Flavor symmetries from modular subgroups in magnetized compactifications

We study the flavor structures of zero-modes, which are originated from the modular symmetry on T²₁ T²₂ and its orbifold with magnetic fluxes. We introduce the constraint on the moduli parameters by ₂=N₁, where ᵢ denotes the complex structure moduli on T²ᵢ. Such a constraint can be derived from the moduli stabilization. The modular symmetry of T²₁ T²₂ is SL (2, Z) 䃑 SL (2, Z) 䃒 Sp (4, Z) and it is broken to ₀ (N) ⁰ (N) by the moduli constraint. The wave functions represent their covering groups. We obtain various flavor groups in these models.