  As the effective field theory of the superstring theory, ten-dimensional
${\cal N}=1$ supersymmetric Yang-Mills theory is induced. We consider the
ten-dimensional space-time ${\cal M}_{10}$ as direct products of our
four-dimensional space-time ${\cal M}_{4}$ and three two-dimensional compact
spaces $X_i\ (i=1,2,3)$, i.e. ${\cal M}_{10} = {\cal M}_4 \times X_2 \times X_2
\times X_3$. In particular, we consider $X_i$ as a torus, the torus orbifold,
and also the blow-up manifold of the torus orbifold with background magnetic
fluxes. We discuss the modular symmetry in the magnetized torus and its
orbifold compactifications. We find that the modular flavor groups and the
representations as well as modular weights of the chiral fields such as quarks
and leptons obtained from the magnetized torus as well as the orbifold
compactifications can be uniquely determined with the magnetic fluxes. We also
discuss the magnetized blow-up manifold compactification in which orbifold
singularities are replaced by parts of sphere as a smooth manifold. We find
that the chiral zero mode numbers on the blow-up manifold as well as the
orbifold can be determined only by the magnetic fluxes including localized
fluxes due to the Atiyah-Singer index theorem, and then the degree of freedom
of localized fluxes gives new additional chiral zero modes. We also find that
the new chiral zero modes correspond to localized modes around the orbifold
singularities.

High Energy Physics - Theory;High Energy Physics - Phenomenology