Tate Trees for Elliptic Fibrations with Rank one Mordell-Weil group

TL;DR

This work constructs a comprehensive Tate-tree framework for elliptic fibrations with rank-one Mordell-Weil, realized as quartics in ${\tt P}^{(1,1,2)}$, to enumerate all canonical and non-canonical Tate forms across Kodaira types. By solving discriminant conditions order-by-order in a local base coordinate $z$ and applying crepant resolutions, the authors map out canonical $I_n$ and $I_n^*$ branches and reveal abundant non-canonical enhancements that generate richer codimension-two matter spectra and $U(1)$ charges. The analysis yields complete codimension-one fiber classifications and explicit matter content for $SU(5)\times U(1)$ models, along with a detailed account of infinite families (the Tate tree tops) and the connections to toric tops/spectral-cover pictures. The results show non-canonical models are common in ${\mathbb P}^{(1,1,2)}$, enabling multiple distinct matter curves and novel charge assignments, which broadens F-theory model-building possibilities beyond canonical Tate forms. The framework thus provides a systematic catalog linking fiber types, section separations, matter loci, and $U(1)$ charges, with direct implications for realistic GUT constructions in F-theory.

Abstract

U(1) symmetries play a central role in constructing phenomenologically viable F-theory compactifications that realize Grand Unified Theories (GUTs). In F-theory, gauge symmetries with abelian gauge factors are modeled by singular elliptic fibrations with additional rational sections, i.e. a non-trivial Mordell-Weil rank. To determine the full scope of possible low energy theories with abelian gauge factors, which allow for an F-theory realization, it is central to obtain a comprehensive list of all singular elliptic fibrations with extra sections. We answer this question for the case of one abelian factor by applying Tate's algorithm to the elliptic fiber realized as a quartic in the weighted projective space P^{(1,1,2)}, which guarantees, in addition to the zero section, the existence of an additional rational section. The algorithm gives rise to a tree-like enhancement structure, where each fiber is characterized by a Kodaira fiber type, that governs the non-abelian gauge factor, and the separation of the two sections. We determine Tate-like forms for elliptic fibrations with one extra section for all Kodaira fiber types. In addition to standard Tate forms that are determined by the vanishing order of the coefficient sections in the quartic (so-called canonical models),the algorithm also gives rise to fibrations that require non-trivial relations among the coefficient sections. Such non-canonical models have phenomenologically interesting properties, as they allow for a richer charged matter content, and thus codimension two fiber structure, than the canonical models that have been considered thus far in the literature. As an application we determine the complete set of codimension one fibers types, matter spectra, both canonical and non-canonical, for SU(5) x U(1) models.

Figures (13)

• Figure 1: The Tate tree for $\mathbb{P}^{(1,1,2)}$, up until and including the $O(z^5)$ discriminant fibers. Black lines are irreducible fiber components of the singular fibers, red nodes show where the sections intersect the irreducible components. Fiber types with green corners are non-canonical models. From each of these, there is another branch of the tree sprouting off, with multiply non-canonical fiber types. Enhancements from type $II, III, IV$ are not shown, but discussed in the text.
• Figure 3: $I_3^{(01)}$ and $I_3^{(0|1)}$ fibers. The black lines correspond to the $\mathbb{P}^1$ fiber components, and the red nodes to the two sections, $\sigma_0$ and $\sigma_1$. Due to the symmetry of the diagram, there are only two distinct distributions of the two sections.
• Figure 4: $III^{(01)}$ and $III^{(0|1)}$ fibers, again with black lines corresponding to the fiber components, and the red dots to the extra sections.
• Figure 5: From left to right, showing the $I_4^{(01)}$, $I_4^{(0|1)}$ and $I_4^{(0||1)}$ fibers, respectively, with sections indicated by the red nodes.
• Figure 6: $IV^{(01)}$ and $IV^{(0|1)}$ fibers.