Noncommutative Gauge Theory and Gravity in Three Dimensions

TL;DR

This paper develops a noncommutative gauge theory framework for three-dimensional gravity using covariant coordinates on fuzzy spaces, linking gauge fields to geometric structures like the vielbein and spin connection.

Contribution

It introduces a novel approach to 3D gravity via noncommutative geometry, constructing gauge theories on fuzzy spheres and hyperboloids and relating them to gravitational actions.

Findings

Constructed noncommutative gauge theories on fuzzy 2-spheres and hyperboloids.

Identified noncommutative vielbein and spin connection within the gauge framework.

Formulated a matrix model action related to 3D gravity.

Abstract

The Einstein-Hilbert action in three dimensions and the transformation rules for the dreibein and spin connection can be naturally described in terms of gauge theory. In this spirit, we use covariant coordinates in noncommutative gauge theory in order to describe 3D gravity in the framework of noncommutative geometry. We consider 3D noncommutative spaces based on SU(2) and SU(1,1), as foliations of fuzzy 2-spheres and fuzzy 2-hyperboloids respectively. Then we construct a U(2)$\times$ U(2) and a GL(2,$\mathbb{C}$) gauge theory on them, identifying the corresponding noncommutative vielbein and spin connection. We determine the transformations of the fields and an action in terms of a matrix model and discuss its relation to 3D gravity.