| Summary: | Abstract N = 8 $$ \mathcal{N}=8 $$ superconformal field theories, such as the ABJM theory at Chern-Simons level k = 1 or 2, contain 35 scalar operators O I J $$ {\mathcal{O}}_{IJ} $$ 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 $$ \mathcal{N}=8 $$ supergravity, has no cubic A 3 couplings in its Lagrangian, where A IJ is the bulk dual of O I J $$ {\mathcal{O}}_{IJ} $$ . 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 A 3 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.
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