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

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 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.

Chern-Simons formulation of noncommutative gravity in three dimensions

“…where R = dw + w * w. The variation with respect to the triad yields (12) while the variation with respect to w yields the noncommutative torsion condition (13). In deriving the eqs.…”

Bicovariant calculus in quantum theory and a generalization of the Gauss law

“…Only for a class of quantum groups {G q } does a structure analogous to the one for Lie groups exist. There have subsequently been several attempts to apply this structure to dynamical systems [8], [9]. In these works, one finds generalizations of the usual classical Lagrangians for gauge theories, including Chern-Simons theory and gravity.…”

“…[12], [13] The continuum limit for these theories is nontrivial. On the other hand, in [8], [9] and what follows one works directly on the continuum. other hand, it can be checked that associativity is recovered for the so-called 'minimal' deformations [8] [9], where Λ ij kℓ Λ kℓ mn = δ i m δ j n (48)…”

Comments on the non-commutative description of classical gravity

“…This result was first achieved in three space-time dimensions [2] using a deformed Chern-Simons action [5] and the quantum Poincaré group ISO q (2, 1) [6]. In the resulting description of 2+1 gravity, the dreibeins and spin-connections are not c-numbers, but instead, obey nontrivial braiding relations.…”

“…In a couple of recent papers [2,3] we obtained a generalization of the gauge theory description of general relativity where the gauge group is replaced by a q-gauge group [4].…”