Towards 4-point correlation functions of any 1/2-BPS operators from supergravity

TL;DR

The paper tackles the computation of four-point functions of arbitrary weight $1/2$-BPS operators in ${\cal N}=4$ SYM within the supergravity regime, focusing on the quartic action for Kaluza-Klein modes from type IIB supergravity on $S^5$. The main technical challenge is that quartic four-derivative couplings could cause unwanted growth in Mellin space, potentially contradicting Rastelli's conjecture for the Mellin representation. By deriving a reduction formula that rewrites vector-harmonic contributions in terms of scalar-harmonic structures and explicitly evaluating the resulting integrals, the authors show that the net four-derivative contribution cancels: ${\mathscr C}_{1234}=0$, confirming the quartic action is of sigma-model type and supporting the Mellin-space conjecture. This result advances the computation of four-point functions for arbitrary weight $1/2$-BPS operators in the supergravity limit and hints at further simplifications in the remaining quartic terms, providing a solid step toward explicit supergravity predictions for these correlators.

Abstract

The quartic effective action for Kaluza-Klein modes that arises upon compactification of type IIB supergravity on the five-sphere S^5 is a starting point for computing the four-point correlation functions of arbitrary weight 1/2-BPS operators in N=4 super Yang-Mills theory in the supergravity approximation. The apparent structure of this action is rather involved, in particular it contains quartic terms with four derivatives which cannot be removed by field redefinitions. By exhibiting intricate identities between certain integrals involving spherical harmonics of S^5 we show that the net contribution of these four-derivative terms to the effective action vanishes. Our result is in agreement with and provides further support to the recent conjecture on the Mellin space representation of the four-point correlation function of any 1/2-BPS operators in the supergravity approximation.