All Holographic Four-Point Functions in All Maximally Supersymmetric CFTs

TL;DR

We present a constructive, universal method to obtain tree-level four-point holographic correlators for all maximally supersymmetric CFTs by exploiting Maximally R-symmetry Violating (MRV) amplitudes. The MRV limit drastically simplifies the Mellin amplitudes, allowing the entire polar part to be fixed, while superconformal Ward identities and the flat-space limit eliminate the need for extra contact terms; from the MRV data, full correlators including all R-symmetry structure are reconstructed across $AdS_4\times S^7$, $AdS_5\times S^5$, and $AdS_7\times S^4$. The authors derive explicit residue structures in Mellin space, provide background-specific formulas, and verify Ward identities and crossing for several nontrivial examples, including new results in $AdS_4\times S^7$. This framework paves the way for loop-level investigations and extensions to less supersymmetric theories, offering a unifying lens on holographic four-point functions in diverse dimensions.

Abstract

We present a constructive derivation of holographic four-point correlators of arbitrary half-BPS operators for all maximally supersymmetric conformal field theories in $d>2$. This includes holographic correlators in 3d ${\cal N}=8$ ABJM theories, 4d ${\cal N}=4$ SYM theory and the 6d ${\cal N}=(2,0)$ theory, dual to tree-level amplitudes in 11D supergravity on $AdS_4 \times S^7$, 10D supergravity on $AdS_5 \times S^5$ and 11D supergravity on $AdS_7 \times S^4$, respectively. We introduce the concept of Maximally R-symmetry Violating (MRV) amplitude, which corresponds to a special configuration in the R-symmetry space. In this limit the amplitude drastically simplifies, but at the same time the entire polar part of the full amplitude can be recovered from this limit. Furthermore, for a specific choice of the polar part, contact terms can be shown to be absent, by using the superconformal Ward identities and the flat space limit.