Flux vacua in type IIB compactifications on orbifolds: their finiteness and minimal string coupling

We perform a detailed study of (supersymmetric) moduli stabilisation in type IIB toroidal orientifolds with fluxes. We provide strong evidence towards the finiteness of the number of inequivalent vacua for a given total 3-form flux charge Nflux. We also find that the minimal value of the string coupling gₛ is given in terms of Nflux and present strong evidence for the asymptotic relation gs,\,min~ c/Nflux^α, with α=1,2, valid for not too small Nflux, where c is an order 1 coefficient and α depends on the orbifold. Imposing tadpole cancellation, Nflux is bounded from the number of orientifold O3-planes. Combined with the flux quantisation, this reduces considerably the allowed vacua while the lower bound of the string coupling is then not far from unity. On the other hand, the presence of negative D3-brane charge induced by magnetised D7-branes breaks supersymmetry and relaxes the bound, allowing significantly smaller values for gₛ.