Abstract
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 $\sim { 10^{48}}$. The distribution of bases peaks around $h^{1, 1}\sim 82$. 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 $\sim 6$ 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)$\times$SU(2), which may act as the non-Abelian part of the standard model gauge group. SU(3)$\times$SU(2) is the third most common connected two-factor product group, following SU(2)$\times$SU(2) and $G_2\times$SU(2), which arise more frequently.
Highlights
Calabi-Yau threefolds has been developed [4,5,6,7,8,9,10]. The upshot of this story is that toric bases seem to provide a good representative sample of the set of all possible base surfaces that support elliptic Calabi-Yau threefolds, and that an important part of the basic physics of each base B is captured by the gauge groups and matter in the “non-Higgsable clusters” that are present for a generic elliptic fibration over B
As found for threefolds, geometrically non-Higgsable gauge groups and matter that are present everywhere in the Weierstrass moduli space of elliptic fibrations over a given base B give a strong guide to the physics and and aid in the classification of 4d F-theory models through the structure of allowed threefold base geometries
We review here some basic aspects of F-theory, related to the geometry of the base B and non-Abelian gauge groups associated with Kodaira singularities in the elliptic fibration
Summary
Calabi-Yau threefolds has been developed [4,5,6,7,8,9,10]. At a simplified level, the upshot of this story is that toric bases seem to provide a good representative sample of the set of all possible base surfaces that support elliptic Calabi-Yau threefolds, and that an important part of the basic physics of each base B is captured by the gauge groups and matter in the “non-Higgsable clusters” that are present for a generic elliptic fibration over B. As found for threefolds, geometrically non-Higgsable gauge groups and matter that are present everywhere in the Weierstrass moduli space of elliptic fibrations over a given base B give a strong guide to the physics and and aid in the classification of 4d F-theory models through the structure of allowed threefold base geometries. Previous explorations of the range of geometries available for 4d F-theory vacua have focused on certain simple classes of elliptic Calabi-Yau fourfolds and threefold bases, in particular threefold bases that are Fano threefolds or P1 bundles over P2 or other base surfaces [12, 15,16,17,18,19].
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