Enhanced gauge symmetry in 6D F-theory models and tuned elliptic Calabi-Yau threefolds

TL;DR

This work develops a systematic, local-tuning framework for 6D F-theory compactifications by classifying admissible enhancements of gauge symmetries and matter on divisors of complex base surfaces. It bridges anomaly cancellation constraints with explicit Weierstrass/Tate realizations, revealing a near-complete match on isolated curves and multi-curve clusters, while identifying swampland configurations where low-energy consistency does not admit known F-theory realizations. The authors introduce a tunings algorithm that, given a base, yields a finite set of candidate elliptic Calabi–Yau threefolds and corresponding 6D SUGRA spectra, and they illustrate this with Kreuzer–Skarke examples and tunings on $b F_{12}$. They also extend the analysis to abelian and exotic matter, discuss generalized $E_8$-type constraints, and discuss potential 4D extensions, thereby framing a comprehensive path toward classifying elliptic Calabi–Yau threefolds and their 6D/F-theory realizations.

Abstract

We systematically analyze the local combinations of gauge groups and matter that can arise in 6D F-theory models over a fixed base. We compare the low-energy constraints of anomaly cancellation to explicit F-theory constructions using Weierstrass and Tate forms, and identify some new local structures in the "swampland' of 6D supergravity and SCFT models that appear consistent from low-energy considerations but do not have known F-theory realizations. In particular, we classify and carry out a local analysis of all enhancements of the irreducible gauge and matter contributions from "non-Higgsable clusters," and on isolated curves and pairs of intersecting rational curves of arbitrary self-intersection. Such enhancements correspond physically to unHiggsings, and mathematically to tunings of the Weierstrass model of an elliptic CY threefold. We determine the shift in Hodge numbers of the elliptic threefold associated with each enhancement. We also consider local tunings on curves that have higher genus or intersect multiple other curves, codimension two tunings that give transitions in the F-theory matter content, tunings of abelian factors in the gauge group, and generalizations of the "$E_8$" rule to include tunings and curves of self-intersection zero. These tools can be combined into an algorithm that in principle enables a finite and systematic classification of all elliptic CY threefolds and corresponding 6D F-theory SUGRA models over a given compact base (modulo some technical caveats in various special circumstances), and are also relevant to the classification of 6D SCFT's. To illustrate the utility of these results, we identify some large example classes of known CY threefolds in the Kreuzer-Skarke database as Weierstrass models over complex surface bases with specific simple tunings, and we survey the range of tunings possible over one specific base.

Figures (4)

• Figure 1: A representation of monomials in $-4K$ (left) and $-6K$ (right) over the local model of a $-3$ curve $\Sigma$ and its two neighbors. Both sets of monomials should be considered as extending infinitely in the positive $x$ and $y$ directions. To write these monomials explicitly, we may establish a local coordinate system $z$, $w$ such that $\Sigma=\{z=0\}$ (corresponding to the ray $(1,0)$ of the toric fan) and one of its two neighbors $\Sigma'$ (corresponding to the ray $(0,1)$) is $\Sigma'=\{w=0\}$. Then a monomial $(a,b)\in -kK$ corresponds concretely to $z^{a+k}w^{b+k}$.
• Figure 2: Some configurations of $-2$ curves associated with Kodaira-type surface singularities associated with degenerate elliptic fibers. The numbers given are the weightings needed to give an elliptic curve with vanishing self-intersection. Labels correspond to Kodaira singularity type and associated Dynkin diagram.
• Figure 3: [Color online.] Tunings of ${\mathfrak{e}}_6$ and ${\mathfrak{e}}_7$ on $-4$ curves. Blue dots mark Hodge numbers of untuned models over toric bases where the ${\mathfrak{so}}(8)$ gauge symmetry on a $-4$ curve can be enhanced to at least an ${\mathfrak{e}}_6$. Orange dots mark Hodge numbers of tuned models over the same bases (without distinguishing between ${\mathfrak{e}}_6$ and ${\mathfrak{e}}_7$. Here the $y$ axis $h^{2,1}$ is plotted versus the $x$ axis $h^{1,1}$.
• Figure 4: [Color online.] Tunings of ${\mathfrak{su}}(2)$ on (chains of) $-2$ curves. As for the previous example, blue dots mark Hodge numbers of untuned models over toric bases with $-2$ curves that can support a ${\mathfrak{su}}(2)$ gauge symmetry. Orange dots mark Hodge numbers of tuned models over the same bases. Here the $y$ axis $h^{2,1}$ is plotted versus the $x$ axis $h^{1,1}$.