Perturbative quantum gauge fields on the noncommutative torus

TL;DR

This paper investigates pure Yang-Mills theory on the noncommutative torus, demonstrating its renormalizability, asymptotic freedom, and convergence properties of non-planar diagrams using standard quantum field theory techniques.

Contribution

It provides a detailed one-loop renormalization analysis of Yang-Mills theory on the noncommutative torus, showing asymptotic freedom and convergence of non-planar diagrams for irrational .

Findings

The theory is ultraviolet divergent but renormalizable.

It exhibits asymptotic freedom at one loop.

Non-planar diagrams are convergent for irrational .

Abstract

Using standard field theoretical techniques, we survey pure Yang-Mills theory on the noncommutative torus, including Feynman rules and BRS symmetry. Although in general free of any infrared singularity, the theory is ultraviolet divergent. Because of an invariant regularization scheme, this theory turns out to be renormalizable and the detailed computation of the one loop counterterms is given, leading to an asymptoticaly free theory. Besides, it turns out that non planar diagrams are overall convergent when $\theta$ is irrational.