Speaker
Description
We present a Chern-Simons theory for the (2+1)-dimensional analog self-dual gravity theory that is based on the gauge group $SL(2,\mathbb{C})_ \mathbb{R}\triangleright\!\!\!< \mathbb{R}^6$. This is formulated by mapping the $3d$ complex self-dual dynamical variable and connection to $6d$ real variables which combines into a $12d$ Cartan connection.
Quantization is given by the application of the combinatorial quantisation program of Chern-Simons theory. The Poisson structure for the moduli space of flat connections on $(SL(2,\mathbb{C})_ \mathbb{R}\triangleright\!\!\!< \mathbb{R}^6)^{n+2g}$ which emerges in the combinatorial description of the phase space on $\mathbb{R} \times \Sigma_{g,n},$ where $\Sigma_{g,n}$ is a genus $g$ surface with $n$ punctures is given in terms of the classical $r$-matrix for the quantum double $D(SL(2,\CC)_\RR)$ viewed as the double of a double $ D(SU(2)\bowtie AN(2))$. This quantum double provides a feature for quantum symmetries of the quantum theory for the model.
