Gauge theory on a space with linear Lie type fuzziness

TL;DR

This paper develops a U(1) gauge theory on a noncommutative space with Lie type fuzziness, introducing an extra gauge component and drawing parallels with lattice and noncommutative gauge theories.

Contribution

It constructs a gauge theory on a Lie type noncommutative space using Fourier translation groups, revealing an additional gauge field component.

Findings

Extra gauge field component observed in the theory.

Analogy with lattice gauge theory and Connes' noncommutative geometry.

Comparison with SU(N) gauge theories in noncommutative spaces.

Abstract

The U(1) gauge theory on a space with Lie type noncommutativity is constructed. The construction is based on the group of translation in Fourier space, which in contrast to space itself is commutative. In analogy with lattice gauge theory, the object playing the role of flux of field strength per plaquette, as well as the action, are constructed. It is observed that the theory, in comparison with ordinary U(1) gauge theory, has an extra gauge field component. This phenomena is reminiscent of similar ones in formulation of SU(N) gauge theory in space with canonical noncommutativity, and also appearance of gauge field component in discrete direction of Connes' construction of the Standard Model.