In [Link](https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-137/issue-1/Quantization-of-Chern-Simons-gauge-theory-with-complex-gauge-group/cmp/1104202513.full), Witten claims (p. 54) that to quantize the moduli space of flat $SL(2,\mathbb{C})$ connections on a torus, one can simply quantize the cotangent bundle of a real torus and take the part invariant by the Weyl group $W$.  

I agree that $T^\*(T\times T)/W$ is the moduli space of flat connections on the torus. However, it seems to me that the correct symplectic form should be the following: if we parametrize $T\times T$ with coordinates $(r\_1, \theta \_1, r\_2, \theta \_2)$ (quotienting the $\theta$ coordinates appropriately) then the symplectic form $\omega$ should be induced by $dr\_1\wedge dr\_2+d\theta \_1 \wedge d\theta \_2$. This restricts to the correct symplectic form on the real torus $T\times T$ obtained by taking the $r\_1 = r\_2 = 1$ subspace (which appears to be the $SU(2)$ character variety with its correct symplectic structure). But this is not the symplectic form induced by the cotangent structure, which would be $\omega'=d\theta \_1 \wedge dr\_1 + d\theta \_2 \wedge dr\_2$. On the other hand $\omega'$ vanishes on the real torus corresponding to $SU(2)$ representations, and does not seem to be the form induced by the character variety, which comes equipped with a natural symplectic form.  

This makes a substantial difference in quantization because $\omega'$ is exact and hence we can take the trivial line bundle for prequantization, and then in a real polarization obtain a Hilbert space $L^2(T\times T)$ (which is what Witten claims to be the quantization of the moduli space). However, if we take $\omega$ as the symplectic form, $\omega$ is non-zero in cohomology, and we will end up with a more complicated quantization.



SL(2,C) Chern-Simons theory in genus 1
Let me call your $\omega$ as $\omega\_I$. The symplectic form you get from the Chern-Simons action is $k\omega\_I+s\omega\_K$, where $\omega\_K$ is one of the Kähler forms on the Hitchin space, which, in particular, is exact. If you choose a real polarization as Witten does, the Hilbert space is $\Gamma(Bun\_GX,Det^{\otimes k})$, where $Det$ is the determinant bundle whose first Chern class $[\omega\_I]$. One should note that the polarization is not the naive vertical polarization on $T^\*Bun\_GX$ since the fibers are not Lagrangian for $k\neq 0$.


Narasimhan-Seshadri identifies $Bun\_GX$ with the character variety for the compact group, which in genus 1 is $T\times T\ /W$. So, the Hilbert space is $\Gamma(T\times T\ /W, Det^{\otimes k})$, precisely what Witten claims after eq. (5.11).


