Orbifold family unification using vectorlike representation on six dimensions

In orbifold family unification on the basis of $SU(N)$ gauge theory on the six-dimensional space-time $M^4\times T^2\mathrm{/}{Z}_m$ ($m=2,3,4,6$), enormous numbers of models with three families of the standard model matter multiplets are derived from a massless Dirac fermion with a vectorlike representation $[N, 3] + [N, N-3]$of$SU(N)$($N = 8, 9$). They contain models with three or more than three neutrino singlets and without any non-Abelian continuous flavor gauge symmetries. The relationship between flavor numbers from a fermion with $[N,N-k]$ and those from a fermion with $[N,k]$ are studied from the viewpoint of charge conjugation.