Shifted orbifold models with magnetic flux

We propose a mechanism to obtain the generation of matter in the standard model. We start from the analysis of the $T^2/Z_N$ shifted orbifold with magnetic flux, which imposes a $Z_N$ symmetry on torus. We also consider several orbifolds such as $(T^2\times T^2)/Z_N$, $(T^2 \times T^2 \times T^2)/(Z_N \times Z_{N'})$ and $(T^2 \times T^2 \times T^2)/(Z_N \times Z_{N'} \times Z_{N"})$. On such orbifolds, we study the behavior of fermions in two different means, one is the operator formalism and the other is to analyze wave functions explicitly. For an interesting result, it is found that the number of zero-mode fermions is related to $N$ of the $Z_N$ symmetry. In other words, the generation of matter relates to the type of orbifolds. Moreover, we find that shifted orbifold models to realize three generations are, in general, severely restricted. For example, the three-generation model on the type of $M^4 \times (T^2 \times T^2)/Z_N$ is unique. One can also construct other types of three-generation orbifold models with rich flavor structure. Those results may bring us a realistic model with desired Yukawa structure.

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