Boundary terms and three-point functions: an AdS/CFT puzzle resolved

N= 8 superconformal field theories, such as the ABJM theory at Chern-Simons level k = 1 or 2, contain 35 scalar operators O_{I J} with Δ = 1 in the 35_v representation of SO(8). The 3-point correlation function of these operators is non-vanishing, and indeed can be calculated non-perturbatively in the field theory. But its AdS₄ gravity dual, obtained from gauged N= 8 supergravity, has no cubic A³ couplings in its Lagrangian, where A^IJ is the bulk dual of O_{I J}. So conventional Witten diagrams cannot furnish the field theory result. We show that the extension of bulk supersymmetry to the AdS₄ boundary requires the introduction of a finite A³ counterterm that does provide a perfect match to the 3-point correlator. Boundary supersymmetry also requires infinite counterterms which agree with the method of holographic renormalization. The generating functional of correlation functions of the Δ = 1 operators is the Legendre transform of the on-shell action, and the supersymmetry properties of this functional play a significant role in our treatment.