Four-point functions of all-different-weight chiral primary operators in the supergravity approximation

TL;DR

The paper advances the holographic computation of four-point functions of $1/2$ BPS operators in ${ m N}=4$ SYM at strong coupling by deriving two new all-different-weight correlators, $ig\bra 2345\big\bra$ and $ig\bra 3456\big\bra$, from type IIB supergravity on ${ m AdS}_5\times S^5$. It introduces substantial algorithmic simplifications for contact and exchange contributions and recasts results in a Mellin-space compatible, compact $ar{D}$-function basis. The authors demonstrate that the interacting parts reduce to a small set of dynamical functions and verify consistency with the recently proposed Mellin-space formula, including nontrivial checks away from extremality. They also address the role of extended operators in the free part and show how their inclusion resolves known discrepancies in planar correlators. Overall, the work strengthens the bridge between AdS/CFT calculations and Mellin-space bootstrap approaches, and lays groundwork for computing further nontrivial correlators at strong coupling.

Abstract

Recently a Mellin-space formula was conjectured for the form of correlation functions of $1/2$ BPS operators in planar $\mathcal{N}=4$ SYM in the strong 't Hooft coupling limit. In this work we report on the computation of two previously unknown four-point functions of operators with weights $\langle 2345 \rangle$ and $\langle 3456\rangle$, from the effective type-IIB supergravity action using AdS/CFT. These correlators are novel: they are the first correlators with all-different weights and in particular $\langle 3456\rangle$ is the first next-next-next-to-extremal correlator to ever have been computed. We also present simplifications of the known algorithm, without which these computations could not have been executed without considerable computer power. The main simplifications we found are present in the computation of the exchange Lagrangian and in the computation of $a$ tensors. After bringing our results in the appropriate form we successfully corroborate the recently conjectured formula.

Figures (3)

• Figure 1: A contact Witten diagram.
• Figure 2: The exchange Witten diagram where a scalar of weight $k$ is exchanged in the $s$ channel.
• Figure 3: An example of a singly-connected graph which has more index loops after splitting: the number of external index loops $L_e$ (loops outside of the lines that connect the nodes) before splitting is $2$, whereas after splitting the total number is $2+1 = 3$.