The co-Riemannian Structure of Smooth Loop Spaces

arXiv:0809.3108v1[math.DG]18Sep2008Theco-RiemannianStructureofSmoothLoopSpacesAndrewStaceySeptember18,2008AbstractWeconstructanaturalco-RiemannianstructureonthemanifoldofsmoothloopsinaRiemannianmanifold.Weshowthatthesmoothloopspaceofastringmanifoldisaper-Hilbert–Schmidtlocallyequivalentco-spinmanifoldandthusadmitsaDiracoperator.1IntroductionIn[Sta]weintroducedtheconceptofaco-orthogonalstructureonaninfinitedimensionalvectorbundleandshowedhowthisledtotheconstructionofaninfinitedimensionalDiracoperator.Inbrief,whilstanorthogonalstructureonavectorbundledefinesaHilbertcompletionofeachfibre,aco-orthogonalstructuredefinesadenseHilbertsubspace.InthispaperweshallshowthatthespaceofsmoothloopsinaRiemannianmanifoldadmitsaco-Riemannianstructure—thatis,aco-orthogonalstructureonitstangentbundle—andthatthisissuitablefortheconstructionoftheDiracoperatortowork.Weclassifiedco-orthogonalandorthogonalstructuresin[Sta]accordingtocertaincriteria—refiningthe“weak–strong”classification.Intermsofthisclassificationweprovethefollowingtheorem.TheoremALetMbeafinitedimensionalRiemannianmanifold.LetLM≔C∞(S1,M)bethespaceofsmoothloopsinM.Thisspaceadmitsanuclear,locallyequivalent,co-RiemannianstructurewithstructuregroupGlres.Thisdefinitionhasthefollowinginterpretation.ThereisabundleofHilbertspaces,sayE→LM,withstructuregroupGlresandavectorbundlemapE→TLM.TherearesimultaneoustrivialisationsofEandofTLMwithrespecttotheirstructuregroupssuchthatE→TLMlocallylooksliketheinclusionofafixedHilbertspaceinthemodelspaceofTLM(thismodelspacebeingLRn)andthatthisinclusionisanuclearmap.Underthislocaltrivialisation,theinnerproductonEisnottakentosomefixedinnerproductonthetypicalfibreofE.ThegroupGlresistherestrictedgenerallineargroupof[PS86].TheoremAreferstoTLMwithitsstandardstructuregroup,namelyLGln.Ifwearepreparedtoworkwithaslightlylargertopologicalgroupthenwecanreplacethewords“locallyequivalent”by“locallytrivial”inTheoremA.ThismeansthatunderthesimultaneoustrivialisationofEandTLM,theinnerproductonEistakentosomefixedinnerproductonthetypicalfibreofE.1Havingshownthis,itisstraightforwardtoshowthatifMisastringmani-foldthenLMhastherequiredstructuretodefineaDiracoperator.TheoremBLetMbeafinitedimensionalstringmanifold.ThenLMisanS1–equivariant,per-Hilbert–Schmidt,locallyequivalent,co-spinmanifold.Again,ifwearepreparedtomodifythestructuregroupofthetangentbundleofLMwehavealocallytrivialco-spinmanifold.Combinedwiththeworkof[Sta]thisyieldsthefollowingimportantcorol-lary.CorollaryCLetMbeafinitedimensionalstringmanifold.ThenLMadmitsanS1–equivariantDiracoperator.Asmentionedabove,aco-orthogonalstructureonaninfinitedimensionalvectorbundleassignstoeachfibreadensesubspacewiththestructureofaHilbertspace—with,almostcertainly,astrictlyfinertopologythanthesubspacetopology.Theclassificationschemeintroducedin[Sta]measureshowmuchitispossibletolocallytrivialisethisstructure,whethertheseHilbertspacesfittogethertoformabundleintheirownright(andwithwhatstructuregroup),andwhetherthemapfromtheHilbertspacetothelargerspacehasanynicepropertiessuchascompactness.Thisclassificationschemeworksequallywellfororthogonalstructuresasco-orthogonalstructures.Toillustratethisclassification,thestandardRiemannianstructureonLMisanuclear,locallytrivial,orthogonalstructurewithstructuregroupOres(therestrictedorthogonalgroup).ThismeansthatthereisabundleofHilbertspaces,sayF→LM,withstructuregroupOresandavectorbundlemapTLM→F.TherearetrivialisationsofFandofTLMwithrespecttotheirstructuregroupssuchthatTLM→FlocallylooksliketheinclusionofLRninafixedHilbertspace(L2Rn)andthatthisinclusionisanuclearmap.Inaddition,thislocaltrivialisationmapstheinnerproducttothestandardoneonL2Rn.Thedifference,therefore,isthelocaltrivialityoftheinnerproduct;thisalsorelatestothefactthatwecanreducethestructuregroupoftheHilbertbundleofOresratherthanGlres.ThepointisthatthereisonlyonereasonableloopspaceonwhichLOnactsisometricallyandthatisL2Rn.Ofcourse,wecouldalwaysreducethestructuregroupofEtoOresbutwecouldn’ttrivialiseEwithrespecttoOresatthesametimeastrivialisingTLMwithrespecttoitsstructuregroup,LGln.ThepurposeofintroducingthelargerstructuregroupthatwementionaboveistoenableustotrivialiseEwithrespecttoOresandTLMwithrespecttothislargergroupsimultaneously.Weshallgivetheconstructioninthreestages.Thefirststage,insection2,isthelinearcase.Astheco-orthogonalstructurethatwewishtoconstructisintendedtohavesomelocaltrivialityproperties,weneedtodecideonareferencestructure.Thatis,weneedtochooseafixedHilbertsubspaceofLRn.Thisisnotdifficult:wetakethoseloopswhichareanalyticonanannulusofradius(r−1,r)forsomefixedr∈(1,∞)andaresquareintegrableontheboundary.WeshalldenotethisspacebyL2rRn.Wealsoneedtoinvestigatewhatgroupactsonthisspace.ItiseasytoshowthatLGln(R)doesnotpreserveanyHilbertsubspaceofLRnbutwecanreduceLGln(R)toLpolOn,thegroupofpolynomialloopsinOn,whichdoes.WealsoshowthattheactionofLpolOnonL2rRnfactorsthroughtherestricted2generallineargroup.ThereisanobviousisomorphismofL2rRnwithL2Rnandwedescribehowthisfitsintothemixture.Thesecondstage,insection3,istheuniversalcase.LetGbeaconnected,compactLiegroup.Let