  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 $D_4 \times
\mathbb{Z}_2$, $(\mathbb{Z}_4 \times \mathbb{Z}_2) \rtimes \mathbb{Z}_2$, and
$(\mathbb{Z}_8 \times \mathbb{Z}_2) \rtimes \mathbb{Z}_2$. 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, where the conventional flavor
symmetry is a normal subgroup of the whole group.

High Energy Physics - Theory;High Energy Physics - Phenomenology