SU(5) Tops with Multiple U(1)s in F-theory

TL;DR

This work develops a systematic, geometry-driven framework for F-theory compactifications with up to two Abelian $U(1)$ factors, realized via ${\rm Bl}_2\mathbb{P}^2[3]$-fibrations and Mordell–Weil rank two. By exploiting both hypersurface and complete-intersection representations, the authors establish a holomorphic zero-section, construct two independent $U(1)$ generators via the Shioda map, and build a class of transversely constrained $G_4$ fluxes that yield calculable chiral spectra. They instantiate this with SU(5) tops realized torically, deriving an explicit Top 4 model with $SU(5)\times U(1)\times U(1)$, analyzing flatness to avoid non-flat fibres, and computing the resulting $SU(5)$-charged matter and singlet chirality. A key finding is that one global model cannot be embedded into a local Higgsed $E_8$ theory due to necessary matter-curve recombination, a phenomenon accommodated by recombination singlets beyond $E_8$, thereby expanding the model-building possibilities in F-theory. The paper also discusses local spectral-cover perspectives and demonstrates that certain global constructions may not have a split spectral-cover description in their local limits, underscoring the intricate interplay between global geometry and local model-building frameworks.

Abstract

We study F-theory compactifications with up to two Abelian gauge group factors that are based on elliptically fibered Calabi-Yau 4-folds describable as generic hypersurfaces. Special emphasis is put on elliptic fibrations based on generic Bl^2 P^2[3]-fibrations. These exhibit a Mordell-Weil group of rank two corresponding to two extra rational sections which give rise to two Abelian gauge group factors. We show that an alternative description of the same geometry as a complete intersection makes the existence of a holomorphic zero-section manifest, on the basis of which we compute the U(1) generators and a class of gauge fluxes. We analyse the fibre degenerations responsible for the appearance of localised charged matter states, whose charges, interactions and chiral index we compute geometrically. We implement an additional SU(5) gauge group by constructing the four inequivalent toric tops giving rise to SU(5) x U(1) x U(1) gauge symmetry and analyse the matter content. We demonstrate that notorious non-flat points can be avoided in well-defined Calabi-Yau 4-folds. These methods are applied to the remaining possible hypersurface fibrations with one generic Abelian gauge factor. We analyse the local limit of our SU(5) x U(1) x U(1) models and show that one of our models is not embeddable into E8 due to recombination of matter curves that cannot be described as a Higgsing of E8. We argue that such recombination forms a general mechanism that opens up new model building possibilities in F-theory.

Figures (12)

• Figure 1: Polygon for $\textmd{Bl}_{2}\mathbb{P}^{2}$
• Figure 2: Topology of the fibre over the six singlet curves $C_{\mathbf 1^{(i)}}$, $i=1,\ldots,6$. Green denotes the zero section $S_0$, blue denotes $S_1$ and red corresponds to $S_2$.
• Figure 3: The fibre structure over three Yukawa points. Green corresponds to the zero section $S_0$, blue to $S_1$ and red to $S_2$.
• Figure 4: The tops over polygon 5 Bouchard:2003bu, where black describes points at $\bar{z}=0$ and red at $\bar{z}=1$.
• Figure 5: Polygon for $\textmd{Bl}_{1}\mathbb{P}^{2}$