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

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.