Noncommutative Geometry and Gauge Theories on Discrete Groups

TL;DR

This paper develops a framework for pure gauge theories on discrete groups, introducing differential calculus, metric properties, and action functionals, with detailed models on small groups and methods to combine discrete and continuous geometries.

Contribution

It presents a systematic approach to gauge theories on discrete groups, including differential calculus, metric analysis, and model construction, advancing noncommutative geometry applications.

Findings

Constructed differential calculus on arbitrary discrete groups

Analyzed metric properties and action functionals for gauge theories

Studied models on 2 and 3 and combined discrete with continuous geometry

Abstract

We build and investigate a pure gauge theory on arbitrary discrete groups. A systematic approach to the construction of the differential calculus is presented. We discuss the metric properties of the models and introduce the action functionals for unitary gauge theories. A detailed analysis of two simple models based on $\z_2$ and $\z_3$ follows. Finally we study the method of combining the discrete and continuous geometry.