Membranes in AdS4 orientifold vacua and their Weak Gravity Conjecture

We study type IIA orientifold compactifications with fluxes that give rise to perturbatively stable, non-supersymmetric AdS$_4$ vacua with D6-brane gauge sectors. Non-perturbative instabilities can be mediated by D8-branes wrapped on the six internal dimensions $X_6$, if they reduce to 4d membranes with a charge $Q$ larger than its tension $T$. The mismatch $Q \neq T$ arises due to i) curvature corrections and ii) the BIon backreaction of D6-branes wrapping 3-cycles of $X_6$. We give a simple expression for the second effect in toroidal orientifolds, and find that only pairs of 3-cycles at SU(2) angles contribute to it. They either contribute towards $Q>T$ or $Q<T$ depending on the 3-cycles separation, allowing to engineer 4d $\mathcal{N}=0$ vacua in tension with the Weak Gravity Conjecture for membranes.