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1、CommunicationsinCommun.Math.Phys.117,353---386(1988)MathematicalPhyss?Springer-VerlagI988TopologicalQuantumFieldTheoryEdwardWitten*SchoolofNaturalSciences,InstituteforAdvancedStudy,OldenLane,Princeton,NJ08540,USAAbstract.Atwistedversionoffourdimensionalsupersymmetricgaugetheoryisformulated.Th
2、emodel,whichrefinesanonrelativistictreatmentbyAtiyah,appearstounderliemanyrecentdevelopmentsintopologyoflowdimensionalmanifolds;theDonaldsonpolynomialinvariantsoffourmanifoldsandtheFloergroupsofthreemanifoldsappearnaturally.Themodelmayalsobeinterestingfromaphysicalviewpoint;itisinasenseagener
3、allycovariantquantumfieldtheory,albeitoneinwhichgeneralcovarianceisunbroken,therearenogravitons,andtheonlyexcitationsaretopological.1.IntroductionOneofthedramaticdevelopmentsinmathematicsinrecentyearshasbeentheprograminitiatedbyDonaldsonofstudyingthetopologyoflowdimensionalmanifoldsvianonline
4、arclassicalfieldtheory[1,2].Donaldson'sworkusesheavilytheself-dualYang-Millsequations,whichwerefirstintroducedbyphysicists[3],anddependsonsomeimportantresultsoriginallyobtainedbymathematicalphysicists,e.g.Taubes'theoremonexistenceofinstantonsoncertainsmoothfourmanifolds[4](aswellashardanalysi
5、sofinstantonmodulispacesI-5]).ThustherehavebeenmanyconjecturesthatDonaldson'sworkmayberelatedtophysicalideasinanintimateway.However,sucharelationhasnotbeenapparentinDonaldson'sdetailedconstructions.ThispicturehaschangedconsiderablybecauseoftheworkofFloeronthreemanifolds[6].Floer'sworkinvolves
6、tunnelingamplitudesin3+1dimensions,andhasbeeninterpretedbyAtiyah[7]intermsofamodifiedversionofsupersymmetricquantumgaugetheory.(Floertheoryhasalsobeenreviewedin[8].)Inthisviewpoint,FloertheorycanbeseenasageneralizationtoinfinitedimensionalfunctionspaceofthesupersymmetricapproachtoMorsetheory[
7、9].*OnleavefromDepartmentofPhysics,PrincetonUniversity.ResearchsupportedinpartbyNSFGrantsNo.80-19754,86-16129,86-20266354E.WittenTheFloerhomologygroupsarethenthegroundstatesofacertainHamiltonianHwhichiscloselyrelatedtophysicalquantumfieldtheo
