On the regularization of the constraints algebra of Quantum Gravity in 2+1 dimensions with non-vanishing cosmological constant

TL;DR

The paper investigates the canonical quantization of 2+1 dimensional Riemannian gravity with a positive cosmological constant within loop quantum gravity. Using a holonomy-flux regularization on a cellular discretization, it shows that while the Gauss constraint algebra remains anomaly-free, the regulated curvature constraint algebra develops a genuine quantum anomaly proportional to $\Lambda$ and the local curvature term $\operatorname{tr}[W]-2$, persisting in the continuum limit. This deformation indicates a deformation of the diffeomorphism-related symmetry, posing a major challenge for Dirac-type quantization and suggesting links to quantum-group structures that arise in gravity with $\Lambda>0$. The results highlight a fundamental obstacle to extending the LQG program to nonzero cosmological constant in 2+1 dimensions and motivate exploring alternative quantization schemes like master constraint methods.

Abstract

We use the mathematical framework of loop quantum gravity (LQG) to study the quantization of three dimensional (Riemannian) gravity with positive cosmological constant (Lambda>0). We show that the usual regularization techniques (successful in the Lambda=0 case and widely applied in four dimensional LQG) lead 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 non-local nature of generalized connections.

Figures (4)

• Figure 1: Orientation of plaquette holonomies chosen for the regularization of $F[N]$.
• Figure 2: On the left: portion of the cellular decomposition $C_{ \Sigma}$ (thin lines) and its dual $C_{ \Sigma}^{\star}$ (thick lines). On the right: the edges of $C_{ \Sigma}^{\star}$ are shifted toward the corresponding nodes. The flux operators necessary for the definition of the regularization of $E[\Lambda N]$ are defined in terms of the latter shifted dual edges.
• Figure 3: On the left: a generic non $SU(2)$ invariant state $\Psi$ in our cellular decomposition. On the right: explicit illustration of the orientation of the paths used in the regularization of the constraints.
• Figure 4: All possible values of the orientations $o_{\eta\gamma}$ of the crossing appearing in equation (\ref{['fluxx']}). The edges $\eta$ regulating the flux operators are the dotted lines while the holonomies are along the continue lines $\gamma$.