We introduce a method for finding flux vacua of type IIB string theory in which the flux superpotential is exponentially small and at the same time one or more complex structure moduli are stabilized exponentially near to conifold points.

We study the topological properties of Calabi-Yau threefold hypersurfaces at large h 1,1 . We obtain two million threefolds X by triangulating polytopes from the Kreuzer-Skarke list, including all polytopes with 240 ≤ h 1,1 ≤ 491. We show that the Kähler cone of X is very narrow at large h 1,1 , and as a consequence, control of the α expansion in string compactifications on X is correlated with the presence of ultralight axions. If every effective curve has volume ≥ 1 in string units, then the typical volumes of irreducible effective curves and divisors, and of X itself, scale as (h 1,1 ) p , with 3 p 7 depending on the type of cycle in question. Instantons from branes wrapping these cycles are thus highly suppressed.

In order to generalize the mechanism of [5] to include conifolds one has to overcome the following obstacle. Introducing fluxes on the conifold cycles generates a conifold superpotential W cf that by itself cannot be tuned to be small. Thus, the total flux superpotential will be small in string units only if the large conifold superpotential is efficiently canceled by a comparably large contribution W bulk generated by fluxes on other cycles, i.e. if 1 See e.g. [37] for an analysis of a de Sitter solution arising from an explicit flux vacuum.

We construct supersymmetric AdS4 vacua of type IIB string theory in compactifications on orientifolds of Calabi-Yau threefold hypersurfaces. We first find explicit orientifolds and quantized fluxes for which the superpotential takes the form proposed by Kachru, Kallosh, Linde, and Trivedi. Given very mild assumptions on the numerical values of the Pfaffians, these compactifications admit vacua in which all moduli are stabilized at weak string coupling. By computing high-degree Gopakumar-Vafa invariants we give strong evidence that the α′ expansion is likewise well-controlled. We find extremely small cosmological constants, with magnitude < 10−123 in Planck units. The compactifications are large, but not exponentially so, and hence these vacua manifest hierarchical scale-separation, with the AdS length exceeding the Kaluza-Klein length by a factor of a googol.