Gravity as a Gauge Theory on Three-Dimensional Noncommutative spaces

TL;DR

This paper explores formulating three-dimensional gravity as a gauge theory within noncommutative spaces, specifically using fuzzy spaces based on SU(2) and SU(1,1), and investigates their connection to classical gravity.

Contribution

It introduces a novel approach to model 3D gravity as a gauge theory on noncommutative fuzzy spaces, extending the gauge symmetry framework to noncommutative geometry.

Findings

Constructed gauge theories on SU(2) and SU(1,1) fuzzy spaces.

Derived curvature and equations of motion for the noncommutative gravity models.

Established links between the noncommutative gauge models and classical 3D gravity.

Abstract

We plan to translate the successful description of three-dimensional gravity as a gauge theory in the noncommutative framework, making use of the covariant coordinates. We consider two specific three-dimensional fuzzy spaces based on SU(2) and SU(1,1), which carry appropriate symmetry groups. These are the groups we are going to gauge in order to result with the transformations of the gauge fields (dreibein, spin connection and two extra Maxwell fields due to noncommutativity), their corresponding curvatures and eventually determine the action and the equations of motion. Finally, we verify their connection to three-dimensional gravity.