D=11 massless superparticle covariant quantization

arXiv:0710.4342v4[hep-th]1Apr2008IFIC-07-65,FTUV-07-2310,hep-th/0710.4342publishedinNucl.Phys.B796,360–401(2008)D=11masslesssuperparticlecovariantquantization,purespinorBRSTchargeandhiddensymmetriesIgorA.BandosDepartamentodeF´ısicaTe´orica,Univ.deValenciaandIFIC(CSIC-UVEG),46100-Burjassot(Valencia),SpainandInstituteforTheoreticalPhysics,NSC“KharkovInstituteofPhysicsandTechnology”,UA61108,Kharkov,UkraineAbstractWeconsiderthecovariantquantizationoftheD=11masslesssuperparticle(M0–brane)inthespinormovingframeortwistor-likeLorentzharmonicsformulation.Theactioninvolvesthesetof16constrained32componentMajoranaspinors,thespinorLorentzharmonicsv−αqparametrizing(ashomogeneouscoordinates,modulogaugesymmetries)thecelestialsphereS9.Therepresenceallowsustoseparatecovariantlythefirstandthesecondclassconstraintsofthemodel.Weshowthat,aftertakingintoaccountthesecondclassconstraintsbymeansofDiracbracketsandafterfurtherreducingthefirstclassconstraintsalgebra,thesystemisdescribedintermsofasimpleBRSTchargeQsusyassociatedtothed=1,n=16supersymmetryalgebra.ThestudyofthecohomologyofthisBRSToperatorrequiresaregularizationbycomplexifyingthebosonicghostsfortheκ–symmetry,λq,andfurtherreductionoftheregularizedcohomologyproblemtotheoneforasimplercomplexBRSTcharge˜Qsusy.ThislatterisessentiallythepurespinorBRSTchargebyBerkovits,butwithacompositepurespinorconstructedfromthecomplexd=9spinorwithzeronorm,˜λq,andthespinorialharmonicsvα−q.Thisexhibitsapossibleoriginofthecomplexity(non-hermiticity)characteristicoftheBerkovitspurespinorapproach.ThesimplestructureofthenontrivialcohomologyoftheM0–braneBRSTchargeQsusyfindsexplanationinthepropertiesthatthesuperparticleactionexhibitsintheso-called‘covariantizedlight–cone’basis,wheretheM0-braneactionisexpressedintermsofκ–symmetryinvariantvari-ables.Thesetofgaugesymmetriesinthisbasisreducestothe[SO(1,1)×SO(9)]⊂×K9BorelsubgroupofSO(1,10).Imposingtheirgeneratorsasconditionsonthesuperparticlewavefunctions,wearriveatthecovariantquantizationintermsofphysicaldegreesoffreedomwhichhintspossiblehiddensymmetriesofD=11supergravity.BesidesSO(16),whichinthetwistorlikeLorentzharmonicformulationisseenalreadyattheclassicallevel,wediscussalsosomeindirectargumentsinfavorofthepossibleE8symmetry.Keywords:Supersymmetry,superparticle,covariantquantization,BRST,Lorentzharmonics,twistors,supergravityPACs:11.30.Pb,11.25.-w,04.65.+e,11.10.KkContents1Introductionandsummary21.1Introduction.........................................21.2Summaryofthemainresults...............................31.3Structureofthepaper...................................42TheM0-braneinthespinormovingframeformulation.Twistor–likeactionanditsgaugesymmetries.52.1TowardsthespinormovingframeactionfortheD=11masslesssuperparticle....52.2Twistor–likespinormovingframeactionofM0–braneanditsgaugesymmetries...62.3OnO(16)gaugesymmetry.................................72.4VectorandspinorLorentzharmonics:movingframeandspinormovingframe....82.4.1OnharmonicsandexplicitparametrizationofSO(1,D−1)/Hcosets....93HamiltonianmechanicsoftheD=11superparticleinthespinormovingframeformulationandtheBRSTchargeQsusy103.1PrimaryconstraintsoftheD=11superparticlemodel(M0–brane)..........103.2DiracbracketsinHamiltonianmechanicsontheSO(1,D−1)groupmanifold....113.3CartanformsandHamiltonianmechanicsontheLorentzgroupmanifold......123.4CanonicalHamiltonianandPoisson/DiracbracketsoftheM0–branemodel.....143.5SecondclassconstraintsoftheD=11superparticlemodel...............153.6Firstclassconstraintsandtheir(nonlinear)algebra...................163.7BRSTchargeforanonlinearsub(super)algebraofthefirstclassconstraints.....173.8ThefurtherreducedBRSTchargeQsusy.........................184BRSTquantizationoftheD=11superparticle.CohomologyofQsusyandtheoriginofthecomplexityoftheBerkovitsapproach194.1QuantumBRSTchargeQsusy...............................194.1.1ThenontrivialcohomologyofQsusyislocatedatλ+qλ+q=0..........194.2Cohomologiesatvanishingbosonicghost........................204.3RelationwiththeBerkovits’spurespinors........................224.4Cohomologyof˜λ+qD−q...................................225M0–braneanditsquantizationinthecovariantizedlight–conebasis.245.1OnBRSTquantizationofM0–braneinthecovariantizedlightconebasis.......255.1.1Hamiltonianmechanicsinthecovariantizedlight–conebasis.............255.1.2BRSTchargeforthefirstclassconstraintsinthecovariantizedlight–conebasis265.2Covariantquantizationofthephysicaldegreesoffreedomandhintsofhiddensym-metries............................................266Conclusionsandoutlook306.1Conclusions.........................................306.2Outlook1:onBRSTchargeforsuperstring.......................316.3Outlook2:SO(16),E8andalthat............................3211Introductionandsummary1.1IntroductionAcovariantquantizationofthemasslessD=11superparticle(see[1,2])hasbeenrecentlyconsidered[3]initstwistor-likeLorentzharmonicsformulation[4](seealso[5,6,7,8]).Thisnew,covariantsupertwistorquantizationledstothelinearizedD=11supergravitymultipletinthespectrum(inagreementwiththelight–co