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Register a new userOperator analysis of physical states on magnetized $T^{2}/Z_{N}$ orbifolds
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We discuss an effective way for analyzing the system on the magnetized twisted orbifolds in operator formalism, especially in the complicated cases $T^{2}/Z_{3}$, $T^{2}/Z_{4}$ and $T^{2}/Z_{6}$. We can obtain the exact and analytical results which can be applicable for any larger values of the quantized magnetic flux M, and show that the (non-diagonalized) kinetic terms are generated via our formalism and the number of the surviving physical states are calculable in a rigorous manner by simply following usual procedures in linear algebra in any case. Our approach is very powerful when we try to examine properties of the physical states on (complicated) magnetized orbifolds $T^{2}/Z_{3}$, $T^{2}/Z_{4}$, $T^{2}/Z_{6}$ (and would be in other cases on higher-dimensional torus) and could be an essential tool for actual realistic model construction based on these geometries.
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We study the CP-violating phase of the quark sector in the $U(8)$ flavor model on $T^2/Z_N , (N=2,3,4,6)$ with non-vanishing magnetic fluxes, where properties of possible origins of the CP violation are investigated minutely. In this system, a non-vanishing value is mandatory in the real part of the complex modulus parameter $tau$ of the two-dimensional torus. On $T^2$ without orbifolding, underlying discrete flavor symmetries severely restrict the form of Yukawa couplings and it is very difficult to reproduce the observed pattern in the quark sector including the CP-violating phase $delta_{rm CP}$. In cases of multiple Higgs doublets emerging on $T^2/Z_2$, the mass matrices of the zero-mode fermions can be written in the Gaussian textures by choosing appropriate configurations of vacuum expectation values of the Higgs fields. When such Gaussian textures of mass matrices are realized, we show that all of the quark profiles, which are mass hierarchies among the quarks, quark mixing angles, and $delta_{rm CP}$ can be simultaneously realized.
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.
We investigate chiral zero modes and winding numbers at fixed points on $T^2/mathbb{Z}_N$ orbifolds. It is shown that the Atiyah-Singer index theorem for the chiral zero modes leads to a formula $n_+-n_-=(-V_++V_-)/2N$, where $n_{pm}$ are the numbers of the $pm$ chiral zero modes and $V_{pm}$ are the sums of the winding numbers at the fixed points on $T^2/mathbb{Z}_N$. This formula is complementary to our zero-mode counting formula on the magnetized orbifolds with non-zero flux background $M eq 0$, consistently with substituting $M = 0$ for the counting formula $n_+ - n_- = (2M - V_+ + V_-)/2N$.
We study the modular symmetry of zero-modes on $T_1^2 times T_2^2$ and orbifold compactifications with magnetic fluxes, $M_1,M_2$, where modulus parameters are identified. This identification breaks the modular symmetry of $T^2_1 times T^2_2$, $SL(2,mathbb{Z})_1 times SL(2,mathbb{Z})_2$ to $SL(2,mathbb{Z})equivGamma$. Each of the wavefunctions on $T^2_1 times T^2_2$ and orbifolds behaves as the modular forms of weight 1 for the principal congruence subgroup $Gamma$($N$), $N$ being 2 times the least common multiple of $M_1$ and $M_2$. Then, zero-modes transform each other under the modular symmetry as multiplets of double covering groups of $Gamma_N$ such as the double cover of $S_4$.
We classify the combinations of parameters which lead three generations of quarks and leptons in the framework of magnetized twisted orbifolds on $T^2/Z_2$, $T^2/Z_3$, $T^2/Z_4$ and $T^2/Z_6$ with allowing nonzero discretized Wilson line phases and Scherk-Schwarz phases. We also analyze two actual examples with nonzero phases leading to one-pair Higgs and five-pair Higgses and discuss the difference from the results without nonzero phases studied previously.
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