Gauge Field Theory on the E_q(2)-covariant Plane

TL;DR

This paper explores gauge theory on the q-deformed Euclidean plane R^2_q using algebraic structures and star products, addressing gauge invariance issues and proposing solutions within noncommutative geometry.

Contribution

It develops two approaches to formulate gauge theory on R^2_q, including a Seiberg-Witten map implementation and analysis of gauge invariance constraints.

Findings

Gauge invariance requires a measure that breaks E_q(2)-invariance.

Two formulations of gauge theory on R^2_q are established.

Possible modifications to preserve invariance are proposed.

Abstract

Gauge theory on the q-deformed two-dimensional Euclidean plane R^2_q is studied using two different approaches. We first formulate the theory using the natural algebraic structures on R^2_q, such as a covariant differential calculus, a frame of one-forms and invariant integration. We then consider a suitable star product, and introduce a natural way to implement the Seiberg-Witten map. In both approaches, gauge invariance requires a suitable ``measure'' in the action, breaking the E_q(2)-invariance. Some possibilities to avoid this conclusion using additional terms in the action are proposed.