We construct a general form for an F-theory Weierstrass model over a general base giving a 6D or 4D supergravity theory with gauge group $(\operatorname{SU}(3) \times \operatorname{SU}(2) \times \operatorname{U}(1)) / \mathbb{Z}_6$ and generic associated matter, which includes the matter content of the standard model. The Weierstrass model is identified by unHiggsing a model with $\operatorname{U}(1)$ gauge symmetry and charges $q \le 4$ previously found by the first author. This model includes two distinct branches that were identified in earlier work, and includes as a special case the class of models recently studied by Cvetic, Halverson, Lin, Liu, and Tian, for which we demonstrate explicitly the possibility of unification through an $\operatorname{SU}(5)$ unHiggsing. We develop a systematic methodology for checking that a parameterized class of F-theory Weierstrass models with a given gauge group $G$ and fixed matter content is generic (contains all allowed moduli) and confirm that this holds for the models constructed here.

中文翻译：

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具有调谐 (SU(3) × SU(2) × U(1))/ℤ6 对称性的通用 F 理论模型

我们在给出具有规范群 $(\operatorname{SU}(3) \times \operatorname{SU}(2) \times \operatorname 的 6D 或 4D 超引力理论的一般基础上构建 F 理论 Weierstrass 模型的一般形式{U}(1)) / \mathbb{Z}_6$ 和通用关联物质，包括标准模型的物质内容。Weierstrass 模型通过对具有 $\operatorname{U}(1)$ 规范对称性和电荷 $q \le 4$ 先前由第一作者发现的模型进行 unHiggsing 来识别。该模型包括在早期工作中确定的两个不同分支，并包括 Cvetic、Halverson、Lin、Liu 和 Tian 最近研究的模型类作为特例，我们通过 $\操作员名称{SU}(5)$ unHiggsing。