Revisiting modular symmetry in magnetized torus and orbifold compactifications
Shota Kikuchi, Tatsuo Kobayashi, Shintaro Takada, Takuya H. Tatsuishi, Hikaru Uchida
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 wave functions as well as massive modes behave as modular forms of weight $1/2$ and represent the double covering group of $\mathrm{\ensuremath{\Gamma}}\ensuremath{\equiv}SL(2,\mathbb{Z})$, $\stackrel{\texttildelow{}}{\mathrm{\ensuremath{\Gamma}}}\ensuremath{\equiv}\stackrel{\texttildelow{}}{SL}(2,\mathbb{Z})$. Each wave function on ${T}^{2}$ with the magnetic flux $M$ transforms under $\stackrel{\texttildelow{}}{\mathrm{\ensuremath{\Gamma}}}(2|M|)$, which is the normal subgroup of $\stackrel{\texttildelow{}}{SL}(2,\mathbb{Z})$. Then, $|M|$ zero modes are representations of the quotient group ${\stackrel{\texttildelow{}}{\mathrm{\ensuremath{\Gamma}}}}_{2|M|}^{\ensuremath{'}}\ensuremath{\equiv}\stackrel{\texttildelow{}}{\mathrm{\ensuremath{\Gamma}}}/\stackrel{\texttildelow{}}{\mathrm{\ensuremath{\Gamma}}}(2|M|)$. We also study the modular symmetry on twisted and shifted orbifolds ${T}^{2}/{\mathbb{Z}}_{N}$. Wave functions are decomposed into smaller representations by eigenvalues of twist and shift. They provide us with reduction of reducible representations on ${T}^{2}$.