MIT OpenCourseWare http ocw mit edu 8 821 String Theory Fall 2008 For information about citing these materials or our Terms of Use visit http ocw mit edu terms 8 821 F2008 Lecture 12 Boundary of AdS Poincare patch wave equation in AdS Lecturer McGreevy October 16 2008 Today 1 the boundary of AdS 2 Poincare patch 3 motivate boundary value problem 4 wave equation in AdS 1 The boundary of AdS We de ned the Lorentzian AdSp 2 as the locus ab X a X b L2 IRp 1 2 where ab X a X b X02 p 1 X i 1 Xi2 Xp2 2 L2 1 The metric is ds2AdS ab X a X b 1 L2 cosh2 d 2 d 2 sinh2 d 2p 1 1 2 Projective boundary Xp 2 of equation 1 Reach the boundary by rescaling X preserving Take a solution V X0 X 1 Let X X then equation 1 becomes ab X a X b L2 2 3 We now take the boundary is ab X a X b 0 X X IRp 1 1 4 Figure 1 Lorentzian AdS The left right axis is the direction At 0 the S p in the lower gure shrinks to zero size like sinh while the radius of the direction depicted in the top gure approaches a constant like cosh This relation can also be read as follows the boundary of AdS is the set of lightrays in IRp 1 2 modulo the rescaling Recall that this is exactly parametrized by points in IRp 1 as 2 1 2 a 1 5 X 1 X 1 X 2 2 We used this fact earlier to make write the SO p 1 2 action of the conformal group on IRp 1 in a linear way The fact that the conformal group of IRp 1 has a nice action on the boundary of AdS is very encouraging 2 Alternative decomposition I 2 Pp 1 X 2 Then we have Fix by imposing 1 X i i 1 2 1 X02 Xp2 2 X AdS S 1 S p 6 Alternative decomposition II 2 0 Let u X0 iXp 1 Then 1 u u X 2 If u 0 set u 1 u X If u 0 set u 1 u X 2 X u is the point at The boundary is compact Then X 2 X 1 2 Penrose diagram one more description of the boundary Let d results in d cosh this variable was called squiggle in lecture The metric in these new coordinates 2 2 2 2 2 2 d p 7 ds cosh d d tan 2 and therefore tanh 0 2 8 2 2 The boundary is 2 IR S p Note that the metric on the boundary is only speci ed up tan Figure 2 The squiggle variable runs from 0 to 2 as goes from 0 to to rescaling i e a Weyl transformation But why do we care about this boundary more than say the conformal boundary of Minkowski space The answer is in the next two subsections 3 1 3 Massless geodesics The massless geodesics are given by the condition ds2 0 which implies 0 ds2 L2 cosh2 d 2 d 2 cosh d d d d d cosh 9 is the time elapsed for a static observer Whether the lightray re ects o the boundary depends on the BC s Hence Cauchy problem problem Figure 3 Massless geodesics 1 4 Massive geodesics The action for a massive relativistic point particle is Z Z q S m ds m g X X X X 10 The equation of motion is S 0 X X X X 0 4 11 where the second equation follows if X s X where s is proper time If we assume 0 the action is Z q S mL d cosh2 2 You will show on problem set 3 that this has an oscillatory solution around 0 it never reaches 2 Poincare patch Pick out X p 1 from among the X i This will break the SO p 1 symmetry of the p sphere Let L X zx 12 X Xp 1 Lz p 2 Xp 2 Xp 1 v Equation 1 and the metric become L L2 v 2 x x L2 z dz 2 dx dx ds2 L2 z2 13 same cancellation as UHP This is the metric which we showed has 1 R g R g 2 p 1 p 2 2L2 14 NOTE it covers part of AdS As z t becomes NULL Poincare horizon CLAIM relation between Poincare patch and global time is state operator correspondence EVIDENCE symmetries SO p 1 IRp 1 and SO p 1 SO 2 2 1 Towards CFT correlators from elds in AdS R Our goal is to evaluate e 0 O CF T e WCF T 0 R O 0 Conjecture e CF T Zstrings in AdS 0 but we cannot compute it The pratical version is the following R 1 1 0 O 2 WCF T 0 ln e CF T extremum z 0 N ISU GRA O O 15 N2 A few comments The supergravity description is valid for large N and large In 15 we ve made the N dependence explicit in units of the AdS radius the Newton constant is G1N N 2 ISU GRA is some dimensionless action 5 Figure 4 Poincare patch anticipating divergences at z 0 we introduce a cuto which will be a UV cuto in the CFT and set boundary conditions at z Eqn 15 is written as if there is just one eld in the bulk Really there is a for every operator O in the dual eld theory We ll say couples to O at the boundary How to match We give four examples 1 Dilaton eld Before near horizon limit we have D3 branes in IR10 the asymptotic value of the dilaton determines the string coupling constant gs e x The YM coupling on D3 s is gY2 M gs Changing we get Z 2 Tr F 16 S 2 gs where the dots stand for all the CP even term in the lagrangian In conclusion we have 1 Zstrings e gs2 R T r F 2 CF T 17 The dilaton couples to all the terms in the lagrangian which are CP invariant 2 RR axion We have that str i gs 2 tranforms under SL 2 C nicely like T r F F This time CP odd terms 6 i gs 2 Therefore 18 3 Stress energy tensor R The tensor T is the response of a local QFT to local change in the metric SQF T T Here we are writing for the metric on the boundary In this case g T 19 4 IIB in AdS5 S 5 Isometry on S 5 SO 6 Kaluza Klein KK gauge elds SO 6 R SU 4 R In this case the correspondence is between these gauge elds and the R current operators a AKK JR a i e Sbdy 2 2 R 20 Aa Ja Useful visualization Figure 5 Feynman graphs in AdS We do the one …