We study orbifolding by the $\mathbb{Z}_2^{\rm (per)}$ permutaion of $T^2_1 \times T^2_2$ with magnetic fluxes and its twisted orbifolds. We classify the possible three-generation models which lead to non-vanishing Yukawa couplings on the magnetized $T^2_1 \times T^2_2$ and orbifolds including the $\mathbb{Z}_2^{\rm (per)}$ permutation and $\mathbb{Z}_2^{\rm (t)}$ twist. We also study the modular symmetry on such orbifold models. As an illustrating model, we examine the realization of quark masses and mixing angles.

中文翻译：

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三代模型分类通过磁化 T2 × T2 的轨道

我们通过 $T^2_1 \times T^2_2$ 的 $\mathbb{Z}_2^{\rm (per)}$ 排列与磁通量及其扭曲的orbifolds 来研究orbifolding。我们对可能的三代模型进行分类，这些模型导致磁化 $T^2_1 \times T^2_2$ 和包括 $\mathbb{Z}_2^{\rm (per)}$ 置换的轨道上的非消失 Yukawa 耦合和 $\mathbb{Z}_2^{\rm (t)}$ 扭曲。我们还研究了这种 orbifold 模型的模对称性。作为说明模型，我们研究了夸克质量和混合角的实现。