The Kauffman bracket skein module $K_t(F\times I)$ (where $t$ is an indeterminant and $F$ is a closed surface) is an associative algebra (the operation being "stacking" links in the $I$ direction). It is a quantum deformation of the coordinate ring of the character variety $\operatorname{Hom}(\pi_1(F),\operatorname{SL}(2))//\operatorname{SL}(2)$. This skein algebra naturally acts on the vector space $\mathcal H_F$, the vector space associated to $F$ by the WRT TQFT.
Is there a direct construction of this vector space (e.g. in terms of generators and relations) which makes the action of $K_t(F\times I)$ manifest?
I know only two constructions of the vector space $\mathcal H_F$, and both are unsatisfactory for the purposes of this question. First, we could pick a complex structure on $F$, and let $\mathcal H_F$ be the global holomorphic sections of a particular line bundle $\mathcal L$ over the $\operatorname{SU}(2)$ character variety of $\pi_1(F)$. In this construction, I don't know of any explicit way to realize the action of $K_t(F\times I)$. Second, we could say the vector space is generated by all $3$-manfolds with boundary $F$, and quotient by the subspace which has zero WFT invariant when paired with all three-manifolds with boundary $-F$. This is unsatisfactory because the relations aren't "local", and they require understanding the WRT invariants for all three-manifolds.
I am looking for a definition of the vector space in the spirit of the definition of $K_t(F\times I)$, i.e. some generators modulo some local "picture" relations.
