Tatsuo Kobayashi

We study the lepton flavor models, whose flavor symmetries are finite subgroups of the modular group such as ${S}_{3}$ and ${A}_{4}$. In our models, couplings are also nontrivial representations of these groups and modular functions of the modulus. We study the possibilities that these models realize realistic values of neutrino masses and lepton mixing angles.

A bstract We study the phenomenological implications of the modular symmetry Γ(3) ≃ A 4 of lepton flavors facing recent experimental data of neutrino oscillations. The mass matrices of neutrinos and charged leptons are essentially given by fixing the expectation value of modulus τ , which is the only source of modular invariance breaking. We introduce no flavons in contrast with the conventional flavor models with A 4 symmetry. We classify our neutrino models along with the type I seesaw model,

We study a flavor model that the quark sector has the S3 modular symmetry while the lepton sector has the A4 modular symmetry. Our model leads to characteristic quark mass matrices which are consistent with experimental data of quark masses, mixing angles and the CP violating phase. The lepton sector is also consistent with the experimental data of neutrino oscillations. We also study baryon and lepton number violations in our flavor model.

A three-generation Pati–Salam model is constructed by compactifying the heterotic string on a particular T6/Z6 Abelian symmetric orbifold with two discrete Wilson lines. The compactified space is taken to be the Lie algebra lattice G2⊕SU(3)⊕SO(4). When one dimension of the SO(4) lattice is large compared to the string scale, this model reproduces many features of a 5d SO(10) grand unified theory compactified on an S1/Z2 orbifold. (Of course, with two large extra dimensions we can obtain a 6d SO(

We study the modular symmetry in magnetized D-brane models on ${T}^{2}$. Non-Abelian flavor symmetry ${D}_{4}$ in the model with magnetic flux $M=2$ (in a certain unit) is a subgroup of the modular symmetry. We also study the modular symmetry in heterotic orbifold models. The ${T}^{2}/{Z}_{4}$ orbifold model has the same modular symmetry as the magnetized brane model with $M=2$, and its flavor symmetry ${D}_{4}$ is a subgroup of the modular symmetry.

A bstract We study a flavor model with A 4 symmetry which originates from S 4 modular group. In S 4 symmetry, Z 2 subgroup can be anomalous, and then S 4 can be violated to A 4 . Starting with a S 4 symmetric Lagrangian at the tree level, the Lagrangian at the quantum level has only A 4 symmetry when Z 2 in S 4 is anomalous. We obtain modular forms of two singlets and a triplet representations of A 4 by decomposing S 4 modular forms into A 4 representations. We propose a new A 4 flavor model of

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 $\mathrm{\ensuremath{\Delta}}(384)$.

We study the modulus stabilization in an ${A}_{4}$ model whose ${A}_{4}$ flavor symmetry is originated from the ${S}_{4}$ modular symmetry. We can stabilize the modulus so that the ${A}_{4}$ invariant superpotential leads to the realistic lepton masses and mixing angles. We also discuss the phenomenological aspect of the present model as a consequence of the modulus stabilization.

We discuss type II seesaw models adopting modular ${A}_{4}$ symmetry in a supersymmetric framework. In our approach, the models are classified by the assignment of ${A}_{4}$ representations and modular weights for leptons and triplet Higgs fields. Then neutrino mass matrix is characterized by modulus $\ensuremath{\tau}$ and two free parameters. Carrying out numerical analysis, we find allowed parameter sets which can fit the neutrino oscillation data. For the allowed parameter sets, we obtain th

We study the geometrical and phenomenological aspects of Z N orbifold models. Conditions on the background moduli and Wilson lines are clarified. We investigate the explicit Z N -invariant states through a modified GSO projection and calculate Yukawa coupling conditions due to space-group and SO(10) invariance. We estimate a contribution of a holomorphic instanton to the couplings.

We study the spontaneous $CP$ violation through the stabilization of the modulus $\ensuremath{\tau}$ in modular invariant flavor models. The $CP$-invariant potential has the minimum only at $\mathrm{Re}[\ensuremath{\tau}]=0$ or $1/2(\mathrm{mod}\text{ }1)$. From this result, we study $CP$ violation in modular invariant flavor models. The physical $CP$ phase is vanishing. The important point for the $CP$ conservation is the $T$ transformation in the modular symmetry. One needs the violation of $T