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1、Commun.Math.Phys.252,189–258(2004)CommunicationsinDigitalObjectIdentifier(DOI)10.1007/s00220-004-1187-3MathematicalPhysicsPerturbativeGaugeTheoryasaStringTheoryinTwistorSpaceEdwardWittenInstituteforAdvancedStudy,Princeton,NJ08540,USAReceived:22April2004/Accepted:3June2004Published
2、online:7October2004–©Springer-Verlag2004DedicatedtoFreemanDysonforhis80thbirthdayAbstract:PerturbativescatteringamplitudesinYang-Millstheoryhavemanyunex-pectedproperties,suchasholomorphyofthemaximallyhelicityviolatingamplitudes.Tointerprettheseresults,weFouriertransformthescatter
3、ingamplitudesfrommomen-tumspacetotwistorspace,andarguethatthetransformedamplitudesaresupportedoncertainholomorphiccurves.ThisinturnisapparentlyaconsequenceofanequivalencebetweentheperturbativeexpansionofN=4superYang-MillstheoryandtheD-inst-antonexpansionofacertainstringtheory,nam
4、elythetopologicalBmodelwhosetargetspaceistheCalabi-YausupermanifoldCP3
5、4.1.IntroductionTheperturbativeexpansionofYang-MillstheoryhasremarkablepropertiesthatarenotevidentuponinspectingtheFeynmanrules.Forexample,thetreelevelscatteringampli-tudesthataremaximallyhelicityviolating(MHV
6、)canbeexpressedintermsofasimpleholomorphicorantiholomorphicfunction.ThiswasfirstconjecturedbyParkeandTay-lorbasedoncomputationsinthefirstfewcases[1];thegeneralcasewasprovedbyBerendsandGiele[2].(UnexpectedsimplicityandselectionrulesinYang-Millsandgravitationalhelicityamplitudeswerefi
7、rstfound,asfarasIknow,byDeWitt[3]forfourparticleamplitudes.)Theseunexpectedsimplificationshaveechoes,inmanycases,inloopamplitudes,especiallyinthesupersymmetriccase.Forasamplingofone-loopresults,seethereview[4],andforsomerecenttwo-loopresultsforthetheorywithmaximalorN=4supersymmetr
8、y(whichwasfirstconstructedin[5]),see[6].N=4superYang-Millstheoryisanimportanttestcaseforperturbativegaugetheory,sinceitisthesimplestcaseand,forexample,hasthesamegluonictreeamplitudesaspureYang-Millstheory.Inthepresentpaper,wewillofferanewperspectiveonexplainingtheseresults.Wewills
9、tudywhathappenswhentheusualmomentumspace
