Classifications of magnetized $T^4$ and $T^4/Z_2$ orbifold models

We study constructions and classifications of three-generation models based on magnetized $T^4$ and $T^4/{Z}_2$ orbifold as candidates of the compact space. We focus on chiral fermion zero-mode wave functions in the extra dimensions. Freedoms of constant gauge fields, called Scherk-Schwarz phases are taken into account. Infinite number of three-generation models are yielded, corresponding to the ways in which the magnetic flux can be turned on. We classify them in a systematic manner, clarifying the relationship between different models. The Higgs sector is also studied by analyzing possible assignments of the magnetic flux and Scherk-Schwarz phases, etc. to left- and right-handed fermions.

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