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.

We study Calabi-Yau threefolds with large Hodge numbers by constructing and counting triangulations of reflexive polytopes. By counting points in the associated secondary polytopes, we show that the number of fine, regular, star triangulations of polytopes in the Kreuzer-Skarke list is bounded above by (14,111 494) ≈ 10 928. Adapting a result of Anclin on triangulations of lattice polygons, we obtain a bound on the number of triangulations of each 2-face of each polytope in the list. In this way we prove that the number of topologically inequivalent Calabi-Yau hypersurfaces arising from the Kreuzer-Skarke list is bounded above by 10 428. We introduce efficient algorithms for constructing representative ensembles of Calabi-Yau hypersurfaces, including the extremal case h 1,1 = 491, and we study the distributions of topological and physical data therein. Finally, we demonstrate that neural networks can accurately predict these data once the triangulation is encoded in terms of the secondary polytope.

No abstract

We study Calabi-Yau threefolds with large Hodge numbers by constructing and counting triangulations of reflexive polytopes. By counting points in the associated secondary polytopes, we show that the number of fine, regular, star triangulations of polytopes in the Kreuzer-Skarke list is bounded above by 14,111 494 ≈ 10 928 . Adapting a result of Anclin on triangulations of lattice polygons, we obtain a bound on the number of triangulations of each 2-face of each polytope in the list. In this way we prove that the number of topologically inequivalent Calabi-Yau hypersurfaces arising from the Kreuzer-Skarke list is bounded above by 10 428 . We introduce efficient algorithms for constructing representative ensembles of Calabi-Yau hypersurfaces, including the extremal case h 1,1 = 491, and we study the distributions of topological and physical data therein. Finally, we demonstrate that neural networks can accurately predict these data once the triangulation is encoded in terms of the secondary polytope.

We construct a vacuum of string theory in which the magnitude of the vacuum energy is < 10 −123 in Planck units. Regrettably, the sign of the vacuum energy is negative, and some supersymmetry remains unbroken.