n=3 Differential calculus and gauge theory on a reduced quantum plane

TL;DR

This paper develops a differential calculus on a reduced quantum plane modeled by 3x3 matrices, exploring cases where the deformation parameter is a root of unity, and applies it to gauge field theories for specific cases.

Contribution

It introduces a 3-nilpotent deformed differential calculus on a matrix algebra viewed as a reduced quantum plane, including cases where the deformation parameter is a root of unity, and constructs associated gauge theories.

Findings

Constructed a 3-nilpotent differential calculus involving a complex parameter q.

Analyzed cases where q is a 3rd or Nth root of unity.

Established gauge field theories for n=2 and n=3 cases.

Abstract

We discuss the algebra of $N\times N$ matrices as a reduced quantum plane. A $3-$nilpotent deformed differential calculus involving a complex parameter $q$ is constructed. The two cases, $q$ $3^{rd}$ and $N^{th}$ root of unity are completely treated. As application, a gauge field theory for the particular cases $n=2$ and $n=3$ is established.