Matrix Realization of Gauge Theory on Discrete Group $Z_2$

TL;DR

This paper develops a matrix algebra framework for gauge theory on the discrete group Z_2, connecting non-commutative geometry models with physical gauge theories and exploring fermionic and Yang-Mills structures.

Contribution

It introduces a matrix algebra representation of functions on Z_2 and formulates gauge theory within this non-commutative geometric setting, linking previous models to a matrix-based approach.

Findings

Matrix algebra representation of Z_2 functions constructed.

Gauge theory on Z_2 developed within the matrix algebra framework.

Connection established between non-commutative geometry models and matrix-based gauge theories.

Abstract

We construct a $2\times 2$ matrix algebra as representation of functions on discrete group $Z_2$ and develop the gauge theory on discrete group proposed by Starz in the matrix algebra. Accordingly, we show that the non-commutative geometry model built by R.Conquereax, G.Esposito-Farese and G.Vaillant results from this approach directly. For the purpose of Physical model building, we introduce a free fermion Lagrangian on $M_4\times Z_2$ and study Yang-Mills like gauge theory.