A Gauge Field Model On SU(2)_Ltimes SU(2)_Rtimes U(1)_Y times π₄(G_(YM))

We construct a gauge field model based on $SU(2)_L\times SU(2)_R\times U(1)_Y \times \pi_4(SU(2)_L\times SU(2)_R\times U(1)_Y) $ from the principle that both the original gauge group $G_{YM}$ and the discrete group $\pi_4(G_{YM})$ should be taken as gauge groups in the sense of non-commutative geometry. We show that the Yukawa coupling and the Higgs mechanism appear as natural results.