Abstract
We give a detailed account of the methods introduced in [1] to calculate holographic four-point correlators in IIB supergravity on AdS5 × S5. 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 one-half BPS single-trace operators can be written as finite 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 “on-shell 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.
Highlights
With Op(x) = Tr X{I1 . . . XIp}(x), Ik = 1, . . . 6, in the symmetric-traceless representation of the SO(6) R-symmetry
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 Witten diagrams
The analogy between AdS correlators and flat space scattering amplitudes becomes manifest in Mellin space: holographic correlators are functions of Mandelstam-like invariants s, t, u, with poles and residues controlled by OPE factorization. (For the AdS5 × S5 background, tree-level correlators are rational functions — this is the Mellin counterpart of the fact that only a finite number of D-functions are needed in position space.) most applications to date of the Mellin technology to holography (e.g., [41,42,43,44,45]) have focussed on the study of individual Witten diagrams in toy models
Summary
The standard recipe to calculate holographic correlation functions follows from the most basic entry of the AdS/CFT dictionary [19,20,21], which states that the generating functional of boundary CFT correlators equals the AdS path integral with boundary sources. The rules of evaluation of Witten diagrams are entirely analogous to the ones for position space Feynman diagrams: we assign a bulk-to-bulk propagator GBB(z, w) to each internal line connecting two bulk vertices at positions z and w; and a bulk-to-boundary propagator GB∂(z, x) to each external line connecting a bulk vertex at z and a boundary point x These propagators are Green’s functions in AdS with appropriate boundary conditions. In [36] a technique was invented that allows, when certain “truncation conditions” for the quantum numbers of the external and exchanged operators are met, to trade the propagator of an exchange diagram for a finite sum of contact vertices In such cases, one is able to evaluate an exchange Witten diagram as a finite sum of D-functions. The formulae for the requisite exchange diagrams have been collected in appendix A
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