Generation structures and Yukawa couplings in magnetized $T^{2g}\mathrm{/}{\mathbb{Z}}_N$ models

We study fermion zero-mode wave functions with various chiralities in magnetized $T^{2g}$, $(g=2,3)$ torus. First, we consider the wave functions satisfying the Dirac equation and the boundary conditions on the magnetized torus. Second, we introduce the $SO(3)$ (or parity) transformations and derive the wave functions under the modular transformation. Additionally, we calculate the Yukawa couplings with consideration for the chirality. Lastly, we briefly review how to construct $T^4\mathrm{/}{\mathbb{Z}}_N$ ($N=2,3,4,6$) and $T^6\mathrm{/}{\mathbb{Z}}_{12}$ twisted orbifold. Also, we explicitly analyze the number of the wave functions in ${\mathbb{Z}}_N$ sectors.