Modular symmetry of localized modes

We study the modular symmetry of localized modes on fixed points of $T^2\mathrm{/}{\mathbb{Z}}_2$ orbifold. First, we find that the localized modes with even (odd) modular weight generally have $\mathrm{\Delta }(6n^2)$ (${\mathrm{\Delta }}^{\mathrm{\prime }}(6n^2)$) modular flavor symmetry. Moreover, when we consider an additional Ansatz, the localized modes with even (odd) modular weight generally enjoy $S_3$ ($S_4^{\mathrm{\prime }}$) modular flavor symmetry, and we show the concrete wave functions of the localized modes.