On a Multisymplectic Formulation of the Classical

arXiv:math-ph/9901013v120Jan1999ONAMULTISYMPLECTICFORMULATIONOFTHECLASSICALBRSTSYMMETRYFORFIRSTORDERFIELDTHEORIESPARTII:GEOMETRICSTRUCTURES.S.P.HRABAK.Abstract.AgeometricmultisymplecticformulationoftheclassicalBRSTsymmetryofconstrainedfirst-orderclassicalfieldtheoriesisdescribed.Toeffectthisweintroducegradedanaloguesofthebundlesandmanifoldsofthemultisymplecticformulationoffirst-orderfieldtheories.TheLagrange-d’Alembertformalismisalsodevelopedintermsofthemultisymplecticframe-work.TheresultisacovariantHamiltonianBFVformalism.Inanaccompanyingprecedingpaper[9]theauthordetailedthehomologicalalgebrawhichprovidedanalgebraicdescriptionoftheMarsden-Weinsteinmulti-symplecticreduction.Thestudyofmultisymplecticgeometry[1,2,3,13,23]aroseinthecontextofthesearchforthegeometricfoundationsofclassicalfieldtheory[17,18,19,20].Weturninthispapertoastudyofthosemultisymplecticman-ifoldswhichformthegeometricfoundationsoffirstorderclassicalfieldtheories.WedevelopageometricmultisymplecticformalismdescribingtheclassicalBRST1symmetryofconstrainedfirst-orderclassicalfieldtheorieswhosesymmetriesarisebyvirtueoftheprolongationofagroupactionbybundlemorphismsoftheconfig-urationbundle.TheessenceofthegeometricformulationistoencodethealgebraicstructuresofthehomologicaldescriptionofMarsden-Weinsteinmultisymplecticre-ductionintogeometricstructuresongraded-multisymplecticmanifolds.1.InthefirstsectionweintroducetheLagrangemultiplierintothemultisym-plecticframework.Weeffectthisbyvirtueoftheintroductionofacertainintegrabledistributionwhoseexistencefollowsbyvirtueoftheassumedfreeandpropergroupactionbybundleautomorphismsontheconfigurationbun-dle.Wemakeuseofthisdistributioninordertodefineaconfigurationbun-dle,multiphasespaceandcovariantphasespaceextendedbytheadditionofaLagrangemultiplieranditscanonicalmomenta.WethengeneralisetheLagrange-d’AlembertformalismtothemultisymplecticcontextwhenceweobtainboththegeneralisedLagrange-d’Alembert-HamiltonianequationsofmotionandtheconservationofthecovariantNoethercurrentsfromasingleelegantgeometricequation.Date:January20,1999.1991MathematicsSubjectClassification.Primary:53Secondary:70.Keywordsandphrases.Multisymplecticgeometry,classicalfieldtheories,BRSTsymmetry,gradedmanifolds,gradedalgebras,Leibnizalgebras.ResearchsupportedbyPPARC.1TheclassicalBRSTsymmetryisaGrassmann-oddsymmetry(symplectomorphism)whichwasoriginallyintroducedwithintheframeworkofthequantisationofconstraineddynamicalsystems,onlylaterwasitsrelationtoreductionintheclassicalcontextunderstood[21,8].12S.P.HRABAK.2.Inthesecondsectionweconstructgradedanaloguesoftheconfigurationbun-dle,themultiphasespaceandthecorrespondingcovariantphasespace,ofthemultisymplecticformalism.Thegeometricconstructionsdescribedherefor-malisetheintroductionoftheGrassmann-odddegreesoffreedomknownintheliteratureasghosts.3.Inthethirdsectionweshallcompletethetranslationofthealgebraicstruc-turesneededinthehomologicaldescriptionofMarsden-Weinsteinmultisym-plecticreductionintogeometricstructuresongradedmultisymplecticmani-folds.WeintroduceacertainAbelianGrassmann-oddbundlemorphismofthegradedconfigurationbundlewhichuponprolongationtothegradedcovariantphasespaceistheclassicalBRSTsymmetry.UponprojectionoftheAbelianGrassmann-oddbundlemorphismtotheconfiguration-bundleoneregainstheoriginalnon-Abeliangaugeautomorphismoftheconfigurationbundle.TheobservablesonthegradedmultisymplecticmanifoldformaZ2×Z2-gradedPoisson-Leibnizalgebra.Weidentifythealgebraicdifferentialcomplexof[9]withapairconsistingofasubalgebraofthealgebraofobservablesandthePoisson-LeibnizderivationgeneratedbyaGrassmann-odd(n-1)-form.TheGrassmann-odd(n-1)-formistheliftedmomentumobservablecorrespondingtotheAbelianGrassmann-oddbundlemorphismofthegradedconfigura-tionbundle.Theobservablesonthereducedmultisymplecticmanifoldareobtainedasthezerothhomologyofthisgeometricdifferentialcomplex.4.InthefourthsectionwecombinetheLagranged’AlembertformalismwiththegradedmultisymplecticformalisminordertoobtainacovariantHamiltonianBFVformalism.5.InthefifthsectionweshallillustratethenewformalismbyderivinganewBRSTalgebraforthewellknownexemplarofYangandMills.ThenoveltyinourcovariantHamiltonianapproachisthattheresultingBRSTalgebraispolynomialinthecanonicalvariables,unlikethenon-covariantapproachwhereonefindsthealgebratoincludespatialderivativesofthecanonicalvariables.6.Weconcludeinthefinalsectionwithanappendixcontainingabriefdescrip-tionofthoseelementsofthemultisymplecticformulationoffirstorderfieldtheoriesprerequisitetotheexpositioninthispaper.WethusobtainacovariantHamiltoniangeometricformulationoftheclassicalBRSTsymmetryforfirstorderfieldtheories.1.AMultisymplecticLagrange-d’AlembertFormalism.TheBFVformulation[11]oftheclassicalBRSTsymmetryincludesapairofdynamicalvariableswhicharetheLagrangemultiplieroftheLagrange-d’AlembertvariationalprincipleandcanonicallyconjugatemomentawhichisprescribedtovanishsothattheintroductionoftheLagrangemultiplierdoesnotintroduceextradynamicaldegreesoffreedom.SinceweshouldliketoconstructamultisymplecticformulationoftheclassicalBRSTsymmetryitisthereforenecessary