2+1 Einstein Gravity as a Deformed Chern-Simons Theory

TL;DR

This paper extends 2+1 Einstein gravity as a Chern-Simons theory by introducing a q-deformed gauge group, resulting in a one-parameter family of formulations with consistent metric and source coupling.

Contribution

It introduces a q-deformed Poincare' gauge symmetry to describe 2+1 Einstein gravity, providing a new family of Hamiltonian formulations with consistent metric structure.

Findings

The deformed theory reproduces standard Einstein gravity when q->1.

Coupling to particles reveals exotic angular momentum.

The framework accommodates a cosmological constant.

Abstract

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 Poincare' gauge group symmetry by a q-deformed Poincare' 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.