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

Jean-Christophe Wallet

TL;DR

Addressing gauge theories on noncommutative space-times, the paper surveys Yang–Mills extensions and matrix-model formulations built from $\star$-products and a derivation-based differential calculus. It spans Moyal spaces $\mathbb{R}^{2n}_\theta$, deformed $\mathbb{R}^3_\lambda$, and Lie-algebraic Minkowski deformations ($\rho$- and $\kappa$-Minkowski), comparing methods based on Hopf-algebra twists with Weyl quantization. A key highlight is the all-orders finiteness of a class of gauge-invariant constructions on $\mathbb{R}^3_\lambda$ with harmonic terms, alongside persistent obstacles such as UV/IR mixing and 1-loop tadpoles in many 4D and Minkowski-space models. The work clarifies the symmetry structures, gauge dynamics, and potential avenues—including twisted traces and braided $L_\infty$-formalisms—for connecting noncommutative gauge theories to quantum gravity frameworks like Group Field Theory.

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 $\mathbb{R}^3_λ$, 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 $κ$-Minkowski.

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

TL;DR

Addressing gauge theories on noncommutative space-times, the paper surveys Yang–Mills extensions and matrix-model formulations built from -products and a derivation-based differential calculus. It spans Moyal spaces , deformed , and Lie-algebraic Minkowski deformations (- and -Minkowski), comparing methods based on Hopf-algebra twists with Weyl quantization. A key highlight is the all-orders finiteness of a class of gauge-invariant constructions on with harmonic terms, alongside persistent obstacles such as UV/IR mixing and 1-loop tadpoles in many 4D and Minkowski-space models. The work clarifies the symmetry structures, gauge dynamics, and potential avenues—including twisted traces and braided -formalisms—for connecting noncommutative gauge theories to quantum gravity frameworks like Group Field Theory.

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 , a deformation of 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 -Minkowski.
Paper Structure (12 sections, 1 theorem, 108 equations, 1 table)

This paper contains 12 sections, 1 theorem, 108 equations, 1 table.

Table of Contents

  1. Introduction
  2. Algebraic and technical aspects
  3. Star-products and quantum symmetries
  4. Differential calculus, connection and curvature
  5. Gauge theory models on Moyal spaces and deformed $\mathbb{R}^3$
  6. Generalising Yang-Mills theory on Moyal spaces
  7. Gauge matrix models on Moyal spaces
  8. Gauge models on $\mathbb{R}^3_\lambda$
  9. Gauge theories on deformations of Minkowski space-time
  10. Yang-Mills type theory on $\rho$-Minkowski space-time
  11. Yang-Mills type theory on $\kappa$-Minkowski space-time
  12. Discussion

Key Result

Theorem 3.1

The amplitudes of the (ribbon) diagrams for any of the gauge theories described by $S^f_{\Omega}(\Phi)$ with $M>0$, $\mu>0$, $\Omega>0$, are finite to all orders in perturbation.

Theorems & Definitions (1)

  • Theorem 3.1: wal-16