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${\cal N} = 8$ superconformal field theories, such as the ABJM theory atChern-Simons level $k=1$ or $2$, contain 35 scalar operators ${\cal O}_{IJ}$with $\Delta=1$ in the ${\bf 35}_v$ representation of SO(8). The 3-pointcorrelation function of these operators is non-vanishing, and indeed can becalculated non-perturbatively in the field theory. But its AdS$_4$ gravitydual, obtained from gauged ${\cal N}=8$ supergravity, has no cubic $A^3$couplings in its Lagrangian, where $A^{IJ}$ is the bulk dual of ${\calO}_{IJ}$. So conventional Witten diagrams cannot furnish the field theoryresult. We show that the extension of bulk supersymmetry to the AdS$_4$boundary requires the introduction of a finite $A^3$ counterterm that doesprovide a perfect match to the 3-point correlator. Boundary supersymmetry alsorequires infinite counterterms which agree with the method of holographicrenormalization. The generating functional of correlation functions of the$\Delta=1$ operators is the Legendre transform of the on-shell action, and thesupersymmetry properties of this functional play a significant role in ourtreatment.

Boundary Terms and Three-Point Functions: An AdS/CFT Puzzle Resolved