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

TL;DR

This work performs a Monte Carlo exploration of toric threefold bases B that support elliptic Calabi–Yau fourfolds for 4d F-theory, aiming to characterize the landscape of geometries and their non-Higgsable gauge content. By constructing bases connected to P^3 through sequences of blow-ups and blow-downs while avoiding (4,6) loci, the authors estimate an enormous connected set with |C| ≈ 10^{48} and a typical base with h^{1,1}(B) ≈ 82. They find non-Higgsable gauge factors accumulate roughly linearly with h^{1,1}(B), with SU(2) and G2 dominating and a frequent SU(3)×SU(2) two-factor product arising on about a majority of bases, suggesting a natural path to MSSM-like structure within the geometric framework. The results reveal that typical 4d F-theory vacua built on these bases exhibit large, intricate non-Higgsable clusters and rich matter content, underscoring both the scale of possible Calabi–Yau fourfolds and the need to extend analyses beyond toric bases and to include G-flux and other dynamics for a complete physical picture.

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.

Figures (22)

• Figure 1: Illustration of two different kinds of blow ups, viewed from above. The left case corresponds to blowing up a point $v_i v_j v_k$. The right case corresponds to blowing up a curve $v_i v_j$.
• Figure 2: Illustration of the flop process, which can happen when $v_i+v_j=v_k+v_l$.
• Figure 3: The distribution of $h^{1, 1}(B)$, including weighting factors. Bold black curve is total distribution (divided by the number of runs, 100), colored curves are some example distributions from individual runs. The total number of samples in each run is normalized to 100,000.
• Figure 4: (Approximate) Hodge numbers for generic elliptically fibered fourfolds over the bases $B$ encountered in two of the random walks through the space of connected bases.
• Figure 5: Mean values of the (approximate) Hodge numbers for generic elliptic Calabi-Yau fourfolds over the threefold bases encountered in each of the independent Monte Carlo runs.