-Cliffordbundles(CℓℓSpine1,3(M,g)andCℓrSpine1,3(M,g))andtheirsections(spinorfields)andtheirrepresentativesinaCliffordbundle(Cℓ(M,g))following[10,14].WerecallinSection3.1someimportantformulasfromthegeneraltheoryofthecovariantderivativesofCliffordandspin-CliffordfieldsandinSection3.2werecallthedefinitionofthespin-Diracoperator∂s(associatedwithaRiemann-Cartanconnection∇)actingonsectionsofaspin-Cliffordbundle.Insection3.3weintroducetherepresentativesofspinorfieldsintheCliffordbundleandtheimportantconceptoftherepresentativeof∂s(denoted∂(s))thatactsontherepresentativesofspinorfields(see[10,14]fordetails).TomakeclearthesimilaritiesanddifferencesbetweenCℓ(M,g)and2CℓℓSpine1,3(M,g),wewrite,inSection4,Maxwellequationinbothformalisms.InSection5.1wefirstfindthecommutatorofthecovariantderivativeofspinorfieldsonaRiemann-Cartanspace(time)andcompareourresultwithonethatcanbefoundin[11],whichseemstoneglectaterm.NextinSection5.2wecalculatethesquareofthespin-DiracoperatoronaRiemann-CartanspacetimeandfindageneralizedLichnerowiczformula.InSection6,takingadvantagethatanyψ∈secCℓℓSpine1,3(M,g)canbewrittenasψ=A1ℓΞwithA∈secCℓ(M,g)and1ℓΞ∈secCℓℓSpine1,3(M,g)wefindtwonoticeableformulas:thefirstrelatesthesquareofthespin-Diracoperator(θa∇sea)actingonψwiththesquareoftheDiracoperator(θa∇ea)actingonA;thesecondformularelatesthesquareofthespin-Diracoperator(θa∇sea)actingonψwiththesquareofthestandardDiracoperator(θaDea).InSection7