DSF− 29/97 UAHEP9710 hep− th/9706190 2+1 Einstein Gravity as a Deformed Chern-Simons Theory

The usual description of 2 + 1 dimensional Einstein gravity as a Chern-Simons (CS) theory is extended to a one parameter family of descriptions of 2 + 1 Einstein gravity. This is done by replacing the Poincaré gauge group symmetry by a q−deformed Poincaré gauge group symmetry, with the former recovered when q → 1. As a result, we obtain a one parameter family of Hamiltonian formulations for 2 + 1 gravity. Although formulated in terms of noncommuting dreibeins and spin-connection fields, our expression for the action and our field equations, appropriately ordered, are identical in form to the ordinary ones. Moreover, starting with a properly defined metric tensor, the usual metric theory can be built; the Christoffel symbols and space-time curvature having the usual expressions in terms of the metric tensor, and being represented by c−numbers. In this article, we also couple the theory to particle sources, and find that these sources carry exotic angular momentum. Finally, problems related to the introduction of a cosmological constant are discussed.