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 WeilPeterson metric is Kahler. Is it known, one way or the other, if doing Kahler quantization gives the same thing? What about a different polarization?

$\begingroup$ Note that due to Tian, If a moduli space admit WeilPetersson metric then it is quasi projective, and its compactification is projective. Since WeilPetersson metric is K\"ahler, and $M$ be a Kahler quantizable then it is projective due to Kodaira. So it is better to work on compactification of moduli spaces which admit WeilPetersson metric. In fact due to Tian, canonical metric of moduli spaces can be introduced by WeilPetersson metric.... $\endgroup$ – user21574 Jun 14 '16 at 21:10

$\begingroup$ ...like moduli space of CalabiYau, moduli space of general type varieties. Moduli space of kpoly stable Fano varieties, see paper of ChiLi. $\endgroup$ – user21574 Jun 14 '16 at 21:10

$\begingroup$ The best is you take a holomorphic map $\pi:X\to Y$ which $X$ and $Y$ are Kahler and moduli space of fibres admit WeilPetersson metric and assume that $Y$ is Quantizable and try to see under which condition $X $ is quantizable. $\endgroup$ – user21574 Jun 14 '16 at 21:17
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 $\omega_{WP}$ WeilPetersson metric.
see page 12 http://arxiv.org/pdf/math/0507076.pdf
and also nice paper of Maryam Mirzakhani pages 89 http://www.ams.org/journals/jams/20072001/S0894034706005261/S0894034706005261.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/geometricquantizationoncoadjointorbits

2$\begingroup$ Thank you but my question is not how to construct the Kahler quantization but if it gets the same result as the quantization methods that use special coordinates or conformal blocks. $\endgroup$ – Eric O. Korman Jun 14 '16 at 23:01