Equivariance on Discrete Space and Yang-Mills-Higgs Model

We introduce the basic equivariant quantity $Q$ in the gauge theory on the noncommutative descrete $Z_2$ space, which plays an important role for the equivariant dimensional reduction. If the gauge configuration of the ground state on the extra dimensional space is described by the equivariant $Q$, then the extra dimensional space is invisible. Especially, using the equivariance principle, we show that the Yang-Mills theory on $R^2\times Z_2$ space is equivalent to the Yang-Mills-Higgs model on $R^2$ space. It can be said that this model is the simplest model of this type.