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 $N_{\mathrm{f}\mathrm{l}\mathrm{u}\mathrm{x}}$. We also find that the minimal value of the string coupling $g_s$ is given in terms of $N_{\mathrm{f}\mathrm{l}\mathrm{u}\mathrm{x}}$ and present strong evidence for the asymptotic relation $g_{s,\mathrm{m}\mathrm{i}\mathrm{n}}\sim c\mathrm{/}{N}_{\mathrm{f}\mathrm{l}\mathrm{u}\mathrm{x}}^{\alpha }$, with $\alpha =1,2$, valid for not too small $N_{\mathrm{f}\mathrm{l}\mathrm{u}\mathrm{x}}$, where $c$ is an order 1 coefficient and $\alpha$ depends on the orbifold. Imposing tadpole cancellation, $N_{\mathrm{f}\mathrm{l}\mathrm{u}\mathrm{x}}$ 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 $g_s$.