Modular forms of finite modular subgroups from magnetized D-brane models

Abstract

We study modular transformation of holomorphic Yukawa couplings in magnetized D-brane models. It is found that their products are modular forms, which are nontrivial representations of finite modular subgroups, e.g., ${S}_{3}$, ${S}_{4}$, $\mathrm{\ensuremath{\Delta}}(96)$ and $\mathrm{\ensuremath{\Delta}}(384)$.