Is there any possibility of a Poisson Geometry version of the Fukaya category? Given a Poisson manifold Y, objects could be manifolds with isolated singularities X which have the property that TX is contained in NX maximally. The naive example would be something like the Poisson structure on R^2 which is (x^2 + y^2)(d/dx^d/dy). Branes would in this case be curves with some nodal singularity at the origin. 


The morphisms could still be from holomorphic disks with respect to the standard complex structure. In principal, it seems like in this example the Fukaya category could be defined in the standard way (although maybe there is something more subtle one should do with the morphisms?). For a brane L passing through the origin there should be some interesting multiplicative structure in the algebra A(L) owing to the fact that the brane is required to remain fixed at the origin.... I would hope that the Hochschild cohomology could be related to the Poisson cohomology of the manifold though I haven't studied my example yet. What obstructions arise when trying to construct this category?



A Poisson Geometry Version of the Fukaya Category
The fundamental technique of symplectic topology is the theory of pseudo-holomorphic curves. One studies maps $u$ from a Riemann surface into a symplectic manifold, equipped with an almost complex structure tamed by the symplectic form, such that $Du$ is complex-linear. Numerous algebraic structures can be built from such maps: Gromov-Witten invariants, Hamiltonian Floer cohomology, Floer cohomology of pairs of Lagrangian submanifolds, and most elaborate of all, $A\_\infty$-structures on Lagrangian Floer cochains (Fukaya categories). 


Though the basic theory of pseudo-holomorphic curves makes sense on more general almost-complex manifolds, the presence of the symplectic structure is vital for Gromov compactness to be applicable. Without this, your curves are liable to vanish into thin air. None of the algebraic structures I mentioned have been developed on almost complex manifolds, nor on Poisson manifolds. It's conceivable that leafwise constructions can be made to work in the Poisson context, but there are basic analytic and geometric questions to be addressed. 


There are situations where one might reasonably hope to find relations between Poisson geometry and symplectic topology, but in those situations it may be wise to go via intermediate constructions. For instance, a version of the derived Fukaya category of $T^{\ast} L$ was shown by Nadler to be equivalent to the derived category of constructible sheaves on $L$, and I'm told that that category is related to deformation quantization of $T^{\ast} L$ - something which truly does belong to Poisson geometry. 


