A Chern-Simons approach to self-dual gravity in (2+1)-dimensions and quantisation of Poisson structure

Abstract

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.