Gauge theories in noncommutative geometry

TL;DR

This review explores the mathematical frameworks for formulating noncommutative gauge theories, focusing on connections, curvatures, and gauge transformations within derivation-based and spectral triple approaches.

Contribution

It provides a comprehensive overview of the fundamental structures enabling noncommutative gauge theories, highlighting two main approaches and illustrating with examples.

Findings

Comparison of derivation-based and spectral triple approaches

Examples demonstrating noncommutative gauge theories

Identification of common features in noncommutative gauge models

Abstract

In this review we present some of the fundamental mathematical structures which permit to define noncommutative gauge field theories. In particular, we emphasize the theory of noncommutative connections, with the notions of curvatures and gauge transformations. Two different approaches to noncommutative geometry are covered: the one based on derivations and the one based on spectral triples. Examples of noncommutative gauge field theories are given to illustrate the constructions and to display some of the common features.