Abstract
\( \mathcal{N}=8 \) superconformal field theories, such as the ABJM theory at Chern-Simons level k = 1 or 2, contain 35 scalar operators \( {\mathcal{O}}_{IJ} \) 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 \( \mathcal{N}=8 \) supergravity, has no cubic A 3 couplings in its Lagrangian, where A IJ is the bulk dual of \( {\mathcal{O}}_{IJ} \). 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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Freedman, D.Z., Pilch, K., Pufu, S.S. et al. Boundary terms and three-point functions: an AdS/CFT puzzle resolved. J. High Energ. Phys. 2017, 53 (2017). https://doi.org/10.1007/JHEP06(2017)053
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DOI: https://doi.org/10.1007/JHEP06(2017)053