Noncommutative Gauge Theories: Yang-Mills extensions and beyond - An overview

TL;DR

This paper reviews various gauge theories on quantum space-times, focusing on Yang-Mills extensions, matrix models, and new quantum Minkowski spaces, highlighting their mathematical structures and differences.

Contribution

It provides an overview of gauge theories on noncommutative spaces, introduces new quantum Minkowski space-times, and compares different formulations of Yang-Mills theories.

Findings

Characterization of 11 new quantum Minkowski space-times.

Construction of a gauge theory on a new quantum space-time.

Presentation of a solvable matrix model on deformed space.

Abstract

The status of several representative gauge theories on various quantum space-times, mainly focusing on Yang-Mills type extensions together with a few matrix model formulations is overviewed. The common building blocks are derivation based differential calculus possibly twisted and noncommutative analog of the Koszul connection. The star-products related to the quantum space-times are obtained from a combination of harmonic analysis of group algebras combined with Weyl quantization. The remaining problems inherent to gauge theories on Moyal spaces in their two different formulations are outlined. A family of gauge invariant matrix models on $\mathbb{R}^3_\lambda$, a deformation of $\mathbb{R}^3$ is presented among which a solvable model. The characterization of 11 new quantum Minkowski space-times through their $*$-algebras is given. A gauge theory of Yang-Mills type is constructed on one recently explored of these space-times and compared to its counterpart built on the popular $\kappa$-Minkowski.