2023/11/16 by Osei, Prince K.
#FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Mathematical Physics (math-ph)
paper · doi:10.48550/arxiv.2311.10220
The (2+1)-dimensional analog self-dual gravity which is obtained via spacetime dimension reduction of the (3+1)-dimensional Holst action without reducing the internal gauge group is studied. A Chern-Simons formulation for this theory is constructed based on the gauge group SL(2,\CC)_\RR\rcross \Rsix and maps the 3d complex self-dual dynamical variable and connection to 6d real variables which combines into a 12d Cartan connection. The Chern-Simons approach leads to a real analogue for the self-dual action based on a larger symmetry group. The quantization process follows the combinatorial quantization method outlined for Chern-Simons theory. In the combinatorial quantization of the phase space the Poisson structure governing the moduli space of flat connections which emerges is obtained using the classical r-matrix for the quantum double D(SL(2,\CC)_\RR) viewed as the double of a double D(SL(2,\RR)\dcross AN(2)). This quantum double gives the structure for quantum symmetries within the quantum theory for the model.