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We study the modular symmetry in magnetized $T^{2g}$ torus and orbifoldmodels. The $T^{2g}$ torus has the modular symmetry$\Gamma_{g}=Sp(2g,\mathbb{Z})$. Magnetic flux background breaks the modularsymmetry to a certain normalizer $N_{g}(H)$. We classify remaining modularsymmetries by magnetic flux matrix types. Furthermore, we study the modularsymmetry 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 orbifoldsbehave as the Siegel modular forms of weight $1/2$ and$\widetilde{N}_{g}(H,h)$, which is the metapletic congruence subgroup of thedouble covering group of $N_{g}(H)$, $\widetilde{N}_{g}(H)$. Then, wavefunctions transform non-trivially under the quotient group,$\widetilde{N}_{g,h}=\widetilde{N}_{g}(H)/\widetilde{N}_{g}(H,h)$, where thelevel $h$ is related to the determinant of the magnetic flux matrix.Accordingly, the corresponding four-dimensional (4D) chiral fields alsotransform non-trivially under $\widetilde{N}_{g,h}$ modular flavortransformation with modular weight $-1/2$. We also study concrete modularflavor symmetries of wave functions on magnetized $T^{2g}$ orbifolds.

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