lecture16 three-point functions, anomalies; expectation values

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1、MITOpenCourseWarehttp://ocw.mit.edu8.821StringTheoryFall2008ForinformationaboutcitingthesematerialsorourTermsofUse,visit:http://ocw.mit.edu/terms.8.821F2008Lecture16:CorrelatorsofmorethantwooperatorsLecturer:McGreevyNovember6,20081IntroThislecturecovers:1.3-poin

2、tfunctions.2.therelationshipbetweenthesubleadingterminthebulkfieldsolutionandtheexpectationvalueofthedualoperator.3.bulkgaugefields.23-PointFunctionsLet’sconsiderabulkgravitytheorywith3scalarfields.��3�1√�S=dD+1x−g((∂φ)2+m2φ2)+bφφφbulkiii1232i=1Theinteractiontermco

3、uldbemodifiedwithsomeothercouplings,e.g.φ21φ2or(∂φ1)2φ2,etc...;suchdifferenceswillmodifythedetailsofthefollowingcalculation.Wewanttosolvetheequationsofmotionperturbativelyinφ0.Thiscouldbejustifiedeitherbysmallbcouplingorbysmall(0)boundaryvaluesφ.i1ThisisjustlikeaFe

4、ynmandiagramexpansion;these“WittenDiagrams”alsokeeptrackofwhichinsertionsareattheboundaryofAdS.��√φ(z,x)=dDx�KΔi(z,x;x�)φ0(x�)+bdDx�dz�−gGΔi(z,x;z�,x�)×ii��×dDxdDxKΔj(z,x;x)KΔk(z,x;x)φ0(x)φ0(x)+···1212j1k2Thediagramwiththreeinsertionsattheboundarywon’tcontribute

5、tothevacuum3-pointfunction,sinceitscontributiontotheon-shellactionwillbeatleastquarticinφ0(andthethree-pointfunctionδ30isobtainedbyactingwithδφ3ontheactionandsettingφ=0).TheΔsthatarerunningaroundaretheweightsoftheprimaryoperatorsforthescalarfieldinsertions.TheGsa

6、ndKsarejustspatialpropagatorsforourtheory.G(z,x;z�,x�)isthebulk-to-bulkpropagator,definedasthenormalizablesolutionto(�−m2)GΔi(z,x;z�,x�)≡√1δ(z−z�)δD(x−x�)i−gwhichisotherwiseregularintheinteriorofAdS.Thebulk-to-boundarypropagatorKisdefinedas:�1��K(z,x;x)≡lim√�n·∂G(

7、z,x;z,x),z�→0γ√whereγistheboundarymetricand�nistheoutwardpointingnormalattheboundary.Thebulk-to-bulkpropagatorandbulk-to-boundarypropagatorarerelatedby1:�ΔKΔ(z,x;x�)=limGΔ(z,x;z�,x�).z�→�2Δ−DOfcourse,thesehaveactual,realliveexpressionsthatcanbeputintermsofhyperg

8、eometricfunctions.Justincaseyoueverneedthem:��Δ��−ΔΔΔ+1Δ1G(z,x;z,x)=cΔηF12,2;Δ+1−2,η2,z2+z�2+(x−x�)2η=,geodesicdistanceinAdS,2zz�2−ΔΓ(Δ)cΔ=.(2Δ−D)πD/2Γ(Δ−D/2)Tocomput