Yang-Mills theory on noncommutative space: does it exist?

TL;DR

This paper investigates the fundamental question of whether noncommutative Yang-Mills theories can be nonperturbatively formulated, focusing on matrix model approaches and their relation to Eguchi-Kawai equivalence.

Contribution

It demonstrates that supersymmetric noncommutative Yang-Mills theories can be straightforwardly defined and discusses potential formulations for non-supersymmetric cases.

Findings

Supersymmetric noncommutative Yang-Mills theory can be defined straightforwardly.

The existence of noncommutative Yang-Mills theory is linked to Eguchi-Kawai equivalence.

Non-supersymmetric theories may be formulated by adjusting ultraviolet and infrared behaviors.

Abstract

I revisit a basic question about the noncommutative Yang-Mills theory: if it exists or not, or more precisely, whether a nonperturbative formulation exists. As the most promising approach, I consider a formulation based on matrix models. It is explained that the existence of the noncommutative Yang-Mills theory is closely related to the Eguchi-Kawai equivalence. I argue that supersymmetric noncommutative Yang-Mills theory can be defined straightforwardly. Non-supersymmetric theories, such as QCD and pure bosonic theories, can presumably be defined, by modifying the ultraviolet and infrared behaviors appropriately.