lecture06 about the n = 4 sym theory

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1、MITOpenCourseWarehttp://ocw.mit.edu8.821StringTheoryFall2008ForinformationaboutcitingthesematerialsorourTermsofUse,visit:http://ocw.mit.edu/terms.8.821F2008Lecture06:SupersymmetricLagrangiansandBasicChecksofAdS/CFTLecturer:McGreevySeptember24,2008Weareonourwaytotalkingaboutrea

2、llyawesomethingsaboutsupercoolstuff.Beforewegetthere,though,weneedtodevelopsomeverypowerfultechnology.Tothatend,todaywewilltalkabout1.SUSYLagrangiansandawhirlwindtourofthebeautiesofsuperspace.2.moreonN=4SYM.3.BacktotheBigPicture:SomebasicchecksofAdS/CFTLookingpastthislecture,we

3、willbetalkingaboutstringsfromgaugetheorynext.1N=4SYMandOtherSupersymmetricLagrangiansRecallthatthefieldcontentofN=4SYMisavectorAµ,gauginiλI=1...4,andsixscalarsXi,allintheadjointofthegaugegroup.TheLagrangiandensity(whichiscompletelydeterminedbytheamountofSUSY,uptotwoparameters,(

4、gYM,ϑ)),is1��L=trF2+(DXi)2+iλ¯D/λg2YM�6−[Xi,Xj]2−λ[X,λ]+λ¯[X,λ¯]])i

5、rentzspinorandSO(6)vector/spinor,supersymmetry).Ifyouwanttoputtheindicesin,eitherleavethatasafunexercise,orcheckoutWeinberg,volume3.Asanexample,F+≡σµνFµν.1.1ASuperspaceDetourTheN=4SYMLagrangianisanexampleofa(highly)supersymmetricLagrangian.Sofar,Ijusttoldyouwhatitwasandthatitw

6、asSUSYinvariant(somethingyoucouldsitdownintheprivacyofyourofficeandcheck,ifyouwanted).It’dbenice,though,ifthereweresomesortofamachinethatonecouldcranktogeneratesupersymmetriclagrangians.Thatcrankablemachineissuperspace.Tounderstandwhysuperspaceisuseful,weshouldthinkaboutwhyfields

7、areusefulforrepresentingtranslationallyinvariantLagrangiansinordinaryQFT.Onereasonisthattherepresentationsofthetranslationgrouponthefieldsareparticularlysimpleφ(x)=eiPˆ·xφ(0)(3)We’dliketointroduceasuperfield(thatcomeswithitsownsupercapitalization)Φ(x,θ),whichisnowafunctionofspac

8、ecoordinatesxand“superspace”coordinatesθ,thatwellrepresentstr