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 non-trivial representations of finite modular subgroups, e.g. $S_3$, $S_4$, $\Delta(96)$ and $\Delta(384)$.
Highlights
The origin of the flavor structure in the quark and lepton sectors is one of the unsolved but important mysteries in particle physics
The key ingredients are the modular forms of weight 2, which are non-trivial representations of finite modular subgroups
Our purpose of this paper is to study a new type of constructions of modular forms for finite modular subgroups
Summary
The origin of the flavor structure in the quark and lepton sectors is one of the unsolved but important mysteries in particle physics. The three generations of leptons are assigned to non-trivial representations of the finite modular subgroup A4. The couplings as well as neutrino masses are assigned to modular forms, which are nontrivial representations under A4 Such an idea was extended in Refs. [23,24,25,26,27] by use of other finite modular subgroups S3 and S4 in addition to A4 In these studies, the key ingredients are the modular forms of weight 2, which are non-trivial representations of finite modular subgroups.. The key ingredients are the modular forms of weight 2, which are non-trivial representations of finite modular subgroups.2 Such modular forms are found for S3 doublet, A4 triplet, and S4 doublet and triplet in Refs.
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