Moyal-Weyl star product and quasiconformal mapping relationship

The deformation quantization procedure can introduce the quantum effect by including the deformation of complex structures on a bi-dimensional Riemann surface Σ without boundary. This is the platform of bi-dimensional conformal models like string theory and two-dimensional gravity. Then, the star product can become a quasiconformal mapping on this surface. This makes the road toward a Teichmuller theory of the Moyal-Weyl star product through the quasiconformal mappings as lifting of universal covering surfaces of any Riemann surface.