lecture09 scale and conformal invariance in field theory

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1、MITOpenCourseWarehttp://ocw.mit.edu8.821StringTheoryFall2008ForinformationaboutcitingthesematerialsorourTermsofUse,visit:http://ocw.mit.edu/terms.8.821F2008Lecture09:PreviewofStringsinN=4SYM;HierarchyofScalingdimensions;ConformalSymmetryinQFTLecturer:McGreevyOctober8,2008

2、1EmergenceofStringsfromGaugeTheoryContinuingfromthepreviouslecturewesubstantiatethecorrespondencebetweentheoperatorsinN=4SYMtheorywiththeexcitationsofsomestringtheory.Wesawthattheprimarychiraloperators(i.e.primaryoperatorsintheconformalfieldtheorywhichalsobelongtotheshortm

3、ultipletsofthesupersymmetry)correspondtotheSUGRAmodesbutwedidn’tdiscusswhathappenswhentheoperatorsinthegaugetheoryareinthelongmultiplet.Thusconsideramoregenericoperator(residinginalongsupersymmetrymultiplet)ofthematrixQFTinthelargeNlimitO(x)=tr(XXXXYXXY....YX)(1)Weconside

4、rthelimitN≫J≫1whereJisthenumberofentitiesintheaboveproduct.Cyclicityofthetraceimpliesthatthestructureoftheaboveoperatorhasthesymmetryofaclosedloop.InfacttheaboveoperatorcorrespondstocreationoperatorforexcitedstringstateswithX′sandY′sthefieldslivingontheworldsheet.Further,i

5、nthelargeJlimit,Jcorrespondstotheangularmomentumofthestringexcitationsandinfactthisrelationcouldbeusedtoreconstructthestringtheory.HavingsketchedthecorrespondencebetweenthevariousoperatorsoftheSYMguagetheorywithSUGRAmodesorstringexcitations,letusorganizethisknowledgetoget

6、abetterperspectiveonwhenitcouldbeuseful.•ThemassoftheSUGRAmodeisgivenbym2=1/L2.UsingtheAdS-CFTcorreSUGRAAdSspondencem2L2=Δ(Δ−4)whereΔisthescalingdimensionofthecorrespondingSUGRAAdSchiralprimaryoperatorintheSYMtheory.ThisimpliesthatΔSUGRA∼N0λ0.√•Movingontotheexcitedstrings

7、tatesm2stringstates∼1/α′∼λ/L2where1/α′isstringAdStension.ThusthecorrespondingscalingdimensionequalsΔstringstates∼N0λ1/4.•Finally,thereareD-branesinthestringtheorywhichcorrespondtobaryonicstatesinthe1gaugetheory.Theirmassisgivenby1√m2∼1/α′g2∼λ/g2L2⇒Δ=Nλ1/4.D−branessAdSD−br

8、aneClearly,thescalingdimensionΔforthedifferentstates(SUGRA,strings,D-branes)hasdistinctivedepende