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 16 Correlators of more than two operators Lecturer McGreevy November 6 2008 1 Intro This lecture covers 1 3 point functions 2 the relationship between the subleading term in the bulk eld solution and the expectation value of the dual operator 3 bulk gauge elds 2 3 Point Functions Let s consider a bulk gravity theory with 3 scalar elds 3 1 D 1 2 2 2 Sbulk d x g i mi i b 1 2 3 2 i 1 The interaction term could be modi ed with some other couplings e g 21 2 or 1 2 2 etc such di erences will modify the details of the following calculation We want to solve the equations of motion perturbatively in 0 This could be justi ed either by small b coupling or by small 0 boundary values i 1 This is just like a Feynman diagram expansion these Witten Diagrams also keep track of which insertions are at the boundary of AdS D i 0 i z x d x K z x x i x b dD x dz gG i z x z x D d x1 dD x2 K j z x x1 K k z x x2 0j x1 0k x2 The diagram with three insertions at the boundary won t contribute to the vacuum 3 point function since its contribution to the on shell action will be at least quartic in 0 and the three point function 3 0 is obtained by acting with 3 on the action and setting 0 The s that are running around are the weights of the primary operators for the scalar eld insertions The Gs and Ks are just spatial propagators for our theory G z x z x is the bulk to bulk propagator de ned as the normalizable solution to 1 m2i G i z x z x z z D x x g which is otherwise regular in the interior of AdS The bulk to boundary propagator K is de ned as 1 K z x x lim n G z x z x z 0 where is the boundary metric and n is the outward pointing normal at the boundary The bulk to bulk propagator and bulk to boundary propagator are related by1 G z x z x z 2 D K z x x lim Of course these have actual real live expressions that can be put in terms of hypergeometric functions Just in case you ever need them 1 1 G z x z x c F1 1 2 2 2 2 c z 2 z 2 x x 2 geodesic distance in AdS 2zz 2 2 D D 2 D 2 To compute 3 point functions we plug into the on shell action S After an integration by parts the action becomes 3 S 1 2 i 1 1 D 1 d b x g i i 2 I dD 1 x g 1 2 3 c t II The proof of this relation follows from Green s second identity Z Z m2 m2 n n U U with G K 2 The rst term I vanishes by properties of the bulk to bulk propagator this is true in general for n 3 This is glossed over in many discussions of this calculation We will not prove it directly but it follows immediately from the result in 3 below Anyway the bulk term is non zero and that s what we ll compute b II 2 D d xdz g 3 K i z x xi 0i xi i 1 So with this in hand we can nd the 3 point function by functional di erentiation using the GKPW formula R R e 0 O e SU GRA S 0 3 O1 x1 O2 x2 O3 x3 0 0 O1 x1 O2 x2 O3 x3 0 0 b 0i xi i 1 II 0 0 3 i dD xdz g K z x xi i 1 To see the structure of the 3 point function we do some work on this integral First let s relabel x We ll also de ne w x 2 w02 w x 2 the coordinates Let wA z x so w0 z and w In this case x0 0 With these new coordinate labels the bulk to boundary propagator becomes w0 Kx w w x 2 Now we do a change of variables inversion Let wA two claims 1 dD 1 w wa w 2 2 Now we make where w 2 w0 2 w g w dD 1 w g w since inversion is an isometry of AdS 2 Kx w x 2 Kx w This is how we found K From these we can nd the transformation of the 3 point correlator II Gfrom xi 3 3 i 1 Some more claims 3 II x i 2 Gfrom xi 3 1 This is the correct transformation of a CFT 3 point function under inversion large conformal transformation denoted I 2 Translation invariance is clear Tb x i x i b We simply rede ne the integration variable w w b to remove the b dependence 3 Special conformal invariance ITb I so that s good 4 This must be of the required form since rotational invariance is clear G3 cijk ij i j xi This is determined up to cijk To nd cijk need to do integral See DZF Translate x3 to 0 Then only two denominator factors in G xi and use Feynman parameters n point functions proceed quite similarly The only new complication is that in general one must evaluate some Witten diagrams with both bulk to boundary and bulk to bulk propagators which don t go away 3 Expectation Values Next we make a valuable observation about expectation values the classical eld The reference is Klebanov Witten hep th 9905104 The solution in response to a source 0 at the boundary in some state is 0 z x lim 0 x O 2 A x O 2 z In this case 0 is the source The function A x is a normalizable uctuation determined by the source and choice of the propagator not just to linear order in 0 CAREFUL The term O 2 in the rst term can be larger than the A x in the second term Use caution when calculating and expanding The claim is that A x R 0 1 1 O x 0 O x e O CF T 2 D 2 D 4 Knowing the one point function O x 0 1 OO 2 0 0 0 OOO allows us to compute all other correlators So this formula circumvents the need for the on shell action S and applies not just in the Euclidean case but also in the real time case We ll give a diagrammatic proof The value of the bulk eld in response to some source can be represented perturbatively by the following collection of diagrams z …