lecture08 't hooft counting

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1、MITOpenCourseWarehttp://ocw.mit.edu8.821StringTheoryFall2008ForinformationaboutcitingthesematerialsorourTermsofUse,visit:http://ocw.mit.edu/terms.8.821F2008Lecture8:LargeNCountingLecturer:McGreevyOctober4,20081IntroductionTodaywe’llcontinueourdiscussionoflargeNscalinginthet

2、’Hooftlimitofquantummatrixmodels.Inparticular,we’llreviewtheNcountingforvacuumdiagramsandlearnhowtodoNcountingforcorrelators.OurperspectiveonthelargeNlimitcontinuestobeoneofconvenience.Itisoneofseveralassumptionsthatenablethesimplestrealizationofholographyandgauge/gravitydu

3、ality.Nexttimewe’llcontinuetalkingaboutsimplifyingassumptionsandturnourattentiontoconformalinvariance.2Vacuumt’HooftcountingLet’srecallourstartingpoint.Wewerethinkingaboutmatrixfieldtheories,fieldtheoriesinwhichallthefieldsareNxNmatricesinadditiontowhateverotherlabelstheymayha

4、ve.Inparticular,wewriteourtheoryschematicallyintermsofonebigfieldΦwhichwethinkofaspotentiallyincludingscalarsφ,gaugefieldsAµ,andfermionsψαallofwhichareNxNmatrices.TheschematicLagrangianis1L=Tr(∂Φ∂Φ−V(Φ))(1)g2whereV(Φ)∼Φ2+Φ4+...andallproductsshouldbethoughtofasmatrixproducts.T

5、hisschematicrepresentationismeanttoconveytwopiecesofinformation.First,thattheLagrangianisinvariantunderU(N)rotationswhichactonthematrixfieldsbyconjugation:Φ→UΦU−1.Second,thatthefieldsarescaledsothatthecouplingg2sitsoutinfrontoftheentireLagrangian.Inparticular,wehavetoverycare

6、fulaboutfieldnormalizationswhendoinglargeNcountingforcorrelationfunctions.Thet’HooftlimitisthelimitwhereN→∞andg2→0whileλ=g2Nremainsfixed.λisthet’Hooftcoupling.NotethatweareadoptingadifferentnormalizationconditionthaninthepreviouslectureonlargeNcounting(thereweusedthefieldΦ˜=(1/

7、g)Φwhichhadkinetictermswiththeusualfieldtheorynormalization).Withourcurrentnormalizationthepropagator�ΦΦ�goeslikeg2=λ.NThepropagatorinthedoublelinenotationappearsinFig.1.Rememberthatforthepurposesof1Figure1:Propagator,λ/NFigure2:Threepointvertex,N/λlargeNcountingweareinstruc

8、tedtothinkonlyaboutcountingpowersofN,sowewillsuppresspositionsoffields,spinindices,