Broken and restored: a holographic constraint for AdS vacua with orbifolds

It has been suggested that families of weakly-coupled AdS vacua with a large-$N$ holographic dual must satisfy non-trivial consistency requirements, which amount to the vanishing of certain cubic couplings, corresponding to (super-)extremal arrangements of scalar operators. While this constraint is known to hold in the simplest incarnation of the DGKT scenario in massive type IIA string theory, i.e. on the $\mathbb{Z}_3\times \mathbb{Z}_3$ orbifold, we find that it is generically violated for type II AdS$_3$ and AdS$_4$ vacua arising from $\mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_2$ and $\mathbb{Z}_2 \times \mathbb{Z}_2$ orbifolds respectively, including scale-separated solutions and DGKT-CFI-type models. In most cases, however, this can be cured by enlarging the orbifold group to a suitable (non-abelian) extension that projects out precisely those scalar operators that would otherwise participate in the constrained cubic couplings. Our results suggest that consistency of the putative holographic dual imposes a non-trivial restriction on the compactification geometry, indicating in particular that O-planes cannot wrap cycles in distinct homology classes.