Modular symmetry in magnetized toroidal orbifold models

The flavor structure of quarks and leptons is one of the most significant mysteries. To understand the flavor structure, a lot of studies from both bottom-up side and top-down side have been carried out. One of interesting bottom-up approaches is assuming flavor symmetry by some non-Abelian discrete groups and realizing the flavor structure. On the other hand, the toroidal orbifold compactification with background magnetic fluxes is one of interesting top-down approaches since we can obtain three-generational chiral fermions and calculate such as Yukawa couplings among them on a magnetized toroidal orbifold. Moreover, the flavor structure depend on the geometry of the compact space. In this sense, the modular symmetry, which some of toroidal orbifolds have as a geometrical symmetry, has been recently attracting for flavor symmetry. Indeed, a lot of modular flavor models in which a finite modular subgroup has been assumed as flavor symmetry have been studied from bottom-up side. Then, it is important to study the modular symmetry in magnetized toroidal orbifold models from top-down side. I will talk about our studies of them. Actually, in a magnetized toroidal orbifold model, we have found the modular flavor group as well as the representations and the modular weights for matter fields, which are assumed in such modular flavor models, are specified.