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

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.

arXiv comment11 pages

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• Statistics of Limit Root Bundles Relevant for Exact Matter Spectra of F-Theory MSSMs
• D7 Moduli Stabilization: The Tadpole Menace
• Brill-Noether-general Limit Root Bundles: Absence of vector-like Exotics in F-theory Standard Models
• Improved statistics for F-theory standard models
• Root bundles: Applications to F-theory Standard Models
• Vafa-Witten Theory: Invariants, Floer Homologies, Higgs Bundles, a Geometric Langlands Correspondence, and Categorification (String Math 2022 Proceedings)