Perturbative Quantum Gauge Fields on the Noncommutative Torus

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 asymptotically free theory. Besides, it turns out that nonplanar diagrams are overall convergent when θ is irrational.