Abstract
We study the modular symmetry in magnetized D-brane models on $T^2$. Non-Abelian flavor symmetry $D_4$ in the model with magnetic flux $M=2$ (in a certain unit) is a subgroup of the modular symmetry. We also study the modular symmetry in heterotic orbifold models. The $T^2/Z_4$ orbifold model has the same modular symmetry as the magnetized brane model with $M=2$, and its flavor symmetry $D_4$ is a subgroup of the modular symmetry.
Highlights
Non-Abelian discrete flavor symmetries play an important role in particle physics
Many models with various finite groups have been studied in order to explain quark and lepton masses and their mixing angles. (See for review [1,2,3].) Those symmetries may be useful for dark matter physics and multi-Higgs models
Using results in Refs. [16,17,18], we compare the modular symmetries in heterotic orbifold models with non-Abelian flavor symmetries and the modular symmetries in the magnetized D-brane models, which have been derived in the previous section
Summary
Non-Abelian discrete flavor symmetries play an important role in particle physics. In particular, many models with various finite groups have been studied in order to explain quark and lepton masses and their mixing angles. (See for review [1,2,3].) Those symmetries may be useful for dark matter physics and multi-Higgs models. (See [5,6].) magnetized D-brane models within the framework of type II superstring theory can lead to similar flavor symmetries [8,9,10,11,12]. The behavior of zero modes under modular transformation was studied in magnetized D-brane models in Ref. The purpose of this paper is to study more how modular transformation is represented by zero modes in magnetized D-brane models, and to discuss relations between modular transformation and non-Abelian flavor symmetries in magnetized D-brane models. Here we study modular symmetry and non-Abelian discrete flavor symmetries in heterotic orbifold models, too. We give brief reviews on non-Abelian discrete flavor symmetries in magnetized D-brane models and heterotic orbifold models in Appendixes A and B, respectively
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