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

TL;DR

This paper investigates the modular symmetry properties of wave functions in magnetized higher-dimensional tori and orbifold models, revealing their behavior as Siegel modular forms and identifying associated modular flavor symmetries.

Contribution

It classifies the remaining modular symmetries after magnetic flux breaking and analyzes the transformation properties of wave functions as Siegel modular forms and their implications for 4D chiral fields.

Findings

Wave functions behave as Siegel modular forms of weight 1/2.

Remaining modular symmetries are classified by flux matrix types.

4D chiral fields transform under quotient modular groups with weight -1/2.

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 $\widetilde{N}_{g}(H,h)$, which is the metapletic congruence subgroup of the double covering group of $N_{g}(H)$, $\widetilde{N}_{g}(H)$. Then, wave functions transform non-trivially under the quotient group, $\widetilde{N}_{g,h}=\widetilde{N}_{g}(H)/\widetilde{N}_{g}(H,h)$, 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 $\widetilde{N}_{g,h}$ modular flavor transformation with modular weight $-1/2$. We also study concrete modular flavor symmetries of wave functions on magnetized $T^{2g}$ orbifolds.