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

Abstract

N = 8 superconformal field theories, such as the ABJM theory at ChernSimons level k = 1 or 2, contain 35 scalar operators O IJ with ∆ = 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 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 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 ∆ = 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.

Anomaly-corrected supersymmetry algebra and supersymmetric holographic renormalization

“…[18,19,[32][33][34][35][36][37] obtained some boundary counterterms for the fermionic sector, but typically these were limited to either lower dimensional spacetime (mainly 3 or 4 dimensions) or to homogeneous solutions which do not depend on the transverse directions. We note that in a context different from this paper, 4D N = 1 SUGRA including the fermionic sector was treated in [38] by a somehow ad hoc approach.…”

“…and 38) where P is the Pontryagin density. Notice that the integral of P is a topological quantity and thus can be a finite counterterm as in the case of the Euler density, as long as there is no other symmetry which prevents its appearance.…”

“…These vielbeins satisfy the relation It follows that the matrix Γ can be used to define the 'radiality' (see e.g. [38]) on the slice, so that a generic spinor ψ on the slice can be split into two by radiality, 22 10) where Γ ± ≡ 1 2 (1 ± Γ). We recall that splitting spinor fields by their radiality is inevitable because different radiality leads to different asymptotic behavior [32,33] as well as the constraints that relate the fermionic fields and their conjugate momenta should be solved in a Lorentz invariant way [36].…”

On the on-shell: the action of AdS4 black holes

“…The first step of holographic renormalization is to cancel the divergences and for supersymmetric theories with scalar fields there are few results in the literature, although recently some interesting works have appeared [11,12]. The divergent boundary counterterms we utilize are similar in form to those presented in [13]: a certain superpotential term cancels the cubic divergences and a term formed from the boundary Ricci scalar coupled to the scalar fields cancels the linear divergences.…”

“…The generalization of (3.21) to include scalar fields has been studied [13] and quite recently revisited to include the constraints imposed by supersymmetry [11,12], following which we generalize the second term in (3.21) with part of the superpotential (3.9) 22) canceling exactly the similar term in (3.15). The precise generalization of the first term in (3.21) is not immediately clear but it should be of the form…”