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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?

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At first sight, the Fukaya category has obvious cyclic symmetry, because the $A_\infty$ structure maps count points in spaces of rigid pseudo-holomorphic polygons subject to Lagrangian boundary conditions, and these spaces depend only on the cyclic order of the Lagrangians. This indeed proves that the cohomological Fukaya category, in which the hom-spaces are Floer cohomology spaces, is cyclically symmetric.

The trouble comes when these Lagrangians don't intersect one another transversely - for instance, the same Lagrangian occurs more than once - because then the morphism spaces and structure maps invoke Hamiltonian perturbations which need not be cyclically symmetric. The problem which Fukaya has solved over the reals (see Matthew Ballard's answer) is, I presume, to find a way to make these perturbations cyclically symmetric whilst also achieving the necessary coherence between them, as well as transversality for compactified moduli spaces of inhomogeneous pseudo-holomorphic polygons (or worse, their abstract perturbations). These are the things which actually define the $A_\infty$-structure.

FOOO worked extremely hard to get geometrically-meaningful units in their Fukaya endomorphism algebras, where other authors are content to define units by tweaking the $A_\infty$-structure algebraically. My hope would be that algebra will also give a cheaper approach to cyclic symmetry, particularly since I'm told that for Costello's theorem to hold, one only needs "derived" cyclic symmetry.

By the way, let's be clear that Costello's theorem, suggestive as it may be, is not about GW invariants! It's about theories over $M_{g,n}$, not over $\overline{M}_{g,n}$.

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Re your last sentence: Yes, I'm aware of this, thus my comment about "something like Gromov-Witten invariants". However, there's the story about the trivialization of circle action and extension to DM boundary.... –  Kevin H. Lin Jan 28 '10 at 1:19
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I believe the correct statement, currently, is that its not established but it is close. Fukaya has a preprint giving a model for Floer cohomology of a Lagrangian that has a pairing at the chain level which is cyclically symmetric. There is work underway to extend it to the whole Fukaya category.

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But note that Fukaya's preprint works over the real numbers only. –  Tim Perutz Jan 27 '10 at 17:01
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Good point Tim. This is one case where characteristic two is not necessarily our best friend. –  Matthew Ballard Jan 27 '10 at 18:54
    
Thank you for the reference! –  Kevin H. Lin Jan 27 '10 at 20:35
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