A Chern-Simons approach to self-dual gravity in ( 2 + 1 ) ...

Abstract

The (2 + 1)-dimensional analogue 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 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 , viewed as the double of a double . This quantum double gives the structure for quantum symmetries within the quantum theory for the model.