Geometric Quantization

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1、GEOMETRICQUANTIZATION1.PRINCIPALBUNDLESANDVECTORBUNDLESLetXbeatopologicalspace.Denition1.1.AberbundleoverXwithberFisacontinuoussurjectionp:E!Xwhichislocallytrivialinthefollowingsense:eachx2XhasanopenneighbourhoodUforwhichthereisahomeomorhismf:p1(U)!UF,whichmakesthefollowingdiagramcommutefp1(U)

2、-UFpproj?+1Uwhereproj1istheprojectionontotherstfactor.Noticethatitfollowsthatp1(x)=Fforeachx2X.WhenXismanifold,itisnaturaltorequireamanifoldstructureonEandsmoothnessofthestructuremaps.Wewilldothatfromnowon.LetGbeaLiegroup.Denition1.2.AprincipalG-bundleisaberbundlep:P!Xwithafreeberwiserig

3、htactionofGonPsuchthatP/G=X.ItfollowsthatGcanbetakentobetheberofthebundle.Amorphismofprincipalbundlesisasmoothmapf:P1!P2whichcommuteswiththerightG-action.AsectionsofaprincipalG-bundeisasmoothmaps:X!Psatisfyingps=identity.Thereisa“cocycleview”onprincipalbundlesoverXasfollows:bydenition,wecanndan

4、opencoveringfUaga2IofXsuchthatPhaslocaltrivialisationsfa:p1(Ua)!UaG.Twolocaltrivializations(Ua,fa),and(Ub,fb)deneasmoothmapgab:UaUb!G,by(f1afb)(x,g)=(x,ggab(x)).Thesefunctionsarecalledtransistionfunctions.Oneeasilyveriesthefollowingconditionssatisedbythetransitionfunctionsofaprincipalbundle:

5、i)forthreelocaltrivializations(Ua,ha),(Ub,hb)and(Ug,hg),gabgbggga=1,onUaUbUg,ii)foreachlocaltrivialization(Ua,ha),gaa=1.12GEOMETRICQUANTIZATIONExercise1.3.Supposewearegiventhefollowingdata:aopencoveringU=fUaga2IofX,andfunctionsgab:UaUb!Gsatisfyingiandii)above.Con-structaprincipalG-bundleoutofthes

6、edata.FixingthecoveringU,whendotwocollectionsoftransitionfunctionsfgabgandfg0gdeneisomorphicvectorabbundles?Whathappensfordifferentcoverings?Ofcourse,wealwayshavethetrivialG-bundleoverX,namelyP:=XG.IngeneralwesaythataG-bundleistrivialifitisisomorphismtothisbundle.Proposition1.4.AG-bundlePistrivial

7、ifandonlyifithasaglobalsection.Denition1.5.Avectorbundleisaberbundlep:E!Xwithtypicalberavectorspace.Alinebundleisavectorbundleofrankone.Proposition1.6(Associatedvectorbundle).Letp:P!XbeaprincipalGb