Equivariance on Discrete Space and Yang-Mills-Higgs Model

TL;DR

This paper explores gauge theories on noncommutative discrete spaces, introducing an equivariant quantity that simplifies the Yang-Mills theory on a combined space to a Yang-Mills-Higgs model on a lower-dimensional space.

Contribution

It introduces the equivariant quantity $Q$ in gauge theory on noncommutative $Z_2$ space and demonstrates the equivalence between Yang-Mills theory on $R^2 \times Z_2$ and a Yang-Mills-Higgs model on $R^2$.

Findings

The equivariant quantity $Q$ is key to dimensional reduction.

Yang-Mills on $R^2 \times Z_2$ is equivalent to a Yang-Mills-Higgs model on $R^2$.

The model provides a simple example of gauge theory reduction.

Abstract

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.