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I recommend M. Bos and V.P. Nair. Coherent State Quantization of Chern-Simons Theory. International Journal of Modern Physics A, A5:959, 1990. Also, I wrote a short review of it: check the second section of my paper on Yang-Mills-Chern-Simons theory (http://arxiv.org/abs/1311.1853) .
The conformal compactification is supposed to belong to the projective space so we still identify points along the rays (equivalence classes under scaling) $$ (u,v,x_i)\sim \lambda (u,v,x_i), \quad \lambda \neq 0$$ Then there is the quadric equation you wrote down – an equation that respects the identification above – so both added variables $u,v$ are pretty ...
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