The systematic construction of families of F-theory models in which $U(1)\times \mathbb{Z}_2$ and $U(1)\times \mathbb{Z}_4$ forms are discussed, motivated by the proposal in the previous paper stating that a discrete $\mathbb{Z}_n$ gauge group tends to be enhanced to, for example, $U(1)\times \mathbb{Z}_n$, in the moduli of multisection geometry, where multisections are split into multisections of smaller degrees. We observed that a discrete $\mathbb{Z}_2$ gauge group is enhanced to $U(1)\times \mathbb{Z}_2$ along certain bisection geometries in four-section geometry. We also present an example of a family of models with $U(1)\times \mathbb{Z}_4$ gauge group as a demonstration of the systematic construction.

中文翻译：

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具有 U(1) × ℤ2, ℤ4 和离散规范群过渡的 F 理论模型

讨论了 $U(1)\times \mathbb{Z}_2$ 和 $U(1)\times \mathbb{Z}_4$ 形式的 F 理论模型族的系统构建，其动机是之前的论文指出离散的 $\mathbb{Z}_n$ 规范群倾向于增强为例如 $U(1)\times \mathbb{Z}_n$，在多段几何的模中，其中 multisections被分成更小度数的多部分。我们观察到离散的 $\mathbb{Z}_2$ 规范群沿四段几何中的某些二分几何增强为 $U(1)\times \mathbb{Z}_2$。我们还展示了一个带有 $U(1)\times \mathbb{Z}_4$ 规范群的模型族的例子，作为系统构建的演示。