6D F-theory models and elliptically fibered Calabi-Yau threefolds over semi-toric base surfaces

TL;DR

This work extends the classification of 6D F-theory bases beyond toric geometries by enumerating all smooth $ abla$C^*-bases that can host elliptic Calabi–Yau threefolds, yielding 162{,}404 bases and a rich set of Calabi–Yau geometries. It develops a concrete enumeration method based on chains of blow-ups on a Hirzebruch base, analyzes the resulting Hodge numbers via the Shioda–Tate–Wazir framework, and reveals both new Calabi–Yau threefolds and bases with nontrivial Mordell–Weil rank, i.e., non-Higgsable $U(1)$ factors. The study shows that the landscape is structurally controlled, with large-$T$ theories dominated by long ${\rak{e}}_8$-based chains and no evidence for a larger bound than $T=193$; it also demonstrates substantial redundancy from $-2$-clusters and identifies 6 Calabi–Yau spaces with previously unknown Hodge data and 13 with enhanced Mordell–Weil rank. Overall, the results extend the known shield of Hodge numbers into a broader, more general base class, while suggesting tractable prospects for fully classifying smooth F-theory bases in higher dimensions. The findings have implications for understanding abelian sectors in F-theory compactifications and for exploring non-toric geometries that realize novel Calabi–Yau threefolds.

Abstract

We carry out a systematic study of a class of 6D F-theory models and associated Calabi-Yau threefolds that are constructed using base surfaces with a generalization of toric structure. In particular, we determine all smooth surfaces with a structure invariant under a single C^* action (sometimes called "T-varieties" in the mathematical literature) that can act as bases for an elliptic fibration with section of a Calabi-Yau threefold. We identify 162,404 distinct bases, which include as a subset the previously studied set of strictly toric bases. Calabi-Yau threefolds constructed in this fashion include examples with previously unknown Hodge numbers. There are also bases over which the generic elliptic fibration has a Mordell-Weil group of sections with nonzero rank, corresponding to non-Higgsable U(1) factors in the 6D supergravity model; this type of structure does not arise for generic elliptic fibrations in the purely toric context.

Figures (17)

• Figure 1: Clusters of intersecting curves that must carry a nonabelian gauge group factor. For each cluster the corresponding gauge algebra is noted and the gauge algebra and number of charged matter hypermultiplet are listed in Table \ref{['t:clusters']}
• Figure 2: Toric base surfaces for 6D F-theory models produced by blowing up a sequence of points on $\mathbb{F}_2$.
• Figure 3: A $\mathbb{C}^*$-base $B$ is characterized by irreducible effective divisors $D_0, D_\infty$ connected by any number of linear chains of divisors $D_{i, j}$ with intersections obeying the cluster rules of clusters. All such bases can be realized as multiple blow-ups of a Hirzebruch surface $\mathbb{F}_m$ that preserve the action of a single $\mathbb{C}^*$ on $B$
• Figure 4: The $\mathbb{C}^*$-base with $N = 11, n_0 = 0, n_\infty = -11$ and 11 $(-1, -1)$ chains has a singular point on $D_\infty$ where the base must be blown up, giving the smooth base with $N = 12, n_\infty = -12$, and 12 $(-1, -1)$ chains.
• Figure 5: Number of distinct $\mathbb{C}^*$-bases as a function of the number of tensor multiplets $T$