This is a question about the Witten--Reshetikhin--Turaev representations of the mapping class group of a closed surface $\Sigma_g$. For simplicity, we'll stick to the case $G=SU(2)$.

These representations are constructed as follows. To a closed surface $\Sigma_g$, we associate the character variety $X=X(\pi_1(\Sigma_g),SU(2))$. There is a natural line bundle $\mathcal L$ over $X$, whose curvature is a natural symplectic form on $X$. We now let $Z(\Sigma_g,k)$ be a certain vector space consisting of sections of $\mathcal L^{\otimes k}$ over $X$. There are various ways of constructing this vector space.

(1) The standard construction for the WRT invariants is to give a holomorphic polarization of $X$ (i.e. equip it with a Kahler structure), which depends on a choice of complex structure on $\Sigma_g$. The space of holomorphic sections of $\mathcal L^{\otimes k}$ then forms $Z(\Sigma_g,k)$. Given two holomorphic structures on $\Sigma_g$, there is a natural isomorphism between the corresponding spaces of holomorphic sections of $\mathcal L^{\otimes k}$, so we get a canonical vector space $Z(\Sigma_g,k)$, and an action of the mapping class group on it.

(2) Alternatively, one can use a "real polarization" of $X$ as in this paper. In this setting, one constructs a Lagrangian fibration $X\to B$ based on a decomposition of $\Sigma_g$ into pairs of pants, and the space $Z(\Sigma_g,k)$ consists of flat sections of $\mathcal L^{\otimes k}$ over the leaves of this fibration for which the monodromy of $\mathcal L^{\otimes k}$ is trivial. In the reference given above, it is shown that the number of such fibers is the same as the dimension of the space of holomorphic sections from (1), so this $Z(\Sigma_g,k)$ has the "correct" dimension. In this construction, the vector spaces associated to different polarizations can also be related by a natural isomorphism, the BKS pairing. Thus there is a resulting representation of the mapping class group on $Z(\Sigma,k)$ here as well.

Question: are these two constructions equivalent in that there is a natural isomorphism between the two spaces I've called $Z(\Sigma_g,k)$ above? Here "natural" means at least that the isomorphism is equivariant with respect to the action of the mapping class group.

Comment: There are 3-manifold invariants coming out of (2) which do NOT agree with the WRT 3-manifold invariants (see this paper), though this is perhaps due to the construction of the canonical elements associated to handlebodies being different, not the quantization of the moduli space being different.

This post imported from StackExchange MathOverflow at 2014-09-02 20:38 (UCT), posted by SE-user John Pardon