minipage}{0.9} We study the modular invariance in magnetized torus models. Modular invariant flavor model is a recently proposed hypothesis for solving the flavor puzzle, where the flavor symmetry originates from modular invariance. In this framework coupling constants such as Yukawa couplings are also transformed under the flavor symmetry. We show that the low-energy effective theory of magnetized torus models is invariant under a specific subgroup of the modular group. Since Yukawa couplings as well as chiral zero modes transform under the modular group, the above modular subgroup (referred to as modular flavor symmetry) provides a new type of modular invariant flavor models with $D_4\times {\mathbb{Z}}_2$, $({\mathbb{Z}}_4\times {\mathbb{Z}}_2)⋊{\mathbb{Z}}_2$, and $({\mathbb{Z}}_8\times {\mathbb{Z}}_2)⋊{\mathbb{Z}}_2$. We also find that conventional discrete flavor symmetries which arise in magnetized torus model are non-commutative with the modular flavor symmetry. Combining both two symmetries we obtain a larger flavor symmetry, which is the semidirect product of the conventional flavor symmetry and the modular flavor symmetry for non-vanishing Wilson line. For the vanishing Wilson line, we have additional ${\mathbb{Z}}_2$ symmetry, i.e., parity, which is the unique common element between the conventional flavor symmetry and the modular flavor symmetry. \end{minipage} } \renewcommand{}{ …

中文翻译：

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磁化圆环上的模块化风味对称性

minipage} {0.9}，我们研究了磁化环面模型中的模不变性。模块化不变风味模型是最近提出的用于解决风味难题的假设，其中风味对称性源自模块化不变性。在这种框架下，诸如汤川耦合之类的耦合常数也在风味的对称性下转化。我们表明，磁化环面模型的低能效理论在模块化组的特定子组下是不变的。由于Yukawa耦合以及手性零模在模块化组下转换，因此上述模块化子组（称为模块化风味对称性）提供了一种新型的模块化不变风味模型，该模型具有$d_4\times {\mathbb{Z}}_2$， $（{\mathbb{Z}}_4\times {\mathbb{Z}}_2）⋊{\mathbb{Z}}_2$和 $（{\mathbb{Z}}_8\times {\mathbb{Z}}_2）⋊{\mathbb{Z}}_2$。我们还发现在磁化圆环模型中出现的常规离散风味对称性与模块化风味对称性是不可交换的。结合这两个对称性，我们获得了更大的风味对称性，这是常规风味对称性和模块化的风味对称性对于不消失的Wilson线的半直接乘积。对于消失的威尔逊线，我们还有其他${\mathbb{Z}}_2$对称性，即奇偶校验，这是常规风味对称性和模块化风味对称性之间唯一的共同要素。\ end {minipage}} \ renewcommand {} {…