Statistics of Base Polytopes in F-theory

TL;DR

The paper develops a Monte Carlo framework to sample toric base polytopes for elliptic Calabi–Yau geometries in F-theory, focusing on triangulation-free base polytopes to coarse-grain the landscape. It introduces redundancy-corrected sampling within fixed boxes, derives practical formulas for counting polytopes, and demonstrates the method on 2d and 3d bases, validating against exact counts in 2d and estimating astronomical numbers of polytopes in 3d (up to ~10^{90} across boxes). The analysis yields Gaussian behavior in weight distributions and reveals robust patterns in non-Higgsable gauge sectors, notably the dominance of E8 factors and the prevalence of rigid gauge groups across the base ensembles. The work provides a scalable, combinatorics-focused view of the F-theory landscape, offering a complementary perspective to fully classified, triangulation-dependent bases and suggesting directions for extending to reflexive polytopes and non-toric geometries. It also highlights the importance of the cosmological measure problem and proposes further exploration of flux choices and phenomenological implications in these polytope ensembles.

Abstract

We propose a new statistical ensemble of toric bases for elliptic Calabi-Yaus used in F-theory models, by focusing on only the convex hull of the base, i.e., the base polytope. This physically motivated coarse-graining greatly simplifies the combinatorial complexity of the part of the 4D F-theory landscape with toric bases. We develop a Monte Carlo approach that randomly samples the base polytopes within fixed boxes, with proper statistical weights. We first apply the algorithm to the set of 2d base polytopes, generating an enlarged set of toric 2d bases that include certain types of codimension-two (4,6) points, and we validate our approach against exact numbers. We then explore the set of 3d base polytopes which fit in a set of ``maximal'' 3d boxes, and estimate the total number of inequivalent 3d base polytopes to be $\sim 10^{85}$--$10^{90}$. We provide statistical data such as the distribution of non-Higgsable gauge groups on these bases. Amusingly, a similar method can also be applied to generate reflexive polytopes in various dimensions. In both the reflexive and base polytope cases, the number of relevant polytopes obeys a Gaussian distribution as a function of the number of vertices, which can be understood in terms of other results on random polytopes in the math literature.

Figures (22)

• Figure 1: The estimated number of 2d base polytopes for a given $h^{1,1}(B_2)$ and its $\log_{10}$, in blue color, with all the combined data across different boxes in Table \ref{['t:2D-box-results']}. The regimes with small or large $h^{1,1}$ are magnified, for the $\log_{10}N_{\rm tot}$. The orange data points denote the exact number of 2d base polytopes for such $h^{1,1}(B_2)$ evaluated by the algorithm in Section \ref{['sec:exact-2D']}.
• Figure 2: The distribution of the weight factors $\log_{10}(w_i)$ in the Monte Carlo sampling of 2d base polytopes with $k=6$, across different boxes, with the data in Table \ref{['t:2D-box-results']}. The horizontal axis is $\log_{10}(w_i)$, while the vertical axis is the number of samplings in each bin. The blue curve is the optimal fitting with a Gaussian function.
• Figure 3: The distribution of the number of vertices $m$ for each sample in the Monte Carlo sampling of 2d base polytopes with $k=6$, across different boxes, with the data in Table \ref{['t:2D-box-results']}. The vertical axis denotes the number of samplings for each $m$. The blue curve is the optimal fitting with a Gaussian function.
• Figure 4: The distribution of $\log_{10}(w_i)$ for the 2d Monte Carlo approach, for a fixed $h^{1,1}(B_2)$. The top-left, top-right, bottom-left and bottom-right figures correspond to $h^{1,1}(B_2)=50$, $100$, $150$ and $180$ respectively.
• Figure 5: The estimated number of 2d base polytopes for a given $h^{1,1}(B_2)$ and its $\log_{10}$, for the $k = 5$ case, with all the combined data across different boxes in Table \ref{['t:2Dboxes-k5']}. We have matched the numbers from the Monte Carlo program (in blue) with the exact numbers (in orange). The regimes with small or large $h^{1,1}(B_2)$ are magnified.