How to Succeed at Holographic Correlators Without Really Trying

TL;DR

The paper develops two complementary frameworks to compute holographic four-point correlators of one-half BPS operators in IIB supergravity on $AdS_5 \times S^5$: a position-space bootstrap that rewrites exchange diagrams as finite sums of $D$-functions and fixes undetermined couplings via the superconformal Ward identity, and a Mellin-space on-shell approach that yields a compact, (likely) unique formula for arbitrary weights. It formulates the general one-half BPS four-point amplitude as a constrained algebraic problem in Mellin space, solving for an explicit ansatz and demonstrating uniqueness in the simplest equal-weight case while carefully treating contour subtleties and the connection to the free-field correlator. The work shows that, under maximal supersymmetry and large-$N$ consistency, holographic four-point functions are governed by a small set of rational structures, reproduce known lower-weight results (and new $p=5$ data), and align with the flat-space limit. This leads to a streamlined, robust bootstrap-like program for AdS/CFT correlators, with potential extensions to other backgrounds and higher-loop analyses that advance practical computations and conceptual understanding of holographic dynamics.

Abstract

We give a detailed account of the methods introduced in [1] to calculate holographic four-point correlators in IIB supergravity on $AdS_5 \times S^5$. 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 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.

Figures (8)

• Figure 1: An exchange Witten diagram.
• Figure 2: A contact Witten diagram.
• Figure 3: Solution to the $\gamma_{ij}$ constraints.
• Figure 4: R-symmetry monomials in $\mathcal{M}$.
• Figure 5: R-symmetry monomials in $\widetilde{\mathcal{M}}$.