8 821 F2008 Lecture 12 Boundary of AdS Poincare patch wave equation in AdS Lecturer McGreevy Scribe Francesco D Eramo 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 defined 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 2 Xi2 Xp 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 figure shrinks to zero size like sinh while the radius of the direction depicted in the top figure 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 2 1 X02 Xp 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 6 0 set u 1 u X If u 6 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 specified 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 reflects off 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 fields in AdS R Our goal is to evaluate he 0 O iCF T e WCF T 0 R O 0 Conjecture he iCF 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 lnhe O iCF T extremum z 0 N ISU GRA 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 cutoff which will be a UV cutoff in the CFT and set boundary conditions at z Eqn 15 is written as if there is just one field in the bulk Really there is a for every operator O in the dual field theory We ll say couples to O at the boundary How to match We give four examples 1 Dilaton field Before near horizon limit we have D3 branes in IR10 the asymptotic value of the dilaton determines the string coupling constant gs he x i 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 he gs2 R T r F 2 iCF 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 fields SO 6 R SU 4 R In this case the correspondence is between these gauge fields and the R current operators a JR a AKK i e Sbdy 2 2 R 20 Aa Ja Useful visualization Figure 5 Feynman graphs in AdS We do the one with two ext legs first Classical field theory in bulk boundary value problem Extr of classical action expand about quadratic solution in powers of …