Flavor symmetries from modular subgroups in magnetized compactifications

We study the flavor structures of zero-modes, which are originated from the modular symmetry on $T_1^2\times T_2^2$ \( {T}_1^2\times {T}_2^2 \) and its orbifold with magnetic fluxes. We introduce the constraint on the moduli parameters by τ₂ = Nτ₁, where τ_i denotes the complex structure moduli on $T_i^2$ \( {T}_i^2 \) . Such a constraint can be derived from the moduli stabilization. The modular symmetry of $T_1^2\times T_2^2$ \( {T}_1^2\times {T}_2^2 \) is $\mathrm{SL}{\left(2,\mathbb{Z}\right)}_{{\tau }_1}\times \mathrm{SL}{\left(2,\mathbb{Z}\right)}_{{\tau }_2}\subset \mathrm{Sp}\left(4,\mathbb{Z}\right)$ \( \textrm{SL}{\left(2,\mathbb{Z}\right)}_{\tau_1}\times \textrm{SL}{\left(2,\mathbb{Z}\right)}_{\tau_2}\subset \textrm{Sp}\left(4,\mathbb{Z}\right) \) and it is broken to Γ₀(N) × Γ⁰(N) by the moduli constraint. The wave functions represent their covering groups. We obtain various flavor groups in these models.