Gauge theories on quantum spaces

TL;DR

This paper reviews the development of gauge theories on noncommutative quantum spaces, discussing mathematical frameworks, various models, and recent efforts to incorporate gravity within these noncommutative geometries.

Contribution

It provides a comprehensive overview of gauge theories on quantum spaces, highlighting different formulations and recent progress in integrating gravity into noncommutative frameworks.

Findings

Different formulations of gauge theories on Moyal and Lie algebra-based quantum spaces

Mathematical tools for constructing gauge theories on noncommutative geometries

Recent approaches to include gravity dynamics in noncommutative models

Abstract

We review the present status of gauge theories built on various quantum space-times described by noncommutative space-times. The mathematical tools and notions underlying their construction are given. Different formulations of gauge theory models on Moyal spaces as well as on quantum spaces whose coordinates form a Lie algebra are covered, with particular emphasis on some explored quantum properties. Recent attempts aiming to include gravity dynamics within a noncommutative framework are also considered.