Wavefunctions and Yukawa Couplings on Resolutions of T2/ZNT^2/\mathbb{Z}_N Orbifolds

We propose matter wavefunctions on resolutions of $T^2/\mathbb{Z}_N$ singularities with constant magnetic fluxes. In the blow-down limit, the obtained wavefunctions of chiral zero-modes result in those on the magnetized $T^2/\mathbb{Z}_N$ orbifold models, but the wavefunctions of $\mathbb{Z}_N$-invariant zero-modes receive the blow-up effects around fixed points of $T^2/\mathbb{Z}_N$ orbifolds. Such blow-up effects change the selection rules and Yukawa couplings among the chiral zero-modes as well as the modular symmetry, in contrast to those on the magnetized $T^2/\mathbb{Z}_N$ orbifold models.Comment: 19 pages, 2 figures, v2: published versio

Modular flavor symmetries of three-generation modes on magnetized toroidal orbifolds

We study the modular symmetry on magnetized toroidal orbifolds with Scherk-Schwarz phases. In particular, we investigate finite modular flavor groups for three-generation modes on magnetized orbifolds. The three-generation modes can be the three-dimensional irreducible representations of covering groups and central extended groups of $\Gamma_N$ for $N=3,4,5,7,8,16$, that is, covering groups of $\Delta(6(N/2)^2)$ for $N=$ even and central extensions of $PSL(2,\mathbb{Z}_{N})$ for $N=$odd with Scherk-Schwarz phases. We also study anomaly behaviors.Comment: 34 page

Texture zeros of quark mass matrices at fixed point τ=ω\tau=\omega in modular flavor symmetry

We study systematically derivation of the specific texture zeros, that is the nearest neighbor interaction (NNI) form of the quark mass matrices at the fixed point $\tau=\omega$ in modular flavor symmetric models. We present models that the NNI forms of the quark mass matrices are simply realized at the fixed point $\tau=\omega$ in the $A_4$ modular flavor symmetry by taking account multi-Higgs fields. Such texture zero structure originates from the $ST$ charge of the residual symmetry $Z_3$ of $SL(2,Z)$. The NNI form can be realized at the fixed point $\tau = \omega$ in $A_4$ and $S_4$ modular flavor models with two pairs of Higgs fields when we assign properly modular weights to Yukawa couplings and $A_4$ and $S_4$ representations to three generations of quarks. We need four pairs of Higgs fields to realize the NNI form in $A_5$ modular flavor models.Comment: 37 page

Quark hierarchical structures in modular symmetric flavor models at level 6

We study modular symmetric quark flavor models without fine-tuning. Mass matrices are written in terms of modular forms, and modular forms in the vicinity of the modular fixed points become hierarchical depending on their residual charges. Thus modular symmetric flavor models in the vicinity of the modular fixed points have a possibility to describe mass hierarchies without fine-tuning. Since describing quark hierarchies without fine-tuning requires $Z_n$ residual symmetry with $n\geq 6$, we focus on $\Gamma_6$ modular symmetry in the vicinity of the cusp $\tau=i\infty$ where $Z_6$ residual symmetry remains. We use only modular forms belonging to singlet representations of $\Gamma_6$ to make our analysis simple. Consequently, viable quark flavor models are obtained without fine-tuning.Comment: 29 page

Modular symmetry in magnetized T2gT^{2g} torus and orbifold models

We study the modular symmetry in magnetized $T^{2g}$ torus and orbifold models. The $T^{2g}$ torus has the modular symmetry $\Gamma_{g}=Sp(2g,\mathbb{Z})$. Magnetic flux background breaks the modular symmetry to a certain normalizer $N_{g}(H)$. We classify remaining modular symmetries by magnetic flux matrix types. Furthermore, we study the modular symmetry for wave functions on the magnetized $T^{2g}$ and certain orbifolds. It is found that wave functions on magnetized $T^{2g}$ as well as its orbifolds behave as the Siegel modular forms of weight $1/2$ and $\widetilde{N}_{g}(H,h)$, which is the metapletic congruence subgroup of the double covering group of $N_{g}(H)$, $\widetilde{N}_{g}(H)$. Then, wave functions transform non-trivially under the quotient group, $\widetilde{N}_{g,h}=\widetilde{N}_{g}(H)/\widetilde{N}_{g}(H,h)$, where the level $h$ is related to the determinant of the magnetic flux matrix. Accordingly, the corresponding four-dimensional (4D) chiral fields also transform non-trivially under $\widetilde{N}_{g,h}$ modular flavor transformation with modular weight $-1/2$. We also study concrete modular flavor symmetries of wave functions on magnetized $T^{2g}$ orbifolds.Comment: 53 page