Holographic Constraint on Scale Separation

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 multiparticle operators, we argue that the cubic scalar bulk couplings for fields dual to operators with extremal arrangements of the conformal dimensions should vanish. We apply this criterion to the 4d N=1 effective supergravity theory describing the simplest DGKT AdS_{4} vacua in type IIA string theory and show that it is nontrivially satisfied. In addition, we calculate explicitly all nonvanishing three-point correlation functions of low-lying scalar operators in the putative 3D CFTs dual to these AdS_{4} string theory backgrounds.