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

Abstract

We study the modular symmetry in magnetized $T^{2g}$ torus and orbifold models. The $T^{2g}$ torus has the modular symmetry ${\Gamma }_g=Sp(2g,\mathbb{Z})$. Magnetic flux background breaks the modular symmetry to a certain normalizer $N_g(H)$. 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 metapletic congruence subgroup of the double covering group of $N_g(H)$, ${\tilde{N}}_g(H)$. Then, wave functions transform non-trivially under the quotient group, ${\tilde{N}}_{g,h}={\tilde{N}}_g(H)/{\tilde{N}}_g(H,h)$, where the level $h$ is related to the determinant of the magnetic flux matrix. Accordingly, the corresponding four-dimensional (4D) chiral fields also transform non-trivially 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.

Comments: 53 pages