Sp(6, Z) modular symmetry in flavor structures: quark flavor models and Siegel modular forms for \(\widetilde{\Delta }\left(96\right)\)

We study an approach to construct Siegel modular forms from Sp(6, Z). Zero-mode wave functions on T⁶ with magnetic flux background behave Siegel modular forms at the origin. Then T-symmetries partially break depending on the form of background magnetic flux. We study the background such that three T-symmetries T_I, T_II and T_III as well as the S-symmetry remain. Consequently, we obtain Siegel modular forms with three moduli parameters (ω₁, ω₂, ω₃), which are multiplets of finite modular groups. We show several examples. As one of examples, we study Siegel modular forms for \(\widetilde{\Delta }\left(96\right)\) in detail. Then, as a phenomenological applicantion, we study quark flavor models using Siegel modular forms for \(\widetilde{\Delta }\left(96\right)\) . Around the cusp, ω₁ = i∞, the Siegel modular forms have hierarchical values depending on their T_I-charges. We show the deviation of ω₁ from the cusp can generate large quark mass hierarchies without fine-tuning. Furthermore CP violation is induced by deviation of ω₂ from imaginary axis.