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1、NuclearPhysicsB223(1983)422-432©North-HollandPublishingCompanyGLOBALASPECTSOFCURRENTALGEBRAEdwardWITTEN*JosephHenryLaboratories,PrincetonUniversity,Princeton,NewJersey08544,USAReceived4March1983AnewmathematicalframeworkfortheWess-Zuminochiraleffectiveactionisd
2、escribed.Itisshownthatthisactionobeysanaprioriquantizationlaw,analogoustoDirac'squantizationofmagneticchange.Itincorporatesincurrentalgebrabothperturbativeandnon-perturbativeanomalies.Thepurposeofthispaperistoclarifyanoldbutrelativelyobscureaspectofcurrentalge
3、bra:theWess-Zuminoeffectivelagrangian[1]whichsummarizestheeffectsofanomaliesincurrentalgebra.Aswewillsee,thiseffectivelagrangianhasunexpectedanalogiestosome2+1dimensionalmodelsdiscussedrecentlybyDeseretal.[2]andtoarecentlynotedSU(2)anomaly[3].Therealsoareconne
4、ctionswithworkofBalachandranetal.[4].FordefinitenesswewillconsideratheorywithSU(3)LXSU(3)RsymmetryspontaneouslybrokendowntothediagonalSU(3).Wewillignoreexplicitsymme-try-breakingperturbations,suchasquarkbaremasses.WithSU(3)L×SU(3)RbrokentodiagonalSU(3),thevacu
5、umstatesofthetheoryareinonetoonecorrespondencewithpointsintheSU(3)manifold.Correspondingly,thelow-energydynamicscanbeconvenientlydescribedbyintroducingafieldU(x~')thattrans-formsinaso-callednon-linearrealizationofSU(3)c×SU(3)R.Foreachspace-timepointx",U(x'~)is
6、anelementofSU(3):a3×3unitarymatrixofdeterminantone.UnderanSU(3)LxSU(3)Rtransformationbyunitarymatrices(A,B),Utrans-formsasU---,AUB-~.TheeffectivelagrangianforUmusthaveSU(3)c×SU(3)Rsymmetry,and,todescribecorrectlythelow-energylimit,itmusthavethesmallestpossible
7、numberofderivatives.Theuniquechoicewithonlytwoderivativesis=~F~fd4xTrO~,UO.Ul,(1)*SupportedinpartbyNSFGrantPHY80-19754.,*22E.Witten/Globalaspectsofcurrentalgebra423whereexperimentindicatesF~=190MeV.TheperturbativeexpansionofUis2i8U=I+~M~ra+.--,(2)a=lwhereM(nor
8、malizedsoTr~b=28~b)aretheSU(3)generatorsand7raaretheGoldstonebosonfields.ThiseffectivelagrangianisknowntoincorporateallrelevantsymmetriesofQCD.Allcurrentalgebratheoremsgoverningthe
