Cost of seven-brane gauge symmetry in a quadrillion F-theory compactifications

We study seven-branes in O(1015) four-dimensional F-theory compactifications where seven-brane moduli must be tuned in order to achieve non-Abelian gauge symmetry. The associated compact spaces B are the set of all smooth weak Fano toric threefolds. By a study of fine-star-regular triangulations of three-dimensional reflexive polytopes, the number of such spaces is estimated to be 5.8×1014 Nbases 1.8×1017. Typically hundreds or thousands of moduli must be tuned to achieve symmetry for h11(B)<10, but the average number drops sharply into the range O(25)-O(200) as h11(B) increases. For some low-rank groups, such as SU(2) and SU(3), there exist examples where only a few moduli must be tuned in order to achieve seven-brane gauge symmetry.