Toric bases for 6D F-theory models

TL;DR

This study exhaustively classifies smooth toric bases that support elliptically fibered Calabi–Yau threefolds for 6D F-theory, enumerating 61,539 bases and linking base geometry to tensor multiplets via $T = h^{1,1}(B) - 1$. It provides explicit Weierstrass parameterizations from toric data and demonstrates that the count of Weierstrass moduli matches the gravitational anomaly constraint $H - V = 273 - 29T$, while identifying a characteristic growth of non-Higgsable gauge content at large $T$, dominated by ${rak e}_8$, ${rak f}_4$, and ${rak g}_2 oxplus {rak su}(2)$ blocks. The work shows that the maximal tensor multiplier value for toric bases is $T = 193$ (with evidence suggesting the bound persists non-toric), and analyzes how blowing up points affects the moduli and spectrum. It also outlines extensions to non-toric bases and to 4D F-theory, highlighting how the base geometry governs gauge structure and moduli, and setting a foundation for broader classifications in string compactifications.

Abstract

We find all smooth toric bases that support elliptically fibered Calabi-Yau threefolds, using the intersection structure of the irreducible effective divisors on the base. These bases can be used for F-theory constructions of six-dimensional quantum supergravity theories. There are 61,539 distinct possible toric bases. The associated 6D supergravity theories have a number of tensor multiplets ranging from 0 to 193. For each base an explicit Weierstrass parameterization can be determined in terms of the toric data. The toric counting of parameters matches with the gravitational anomaly constraint on massless fields. For bases associated with theories having a large number of tensor multiplets, there is a large non-Higgsable gauge group containing multiple irreducible gauge group factors, particularly those having algebras e_8, f_4 and (g_2 + su(2)) with minimal (non-Higgsable) matter.

Figures (13)

• 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 matter content are listed in Table \ref{['t:clusters']}
• Figure 2: Results of blowing up a point on the intersection structure of various configurations of irreducible effective curves. Arrow indicates the map $\pi:B' \rightarrow B$ from the blown-up space in each case.
• Figure 3: Toric diagram and corresponding loop of connected curves representing irreducible effective divisors for Hirzebruch surface $\mathbb{F}_2$.
• Figure 4: Blow-up and blow-down transitions connect the surfaces that can be used as F-theory bases. Examples of blow-ups connecting several toric bases are shown. Dashed (red) vectors and circled vertices represent points blown up on $\mathbb{F}_1$ and $\mathbb{F}_2$ that give a common toric base with $h^{1, 1} (B) = 3$.
• Figure 5: The number of distinct toric bases for F-theory compactifications for different numbers $T$ of tensor multiplets. There are 61,539 toric bases including those with $-9, -10, -11$ curves that must be blown up (upper blue data), and 34,868 truly toric bases not including such curves (lower purple data). The largest number of tensor multiplets is $T = 193$.