How to succeed at holographic correlators without really trying
Journal of High Energy Physics
We give a detailed account of the methods introduced in  to calculate holographic four-point correlators in IIB supergravity on AdS 5 × S 5 . Our approach relies entirely on general consistency conditions and maximal supersymmetry. We discuss two related methods, one in position space and the other in Mellin space. The position space method is based on the observation that the holographic four-point correlators of onehalf BPS single-trace operators can be written as finite sums of contact
... sums of contact Witten diagrams. We demonstrate in several examples that imposing the superconformal Ward identity is sufficient to fix the parameters of this ansatz uniquely, avoiding the need for a detailed knowledge of the supergravity effective action. The Mellin space approach is an "onshell method" inspired by the close analogy between holographic correlators and flat space scattering amplitudes. We conjecture a compact formula for the four-point correlators of one-half BPS single-trace operators of arbitrary weights. Our general formula has the expected analytic structure, obeys the superconformal Ward identity, satisfies the appropriate asymptotic conditions and reproduces all the previously calculated cases. We believe that these conditions determine it uniquely. Contents A Formulae for exchange Witten diagrams 37 B Simplification of contact vertices 40 C The p = 2 case: a check of the domain-pinching mechanism 42 D The p = 3, 4, 5 results from the position space method 46 1 On the other hand, two-and three point functions of one-half BPS operators obey non-renormalization theorems [2-10] and are easily evaluated in free field theory. A non-renormalization theorem also holds for extremal and next-to-extremal correlators      , defined respectively by the conditions p1 = p2 + p3 + p4 and p1 = p2 + p3 + p4 − 2. 2 Four-point functions are the current frontier of the N = 4 integrability program -see, e.g.,    and references therein for very interesting recent progress. 3 As we have remarked in the previous footnote, the extremal and next-to-extremal correlators do not depend on λ and can thus be evaluated at λ = 0 from Wick contractions in free field theory, yielding some simple rational functions of the cross ratios. It has been shown that the holographic calculation at λ = ∞ gives the same result    . 4 Because of selection rules, the number of diagrams is vastly smaller for correlators near extremality, which explains why an explicit calculation is possible in those cases. 11 We use capital letters because the symbol u is already taken to denote the Mandelstam invariant, (3.17). 12 The disconnected term G disc will of course vanish unless the four operators are pairwise identical. 13 In fact for fixed n and , there are in general multiple conformal primaries of this schematic form, which differ in the way the derivatives are distributed.