Modular symmetry by orbifolding magnetized $T^2\times T^2$: realization of double cover of ${\Gamma }_N$

Abstract

We study the modular symmetry of zero-modes on $T_1^2\times T_2^2$ and orbifold compactifications with magnetic fluxes, $M_1,M_2$, where modulus parameters are identified. This identification breaks the modular symmetry of $T_1^2\times T_2^2$, $SL(2,\mathbb{Z}{)}_1\times SL(2,\mathbb{Z}{)}_2$ to $SL(2,\mathbb{Z})\equiv \Gamma$. Each of the wavefunctions on $T_1^2\times T_2^2$ and orbifolds behaves as the modular forms of weight 1 for the principal congruence subgroup $\Gamma$($N$), $N$ being 2 times the least common multiple of $M_1$ and $M_2$. Then, zero-modes transform each other under the modular symmetry as multiplets of double covering groups of ${\Gamma }_N$ such as the double cover of $S_4$.

Comments: 27 pages