TITLE: Geometric quantization of Teichmuller space
QUESTION [5 upvotes]: The quantizations of Teichmuller space I have seen are via special coordinates (e.g. the paper of Chekhov and Fock) or conformal blocks.  Does one get an equivalent quantization by geometric quantization?  For example the Weil-Peterson metric is Kahler.  Is it known, one way or the other, if doing Kahler quantization gives the same thing?  What about a different polarization?

REPLY [2 votes]: One of best ways for quantization of Kahler varieties is to use Kahler reduction since quantization commutes with reduction. 
so you can write Teichmuller space $\mathcal T(M)$ by symplectic reduction, 
$$\mu^{-1}(0)/Diff^+M\cong\mathfrak{Met}_{-1}(M)/Diff^+M=\mathcal T(M)$$
which $\mathfrak{Met}_{-1}M$ is the space of reimannian metrics of constant scalar curvature $-1$. Hence has $\omega_{WP}$ Weil-Petersson metric. 
see page 12 http://arxiv.org/pdf/math/0507076.pdf
and also nice paper of Maryam Mirzakhani pages 8-9 http://www.ams.org/journals/jams/2007-20-01/S0894-0347-06-00526-1/S0894-0347-06-00526-1.pdf
For example for quantization of coadjoint orbit, you need to quantize $T^*G$, since coadjoint orbit is just symplectic quotient of $T^*G$. See my master project presentation http://fr.slideshare.net/HassanJolany/geometric-quantization-on-coadjoint-orbits