Gauge fields on noncommutative geometries with curvature

TL;DR

This paper constructs a $U_1$ gauge field on a curved noncommutative space, exploring how coordinate dependence affects gauge theories and their quantization, including gauge fixing and BRST invariance.

Contribution

It extends the noncommutative geometry framework to include gauge fields with curvature, analyzing their quantization and gauge fixing procedures.

Findings

Gauge fields exhibit coordinate dependence due to curvature.

The model reduces to a 2D system with gauge and scalar degrees of freedom.

Quantum action maintains BRST invariance.

Abstract

It was shown recently that the lagrangian of the Grosse-Wulkenhaar model can be written as lagrangian of the scalar field propagating in a curved noncommutative space. In this interpretation, renormalizability of the model is related to the interaction with the background curvature which introduces explicit coordinate dependence in the action. In this paper we construct the $U_1$ gauge field on the same noncommutative space: since covariant derivatives contain coordinates, the Yang-Mills action is again coordinate dependent. To obtain a two-dimensional model we reduce to a subspace, which results in splitting of the degrees of freedom into a gauge and a scalar. We define the gauge fixing and show the BRST invariance of the quantum action.