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

TL;DR

This paper resolves a puzzle in AdS/CFT correspondence by showing that boundary supersymmetry necessitates specific counterterms, enabling the gravity dual to reproduce the non-zero three-point functions of certain operators in superconformal field theories.

Contribution

It demonstrates that boundary supersymmetry requires finite and infinite counterterms in the gravity dual, explaining the non-zero three-point functions despite the absence of cubic couplings.

Findings

Finite A^3 counterterm reproduces the field theory three-point function.

Infinite counterterms are required by boundary supersymmetry.

The generating functional is the Legendre transform of the on-shell action.

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 $\Delta=1$ 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 $\Delta=1$ operators is the Legendre transform of the on-shell action, and the supersymmetry properties of this functional play a significant role in our treatment.