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

TL;DR

<p>We resolve a long-standing puzzle in AdS$_4$/CFT$_3$ by showing that the nonvanishing 3-point function of dimension-1 operators in ${ m N}=8$ SCFTs, computed nonperturbatively in field theory, is reproduced holographically not by a bulk cubic scalar coupling (which is absent in ${ m N}=8$ gauged supergravity) but by a finite cubic boundary counterterm that preserves supersymmetry. The essential construction relies on extending bulk SUSY to the boundary, employing a Bogomolny-type argument to identify the required boundary term, and implementing Legendre transformations for alternate quantization of the bulk scalars dual to ${ m Δ}=1$ operators. The generating functional for CFT correlators is the Legendre transform of the on-shell action, and supersymmetry imposes precise cancellations between bulk variations and boundary counterterms, including infinite holographic renormalization terms. The resulting 2- and 3-point functions, computed on the gravity side and fixed by $c_T$, precisely match the field theory predictions for ABJM/ABJ theories in the holographic limit, providing a stringent check of AdS/CFT in this maximally supersymmetric context.</p>

Abstract

${\cal N} = 8$ superconformal field theories, such as the ABJM theory at Chern-Simons level $k=1$ or $2$, contain 35 scalar operators ${\cal O}_{IJ}$ with $Δ=1$ in the ${\bf 35}_v$ 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 AdS$_4$ gravity dual, obtained from gauged ${\cal N}=8$ supergravity, has no cubic $A^3$ couplings in its Lagrangian, where $A^{IJ}$ is the bulk dual of ${\cal O}_{IJ}$. So conventional Witten diagrams cannot furnish the field theory result. We show that the extension of bulk supersymmetry to 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 $Δ=1$ operators is the Legendre transform of the on-shell action, and the supersymmetry properties of this functional play a significant role in our treatment.