In http://arxiv.org/abs/hep-th/9612216 , Rozansky and Witten have associated to every holomorphic symplectic manifold $X$ a 3d TQFT (three dimensional Topological Quantum Field Theory), by topological twisting of a 3d sigma model of target $X$. The 3d sigma model is not renormalizable and so not well-defined in the UV but nevertheless, the 3d topological theory is well-defined. It is a kind of odd Chern-Simons theory.

One can show that if we take $X = T^*Y$ for $Y$ a complex manifold ( $T^*Y$ is naturally holomorphic symplectic), then the dimensional reduction over a circle of the 3d Rozansky-Witten TQFT of target $X$ is the B-model 2d TQFT of target $Y$. The question is: is there a similar story for the A-model?

In other words: Let $Z$ be a (real) symplectic manifold, the A-model of target $Z$ is a 2d TQFT. Is there a 3d TQFT constructed from $Z$ whose dimensional reduction over a circle is the A-model 2d TQFT of target $Z$ ?

This question is motivated by the fact that the existence of the Rozansky-Witten TQFT in some sense "explains" the existence of a (derived) tensor product over the derived category of coherent sheaves $D(Y)$ of a complex manifold $Y$. If one thinks about $D(Y)$ as the category of branes of the B-model, this structure seems to have no physical meaning ($D(Y)$ is what is associated to a point by the TQFT but there is no smooth cobordism between 2 points on the one and and 1 point on the other hand). But in the Rozansky-Witten theory on $X=T^*Y$, $D(Y)$ is the category of line defects, it is what is associated to a circle by the TQFT and the tensor product becomes natural because there exists a cobordism between 2 circles on the one hand and 1 circle and the other hand (a pair of pants). Mathematically, the (derived) tensor product is a rather obvious operation on $D(Y)$. If $Z$ is a symplectic manifold, the category of branes of the A-model of target $Z$ is the Fukaya category $Fuk(Z)$ of $Z$. But now, mathematically, there is no obvious tensor product on $Fuk(Z)$. Nevertheless, such a thing should exist if mirror symmetry is true, i.e. if there exists a complex manifold $Y$ such that $Fuk(Z)=D(Y)$. Maybe that if the A-model analogue of the Rozansky-Witten theory exists, it will be easier to see what is the tensor product like operation on the Fukaya category.