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

A bstract 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 ∼ $$ c/{N}_{\textrm{flux}}^{\alpha } $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>c</mml:mi> <mml:mo>/</mml:mo> <mml:msubsup> <mml:mi>N</mml:mi> <mml:mtext>flux</mml:mtext> <mml:mi>α</mml:mi> </mml:msubsup> </mml:math> , 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 .