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1、MITOpenCourseWarehttp://ocw.mit.edu8.821StringTheoryFall2008ForinformationaboutcitingthesematerialsorourTermsofUse,visit:http://ocw.mit.edu/terms.8.821F2008Lecture17:Moreon3-pointfunctions,thechiralanomaly,andWilsonloopsLecturer:McGreevyNovember24,20081Introduction1
2、.Finish�OO∗J�2.Anomalies3.Wilsonloops2Calculationofa3-pointfunctionConsideraCFTwithaconservedU(1)currentJ.TheoperatorOΔisanoperatorintheCFTwithscalingdimensionΔthatcouplestoachargedscalarfieldφinthebulk,whilethecomplexconjugateofthisbulkscalarfieldcouplestoO∗.Supposet
3、hatthereisamasslessvectorfieldAinΔthebulkthatcouplestoJµ.Theminimalcouplingofthischargedscalarfieldtothevectorfieldleadstoavertex(−iηgAB(w)),whichgivestheleadingcontributiontothethree-pointfunctionGµ≡�O(x)O∗(x)Jµ(x)�123Δ1Δ23��←→�dDwdw0ABΔ∂Δµ=D+1(−iηg(w))K(w
4、�x1)∂wBK(w
5、
6、�x2)KA(w
7、�x3).(1)w0Thedouble-arrowmeanstakethederivativeactingtotherightminusthederivativeactingtotheleft.KΔisthebulk-to-boundarypropagatorforthechargedscalarfieldandKµ(w
8、�x)istheA3bulk-to-boundarypropagatorforthegaugeboson.Thebulk-to-boundarypropagatorforthevectorfie
9、ldsolvesthebulkMaxwellequationsandgoestoadeltafunctionattheboundary(itgoestoδ(w�−�x)asw0→0).TheWittentrickthatweusedtodeducethebulk-to-boundarypropagatorforthescalarfieldcanalsobeusedtoobtainthebulk-to-boundarypropagatorofthevectorfield.Weclaimthatitimplies:wD−1KB(w
10、�
11、x)=C0JB(w−�x),(2)AD(w−�x)2(D−1)A1wereCDissomenormalizationconstantandJBisaJacobianforinversion–itisgivenbyAxAxBJB(x)≡δB−2,(3)AAx2�2∂x�AwhichcanbedeterminedfromJB=xA,withxA=x.ThisJacobianarisesbecausefortheA∂x�(x�)2BWittentrick,onefirstfindsthebulk-to-boundarypropagato
12、rwiththesingularityatinfinityandthenperformsaninversiontoputthesingularityatsomefinitevalue.Usingtheinversiontrickanddoingthewintegralyieldsµµ(Δ−D/2)Γ(D/2)Γ(Δ)G123=S(x1,x2,x3)ηπD/2Γ(Δ−D/2).(4)Sµistheuniquecombinationofx1,x2,andx3allowedbyconformalsymmetryandisgivenby�
13、µµ�µ1x13x231S(x1,x2,x3)=2Δ−D+2x2−x2D−2D−2.(5)x121323x13x23UsingtheWardidentity,∂µ∗µG123=i(δ(x13)−δ(x23))�OO�,(6)∂x3wecandeterminethenormal
