Classifying Fibers and Bases in Toric Hypersurface Calabi-Yau Threefolds

TL;DR

The paper conducts a comprehensive classification of toric elliptic and genus-one fibrations across nearly half a billion reflexive 4D polytopes from the Kreuzer–Skarke database, revealing 2.26 billion fibrations that collapse to about 2.25 billion automorphism-inequivalent classes. By combining a lattice-based fiber-detection algorithm with Nagell’s/Wierstrass techniques and F-theory anomaly constraints, the authors quantify distributions of fiber types, toric bases, and base-fiber pairings, and they expose rich structure including non-toric divisors hosting SU(N) factors, multiple U(1) symmetries, and SCFT sectors arising from non-flat fibers. The results highlight the utility of the F-theory framework for organizing and interpreting Calabi–Yau threefolds, uncovering ubiquitous fibration structures, typical bases (e.g., base 72 with fiber F6), and extreme examples with large Hodge numbers (notably h^{1,1}=491). Overall, the work demonstrates that toric hypersurface CY3s provide a broad, physically informative slice of the elliptic/genus-one CY landscape and motivate further exploration of non-toric bases and higher-section genus-one fibrations.

Abstract

We carry out a complete analysis of the toric elliptic and genus-one fibrations of all 474 million reflexive polytopes in the Kreuzer-Skarke database. Earlier work with Huang showed that all but 29,223 of these polytopes have such a fibration. We identify 2,264,992,252 distinct fibrations, and determine the fiber and base structure in each case; after accounting for automorphisms of the ambient polytope, these fibrations furnish 2,250,744,657 equivalence classes. We summarize generic features and identify exotic special cases among these fibrations. These fibrations illustrate many features that have been explored in the context of 6D F-theory, including gauge groups hosted on non-toric divisors, automatic enhancement of gauge groups, and implicit non-toric bases and high-rank 6D SCFTs associated with nonflat fibers, as well as novel geometric features such as singular bases for genus-one fibrations with multisections. This analysis illustrates the power of elliptic and genus-one fibrations, and the geometro-physical language of F-theory as a tool for understanding the structure of Calabi-Yau threefolds.

Figures (15)

• Figure 1: The sixteen two-dimensional reflexive polytopes. Vertices are connected by black lines, the origin is indicated by a cross, and all other points are indicated by a black dot.
• Figure 2: The average (a), maximum (b), and minimum (c) number of automorphism-inequivalent fibrations over a polytope giving Calabi-Yau threefolds $X$ with Hodge numbers $h^{1, 1}(X)$ and $h^{2, 1}(X)$ are plotted. The black triangles indicate zero structures, the blue colors indicate smaller number of structures, up to 10, and the color bar on the right indicates the color corresponding to larger number of fibrations (more red means larger number of structures).
• Figure 3: The density of inequivalent fibrations by each of the sixteen fiber types.
• Figure 4: The total number of automorphism-inequivalent fibrations of each fiber type as a function of $h^{1,1}(X)$. Note that at large $h^{1,1} (X)$, above $h^{1,1} (X)\sim 400$, all polytopes with a fiber $F_{13}$ also have a fiber $F_{10}$.
• Figure 5: The mean, minimum, maximum and $1\sigma$ deviation of the number of inequivalent fibrations over a base $B$ with a given $h^{1,1}(B)$ are plotted. Note that there are more structures fibered over bases with small $h^{1, 1}(B).$