Abelian F-theory Models with Charge-3 and Charge-4 Matter

TL;DR

The paper addresses how to realize abelian F-theory models with higher charge matter, focusing on charges $q=3$ and $q=4$. It develops a systematic $q=3$ tuning using a procedure akin to SU(N) and Sp(N) tunings, revealing non-UFD Weierstrass structures tied to the vanishing order of the generating section components, and connects this to the normalized intrinsic ring framework. It also constructs a $q=4$ model by Higgsing a $ ext{U}(1) imes ext{U}(1)$ setup, marking the first published example with charge $4$ in F-theory and illustrating how division by section components signals non-UFD behavior. The work relates these abelian constructions to Morrison-Park and KMOPR forms via birational equivalences, and discusses unHiggsings to non-abelian groups along with conjectures about models with $q>4$, thereby advancing understanding of high-charge spectra and their geometric realization in Calabi–Yau fibrations.

Abstract

This paper analyzes U(1) F-theory models admitting matter with charges $q=3$ and $4$. First, we systematically derive a $q=3$ construction that generalizes the previous $q=3$ examples. We argue that U(1) symmetries can be tuned through a procedure reminiscent of the SU(N) and Sp(N) tuning process. For models with $q=3$ matter, the components of the generating section vanish to orders higher than 1 at the charge-3 matter loci. As a result, the Weierstrass models can contain non-UFD structure and thereby deviate from the standard Morrison-Park form. Techniques used to tune SU(N) models on singular divisors allow us to determine the non-UFD structures and derive the $q=3$ tuning from scratch. We also obtain a class of a $q=4$ models by deforming a prior U(1)$\times$U(1) construction. To the author's knowledge, this is the first published F-theory example with charge-4 matter. Finally, we discuss some conjectures regarding models with charges larger than 4.

Figures (1)

• Figure 1: Illustration of the fibers at $a_1=b_1=0$ loci for the $\mathbb{P}^2$ form of the fibrations. The fiber splits into two components. One component is given by $u=0$, and the other is denoted as $c$. The zero section $P$ wraps the $u=0$ component of the fiber, indicated by the filling of the $u=0$ component in the illustration.