Let X be a compact symplectic manifold. There is an idea, I think probably originally due to Kontsevich, that we should be able to get Gromov-Witten invariants of X out of the Fukaya category of X. One possible approach to doing this is via the theorem proved by Costello (I think there is also a similar(?) result of Kontsevich-Soibelman?) that a Calabi-Yau category determines a TCFT, which then should determine the Gromov-Witten invariants of X --- or at least something like the Gromov-Witten invariants of X. But in order for this to even get started, we need the Fukaya category of X to be a Calabi-Yau category (you can find the definition of CY category in Costello's paper, at the beginning of section 2).

Hence: Is the Fukaya category of a compact symplectic manifold known to be a Calabi-Yau category? What is the trace map supposed to be?