Revisiting modular symmetry in magnetized torus and orbifold compactifications

Abstract

We study the modular symmetry in $T^2$ and orbifold comfactifications with magnetic fluxes. There are $|M|$ zero-modes on $T^2$ with the magnetic flux $M$. Their wavefunctions as well as massive modes behave as modular forms of weight $1/2$ and represent the double covering group of $\Gamma \equiv SL(2,\mathbb{Z})$, $\tilde{\Gamma }\equiv \widetilde{SL}(2,\mathbb{Z})$. Each wavefunction on $T^2$ with the magnetic flux $M$ transforms under $\tilde{\Gamma }(2|M|)$, which is the normal subgroup of $\widetilde{SL}(2,\mathbb{Z})$. Then, $|M|$ zero-modes are representations of the quotient group ${\tilde{\Gamma }}_{2|M|}^{\prime }\equiv \tilde{\Gamma }/\tilde{\Gamma }(2|M|)$. We also study the modular symmetry on twisted and shifted orbifolds $T^2/{\mathbb{Z}}_N$. Wavefunctions are decomposed into smaller representations by eigenvalues of twist and shift. They provide us with reduction of reducible representations on $T^2$.

29 pages