Complete loop quantization of a dimension 1+2 Lorentzian gravity theory

TL;DR

This paper achieves a complete loop quantum gravity quantization of a 1+2 dimensional Lorentzian gravity theory by fixing a gauge to obtain a compact residual gauge group, enabling standard LQG techniques.

Contribution

It introduces a gauge fixing that reduces the gauge group to SU(2), facilitating the LQG quantization of a Lorentzian gravity model in 3D space-time.

Findings

Successfully applies LQG techniques to a 3D Lorentzian gravity model.

Demonstrates the residual gauge group is compact, enabling standard quantization.

Provides a framework for future quantum gravity research in lower dimensions.

Abstract

De Sitter Chern-Simons gravity in D = 1 + 2 spacetime is known to possess an extension with a Barbero-Immirzi like parameter. We find a partial gauge fixing which leaves a compact residual gauge group, namely SU(2). The compacticity of the residual gauge group opens the way to the usual LQG quantization techniques. We recall the exemple of the LQG quantization of SU(2) CS theory with cylindrical space topology, which thus provides a complete LQG of a Lorentzian gravity model in 3-dimensional space-time.