A Holographic Constraint on Scale Separation

TL;DR

This paper introduces a new holographic consistency condition linking AdS effective theories with dual CFTs, and verifies it in specific string theory models, providing insights into scale separation constraints.

Contribution

It proposes a novel criterion for the compatibility of gravitational effective theories with holographic duals, focusing on cubic scalar couplings and their vanishing in extremal cases.

Findings

Cubic scalar couplings for extremal operators should vanish according to the criterion.

The criterion is satisfied in the 4d $ ext{AdS}_4$ DGKT vacua in type IIA string theory.

Explicit three-point functions of scalar operators are computed in the dual 3d CFTs.

Abstract

We propose a new consistency condition for the compatibility of a gravitational effective field theory in AdS with a dual holographic description in terms of a family of large-$N$ CFTs. Using large-$N$ factorization of correlation functions combined with a properly defined notion of single- and multi-particle operators, we argue that the cubic scalar bulk couplings for fields dual to operators with extremal arrangements of the conformal dimensions, i.e. $\Delta_i=\Delta_j+\Delta_k$, should vanish. We apply this criterion to the 4d $\mathcal{N}=1$ effective supergravity theory describing the simplest DGKT AdS$_4$ vacua in type IIA string theory and show that it is non-trivially satisfied. In addition, we calculate explicitly all non-vanishing three-point correlation functions of low-lying scalar operators in the putative 3d CFTs dual to these AdS$_4$ string theory backgrounds.