Gauge-Fermion Unification and Flavour Symmetry

After we study the 6-dimensional ${\cal N} = (1, 1)$ supersymmetry breakingand $R$ symmetry breaking on $M^4\times T^2/Z_n$, we construct two ${\cal N} =(1, 1)$ supersymmetric $E_6$ models on $M^4\times T^2/Z_3$ where $E_6$ isbroken down to $SO(10)\times U(1)_X$ by orbifold projection. In Model I, threefamilies of the Standard Model fermions arise from the zero modes of bulkvector multiplet, and the $R$ symmetry $U(1)_F^{I} \times SU(2)_{{\bf 4}_-}$can be considered as flavour symmetry. This may explain why there are threefamilies of fermions in the nature. In Model II, the first two families comefrom the zero modes of bulk vector multiplet, and the flavour symmetry issimilar. In these models, the anomalies can be cancelled, and we have very goodfits to the SM fermion masses and mixings. We also comment on the ${\cal N}=(1,1)$ supersymmetric $E_6$ models on $M^4\times T^2/Z_4$ and $M^4\times T^2/Z_6$,SU(9) models on $M^4\times T^2/Z_3$, and SU(8) models on $T^2$ orbifolds.Comment: Latex, 33 pages, minor change