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

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