We present a complete quantization of Lorentzian D=1+2 gravity with cosmological constant, coupled to a set of topological matter fields. The approach of Loop Quantum Gravity is used thanks to a partial gauge fixing leaving a residual gauge invarianc
e under a compact semi-simple gauge group, namely Spin(4) = SU(2) x SU(2). A pair of quantum observables is constructed, which are non-trivial despite of being null at the classical level.
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 g
auge 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.
D = 2+1 gravity with a cosmological constant has been shown by Bonzom and Livine to present a Barbero-Immirzi like ambiguity depending on a parameter. We make use of this fact to show that, for positive cosmological constant, the Lorentzian theory ca
n be partially gauge fixed and reduced to an SU(2) Chern-Simons theory. We then review the already known quantization of the latter in the framework of Loop Quantization for the case of space being topogically a cylinder. We finally construct, in the same setting, a quantum observable which, although non-trivial at the quantum level, corresponds to a null classical quantity.
In this paper we consider the quantization of the 2d BF model coupled to topological matter. Guided by the rigid supersymmetry this system can be viewed as a super-BF model, where the field content is expressed in terms of superfields. A canonical an
alysis is done and the constraints are then implemented at the quantum level in order to construct the Hilbert space of the theory under the perspective of Loop Quantum Gravity methods.
As a preparation for its quantization in the loop formalism, the 2-dimensional gravitation model of Jackiw and Teitelboim is analysed in the classical canonical formalism. The dynamics is of pure constraints as it is well-known. A partial gauge fixin
g of the temporal type being performed, the resulting second class constraints are sorted out and the corresponding Dirac bracket algebra is worked out. Dirac observables of this classical theory are then calculated.
