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 exhaustion of the finite number of inequivalent vacua for a given total 3-form flux charge N_flux. We also find that the minimal value of the string coupling g_s is given in terms of N_flux and present strong evidence for the asymptotic relation g_{s, min} ∼ $c/N_{\text{flux}}^{\alpha }$ \( c/{N}_{\textrm{flux}}^{\alpha } \) , with α = 1, 2, valid for not too small N_flux, where c is an order 1 coefficient and α depends on the orbifold. Imposing tadpole cancellation, N_flux is bounded from the number of orientifold O3-planes. Combined with the flux quantisation, this forbids orientifold vacua without anti-brane charge. 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 g_s.