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MIT 8 821 - Correlators of more than two operators

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MIT OpenCourseWarehttp://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 Novemb er 6, 2008 1 Intro This lecture covers: 1. 3-point functions. 2. the relationship between the subleading term in the bulk field solution and the expectation value of the dual operator. 3. bulk gauge fields. 2 3-Point Functions Let’s consider a bulk gravity theory with 3 scalar fields. �� 3� Sbulk =1 dD+1 x√−g � ((∂φi)2 + m 2 i φ2 i ) + bφ1φ2φ32 i=1 The interaction term could be modified with some other couplings, e.g. φ21φ2 or (∂φ1)2φ2, etc... ; such differences will modify the details of the following calculation. We want to solve the equations of motion perturbatively in φ0 . This could be justified either by small b coupling or by small (0)boundary values φi . 1� � � � � � This is just like a Feynman diagram expansion; these “Witten Diagrams” also keep track of which insertions are at the boundary of AdS. dD dDφi(z, x) = x�KΔi (z, x; x�)φ0(x�) + b x�dz�√−gGΔi (z, x; z�, x�) ×i × dD x1 dD x2KΔj (z, x; x1)KΔk (z, x; x2)φj 0(x1)φk0(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 φ0 (and the three-point function is obtained by acting with δ3 on the action and setting φ0 = 0). The Δs that are running around δφ3 are the weights of the primary operators for the scalar field insertions. The Gs and Ks are just spatial propagators for our theory. G(z, x; z�, x�) is the bulk-to-bulk propagator, defined as the normalizable solution to 1(� − m 2)GΔi (z, x; z�, x�) ≡ √−gδ(z − z�)δD(x − x�)i which is otherwise regular in the interior of AdS. The bulk-to-boundary propagator K is defined as: 1 K(z, x; x�) ≡ lim �n ∂G(z, x; z�, x�), z�→0 √γ · where √γ 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: �Δ KΔ(z, x; x�) = lim GΔ(z, x; z�, x�). z�→� 2Δ − D Of course, these have actual, real live expressions that can be put in terms of hypergeometric functions. Just in case you ever need them: Δ Δ + 1 Δ 1 GΔ(z, x; z�, x�) = cΔη−ΔF1 , ; Δ + 1 − , ,2 2 2 η2 η = z2 + z�2 + (x − x�)2 , geodesic distance in AdS,2zz� 2−ΔΓ(Δ) cΔ = .(2Δ − D)πD/2Γ(Δ − D/2)To compute 3-point functions, we plug φ into the on-shell action S[φ]. After an integration by parts, the action becomes: 3� � S[φ] = 1 � dD+1x∂µ(√−gφi∂µφi) − bdD+1 x√−gφ1φ2φ3 + c.t.2 2 i=1 = I + II The proof of this relation follows from “Green’s second identity” Z` ´Z φ(� − m 2)ψ + ((� − m 2)φ)ψ) = (φn ∂ψ + (n ∂φ)ψ) ∀φ, ψ · · U ∂U with φ = G, ψ = K. 2 1� � � The first 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. � 3II = − bdDxdz√−g �� KΔi (z, x; xi)φ0 i (xi) � 2 i=1 So with this in hand we can find the 3-point function by functional differentiation using the GKPW formula. � R � R e φ0O = e− SUGRA S[φ0] 3� δ = ⇒ �O1(x1)O2(x2)O3(x3)�φ0=0 δφ0(xi)(II)|φ0=0 i=1 i � 3�O1(x1)O2(x2)O3(x3)�φ0=0 = b dDxdz√−g � KΔi (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 the coordinates. Let wA = (z, �x), so w0 = z and w� = �x. We’ll also define (w − �x)2 ≡ w02 + ( w� − �x)2 . In this case, x0 ≡ 0. With these new coordinate labels, the bulk-to-boundary propagator becomes: � �Δ KΔ(w) = w0 .x (w − �x)2 wa� 2Now we do a change of variables, inversion. Let wA = w�2 , where w�2 = w0�2 + w�� . Now we make two claims: 1. dD+1w −g(w) = dD+1w� −g(w�), since inversion is an isometry of AdS. 2. Kx(w) = |x�|2ΔKx� (w�). This is how we found K. From these, we can find the transformation of the 3-point correlator. 3Gfrom II 2ΔGfrom II ⇒ 3 (xi) = |x�i|3 (x�i) i=1 Some more claims: 33 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 : xµi → xµi + bµ. We simply redefine the integration variable w˜µ = wµ − bµ to remove the bµ dependence. 3. Special conformal invariance = ITbI, so that’s good. 4. This must be of the required form, since rotational invariance is clear. G3 = � cijk (xi)Δiji>j This is determined up to cijk. To find 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). Expectation Values Next we make a valuable observation about expectation values & the classical field. The reference is Klebanov-Witten, hep-th/9905104. The solution in response to a source φ0 at the boundary (in some state) is: φ[φ0](z, x)] lim �Δ− � φ0(x) + O(�2) � + �Δ+ � A(x) + O(�2) � → z �→In this case φ0 is the source. The function A(x) is a normalizable fluctuation, determined by the source and choice of the propagator (not just to linear order in φ0). CAREFUL: The term O(�2) in the first term can be larger than the A(x) in the second term! Use caution when calculating and expanding! The claim is that: A(x) = 2Δ1 − D �O(x)�φ0 = 2Δ1 − D �O(x)e R φ0O�CF T 4� � Knowing the one-point function �O(x)�φ0 = � φ0�OO� + 1 2 � φ0φ0�OOO� + · · · allows us to compute all other correlators. So this formula circumvents the need for the on-shell action S[φ], and


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