The Cost of Seven-brane Gauge Symmetry in a Quadrillion F-theory Compactifications

TL;DR

This paper estimates the vast number of F-theory compactifications requiring tuning of moduli to achieve non-abelian gauge symmetry, revealing that the tuning complexity varies significantly with the properties of the compactification space.

Contribution

It provides a quantitative estimate of the number of F-theory bases and analyzes the moduli tuning required for gauge symmetry in a large class of compactifications.

Findings

Number of bases estimated between 5.8×10^{14} and 1.8×10^{17}.

Average moduli tuned decreases as h^{11}(B) increases.

Few moduli needed for certain low-rank gauge groups like SU(2) and SU(3).

Abstract

We study seven-branes in $O(10^{15})$ 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 $5.8\times 10^{14}\lesssim N_\text{bases}\lesssim 1.8\times 10^{17}$. Typically hundreds or thousands of moduli must be tuned to achieve symmetry for $h^{11}(B)<10$, but the average number drops sharply into the range $O(25)$-$O(200)$ as $h^{11}(B)$ 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.