On the regularization of the constraint algebra of quantum gravity in 2 + 1 dimensions with a nonvanishing cosmological constant

Abstract

We use the mathematical framework of loop quantum gravity (LQG) to study the quantization of three-dimensional (Riemannian) gravity with a positive cosmological constant (Λ > 0). We show that the usual regularization techniques (successful in the Λ = 0 case and widely applied in four-dimensional LQG) led to a deformation of the classical constraint algebra (or anomaly) proportional to the local strength of the curvature squared. We argue that this is an unavoidable consequence of the nonlocal nature of generalized connections.