Noncommutative Gauge Theory and Gravity in Three Dimensions

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.