On a non-geometric approach to noncommutative gauge theories

TL;DR

This paper introduces a non-geometric method to construct noncommutative gauge theories, revealing new conserved currents and deriving noncommutative gauge structures without relying on traditional geometric frameworks.

Contribution

It generalizes Deser's non-geometrical approach to noncommutative settings, identifying non-local conserved currents and deriving noncommutative gauge theories from them.

Findings

Discovery of a non-local conserved current in free theories.

Derivation of noncommutative gauge theories from current consistency.

Emergence of the U(N) restriction from Lie algebra conditions.

Abstract

In this work, we generalize the non-geometrical construction of gauge theories, due to S. Deser, to a noncommutative setting. We show that in a free theory, along with the usual local N\"{o}ther current, there is another conserved current, which is non-local. Using the latter as a source for self-interaction, after a well-defined consistency procedure, we arrive at noncommutative gauge theories. In the non-abelian case, the standard restriction, namely that the theory should be $U(N)$ in the fundamental representation, emerges as a consequence of the requirement that the non-local current be Lie algebra valued.