lecture12 geometry of ads (cont.); poincaré patch; wave equation in ads

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1、MITOpenCourseWarehttp://ocw.mit.edu8.821StringTheoryFall2008ForinformationaboutcitingthesematerialsorourTermsofUse,visit:http://ocw.mit.edu/terms.8.821F2008Lecture12:BoundaryofAdS;Poincar´epatch;waveequationinAdSLecturer:McGreevyOctober16,2008Today:1.theboundaryofAdS2.Po

2、incar´epatch3.motivateboundaryvalueproblem4.waveequationinAdS.1TheboundaryofAdSWedefinedtheLorentzianAdSasthelocus{ηXaXb=−L2}⊂IRp+1,2,wherep+2abXp+1ηXaXb=−X2+X2−X2=−L2(1)ab0ip+2i=1Themetricisds2=ηXaXb

3、=L2−cosh2ρdτ2+dρ2+sinh2ρdΩ2(2)AdSab(1)p1.1ProjectiveboundaryTakeaso

4、lutionV=X0,X~,Xp+2ofequation(1).ReachtheboundarybyrescalingX,preserving(1).LetX=λX˜,thenequation(1)becomesL2ηX˜aX˜b=−(3)abλ2Wenowtakeλ→∞,theboundaryis{ηX˜aX˜b=0}/{X˜∼λX˜}≃IRp,1(4)ab1Figure1:LorentzianAdS:Theleft-rightaxisistheρdirection.Atρ=0,theSpinthelowerfigureshrinkst

5、ozerosize(likesinhρ),whiletheradiusoftheτdirection,depictedinthetopfigure,approachesaconstant(likecoshρ).Thisrelationcanalsobereadasfollows:theboundaryofAdSisthesetoflightraysinIRp+1,2,modulotherescaling.RecallthatthisisexactlyparametrizedbypointsinIRp,1as:aµ1212ρ=κX,(1

6、−X),(1+X).(5)22WeusedthisfactearliertomakewritetheSO(p+1,2)actionoftheconformalgrouponIRp,1inalinearway.ThefactthattheconformalgroupofIRp,1hasaniceactionontheboundaryofAdSisveryencouraging.2AlternativedecompositionI2Pp+12Fixλbyimposing1=X~=X.Thenwehavei=1iX2+X2=X~2=1⇒∂Ad

7、S=S1×Sp(6)0p+2AlternativedecompositionIILetu±=X0±iXp+1.Then(1)⇒−u+u−+X~2=0.Ifu+=�0setu+=1⇒u−=X~2Ifu−=�0setu−=1⇒u+=X~2ThenX~˜=X~.uis’thepointat∞’.Theboundaryiscompact.X~2−1.2Penrosediagram(onemoredescriptionoftheboundary)dρLetdΘ=(thisvariablewascalled‘squiggle’inlecture).

8、Themetricinthesenewcoordinatescoshρresultsin22222Θ2ds=coshρ−dτ+dΘ+tandΩp(7)2andthereforeΘρtan=tanhΘ∈[0,π/2](8)22Theboundaryis{Θ=π/2}∼IR×Sp.NotethatthemetricontheboundaryisonlyspecifiedupFigure2:ThesquigglevariableΘrunsfrom0toπ/2asρgoesfrom0to∞torescaling,i.e.aWeyltransf

9、ormation.ButwhydowecareaboutthisboundarymorethansaytheconformalboundaryofMinkowskispace?Theanswerisinth