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

The quartic effective action for Kaluza-Klein modes that arises upon compactification of type IIB supergravity on the five-sphere S⁵ 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 S⁵ 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.