Zero-modes in magnetized $T^6\mathrm{/}{\mathbb{Z}}_N$ orbifold models through $Sp(6,\mathbb{Z})$ modular symmetry

We study of fermion zero-modes on magnetized $T^6\mathrm{/}{\mathbb{Z}}_N$ orbifolds. In particular, we focus on non-factorizable orbifolds, i.e. $T^6\mathrm{/}{\mathbb{Z}}_7$ and $T^6\mathrm{/}{\mathbb{Z}}_{12}$ corresponding to $SU(7)$ and $E_6$ Lie lattices respectively. The number of degenerated zero-modes corresponds to the generation number of low energy effective theory in four dimensional space-time. We find that three-generation models preserving 4D $\mathcal{N}=1$ supersymmetry can be realized by magnetized $T^6\mathrm{/}{\mathbb{Z}}_{12}$, but not by $T^6\mathrm{/}{\mathbb{Z}}_7$. We use $Sp(6,\mathbb{Z})$ modular transformation for the analyses.