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

Abstract

N = 8 superconformal field theories, such as the ABJM theory at Chern-Simons level k = 1 or 2, contain 35 scalar operators O-IJ with Delta = 1 in the 3 5(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 AdS(4) gravity dual, obtained from gauged N = 8 supergravity, has no cubic A(3) couplings in its Lagrangian, where A(IJ) is the bulk dual of O-IJ. So conventional Witten diagrams cannot furnish the field theory result. We show that the extension of bulk supersymmetryto the AdS(4) 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 Delta = 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.