Tate's Algorithm for F-theory GUTs with two U(1)s

TL;DR

This work provides a systematic Tate’s-algorithm treatment for elliptic fibrations with two extra rational sections, yielding a large class of $SU(5)\times U(1)^2$ F-theory models. By encoding the fibration as a cubic in $\mathbb{P}^2$ and employing a detailed resolution and Shioda-map framework, the authors derive canonical and non-canonical $I_n$ and exceptional fibers, including novel $I_5$ realizations with multiple charged ${\mathbf{10}}$ and ${\mathbf{5}}$ curves. They also connect these geometric constructions to known toric tops, produce a comprehensive table of Tate-like forms for two $U(1)$s, and extend the analysis to exceptional fibers ($IV^*$, $III^*$, $II^*$) with both split and non-split possibilities. The results broaden the landscape of globally consistent F-theory GUTs with multiple abelian factors and offer new phenomenologically interesting charge configurations for matter fields. Overall, the paper provides a robust, geometry-driven roadmap for engineering $SU(5)$-based GUTs with two extra $U(1)$ symmetries and lays groundwork for exploring proton-decay suppression and other phenomenological features via abelian charges.

Abstract

We present a systematic study of elliptic fibrations for F-theory realizations of gauge theories with two U(1) factors. In particular, we determine a new class of SU(5) x U(1)^2 fibrations, which can be used to engineer Grand Unified Theories, with multiple, differently charged, 10 matter representations. To determine these models we apply Tate's algorithm to elliptic fibrations with two U(1) symmetries, which are realized in terms of a cubic in P^2. In the process, we find fibers which are not characterized solely in terms of vanishing orders, but with some additional specialization, which plays a key role in the construction of these novel SU(5) models with multiple 10 matter. We also determine a table of Tate-like forms for Kodaira fibers with two U(1)s.

Figures (7)

• Figure 1: The type $I_2$ and type $III$ singular fibers with the possible locations of the three marked points denoted by the blue nodes. Respectively these are $I_2^{(ijk)}$, $I_2^{(i|jk)}$, $II^{(ijk)}$ and $II^{(i|jk)}$ fibers.
• Figure 2: The type $I_3$ singular fibers with the locations of the three marked points denoted by the blue nodes. Respectively these are $I_3^{(ijk)}$, $I_3^{(ij|k)}$ and $I_3^{(i|j|k)}$ fibers.
• Figure 3: The $I_4$ singular fibers and the decorations detailing where the rational sections can intersect. The fibers shown are $I_4^{(ijk)}$, $I_4^{(ij|k)}$, $I_4^{(ij||k)}$ and $I_4^{(i|j|k)}$ fibers.
• Figure 4: The $IV$ fibers. We denote by the blue nodes the components of the fiber which are intersected by the sections. In the order, the fiber shown are $IV^{(ijk)}$, $IV^{(i|jk)}$ and $IV^{(i|j|k)}$ fibers.
• Figure 5: The $I_5$ singular fibers. The possible intersections of the sections with the singular fibers are denoted by the positions of the blue nodes. The fibers shown in the first row are $I_5^{(ijk)}$, $I_5^{(ij|k)}$ and $I_5^{(ij||k)}$, whereas the fibers shown in the second row are, respectively, $I_5^{(i|j|k)}$ and $I_5^{(i|j||k)}$.