A Monte Carlo exploration of threefold base geometries for 4d F-theory vacua

We use Monte Carlo methods to explore the set of toric threefold bases that support elliptic Calabi-Yau fourfolds for F-theory compactifications to four dimensions, and study the distribution of geometrically non-Higgsable gauge groups, matter, and quiver structure. We estimate the number of distinct threefold bases in the connected set studied to be ∼10⁴⁸. The distribution of bases peaks around h^1,1∼82. All bases encountered after "thermalization" have some geometric non-Higgsable structure. We find that the number of non-Higgsable gauge group factors grows roughly linearly in h^1,1 of the threefold base. Typical bases have ∼6 isolated gauge factors as well as several larger connected clusters of gauge factors with jointly charged matter. Approximately 76% of the bases sampled contain connected two-factor gauge group products of the form SU(3)×SU(2), which may act as the non-Abelian part of the standard model gauge group. SU(3)×SU(2) is the third most common connected two-factor product group, following SU(2)×SU(2) and G₂×SU(2), which arise more frequently.