Modular symmetry in magnetized $T^{2g}$ torus and orbifold models

We study the modular symmetry in magnetized $T^{2g}$ torus and orbifold models. The $T^{2g}$ torus has the modular symmetry ${\Gamma }_g=Sp(2g,Z)$. The magnetic flux background breaks the modular symmetry to a certain normalizer $N_g\left(H\right)$. We classify remaining modular symmetries by magnetic flux matrix types. Furthermore, we study the modular symmetry for wave functions on the magnetized $T^{2g}$ and certain orbifolds. It is found that wave functions on magnetized $T^{2g}$ as well as its orbifolds behave as the Siegel modular forms of weight $1/2$ and ${\tilde{N}}_g(H,h)$, which is the metaplectic congruence subgroup of the double covering group of $N_g\left(H\right)$, ${\tilde{N}}_g\left(H\right)$. Then, wave functions transform nontrivially under the quotient group, ${\tilde{N}}_{g,h}={\tilde{N}}_g\left(H\right)/{\tilde{N}}_g(H,h)$, where the level $h$ is related to the determinant of the magnetic flux matrix. Accordingly, the corresponding four-dimensional chiral fields also transform nontrivially under ${\tilde{N}}_{g,h}$ modular flavor transformation with modular weight $-1/2$. We also study concrete modular flavor symmetries of wave functions on magnetized $T^{2g}$ orbifolds.