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?

  • $\begingroup$ This sounds very interesting to me. However I am a bit perplexed by the objects of your category being (sub)manifolds with isolated singularities. In the Fukaya-type categories I've seen, the objects are non-singular Lagrangian submanifolds, or immersed Lagrangian submanifolds... $\endgroup$ Mar 22 '10 at 20:29
  • $\begingroup$ I was just thinking of one way to generalize the condition of Lagrangian submanifold. Since in the above example the bilinear form is zero on the full vector space,the nodal sing somehow was meant to generalize being a maximal subspace on which the form vanishes. It seems like it might be better to think about leafwise constructions... or maybe not think about it at all. $\endgroup$ Mar 24 '10 at 18:14

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.

  • $\begingroup$ Thanks Tim. I was hoping that it could be enough to say the complex structure is something like Poisson semicompatible replacing the usual positivity with a semipositivity condition on the degenerate locus. It was your last point about quantization which lead me to think about this in the first place actually. I was trying to understand what conditions one could put on a Poisson structure so that one could have a chance to define Gromov Witten invariants. $\endgroup$ Mar 22 '10 at 18:40
  • $\begingroup$ I'd say that if you can prove a compactness theorem, then you have a chance. Non-compact symplectic leaves look like trouble to me... $\endgroup$
    – Tim Perutz
    Mar 22 '10 at 18:51
  • $\begingroup$ I'm far from knowledgeable about Fukaya categories, but I had the vague impression that a game akin to what Dan is asking about is happening also in the cotangent bundle case: one would rather work with a compact manifold, but the natural compactification is only Poisson. So instead one tries morally to reproduce the noncompact symplectic manifold by working on the compact manifold but with boundary conditions. [I thought in Nadler's stuff, Hamiltonian isotopies have boundary conditions at infinity?] A boundary condition at the origin seems implicit in Dan's discussion of the plane? $\endgroup$ Mar 23 '10 at 3:56
  • $\begingroup$ @Thomas. Nadler's picture also fits into a general class of good boundary conditions involving Lagrangians with Legendrian boundary on a convex contact boundary. I don't see that having a Poisson compactification can be a meaningful source of intuition about holomorphic curves (rather than about e.g. constructible sheaves) because we know nothing at all about holomorphic curves in Poisson manifolds. For instance, Dan's punctured plane has a degenerate symplectic form; it's not clear that viewing it as Poisson is relevant. $\endgroup$
    – Tim Perutz
    Mar 23 '10 at 15:02
  • $\begingroup$ Ah, ok...I was secretly (ignorantly!) imagining that boundary conditions in the Fukaya category should work similarly to what I'd imagine one would want to do in D-module--land...thanks for the explanation! $\endgroup$ Mar 24 '10 at 1:38

How about looking at the source-simply connected symplectic groupoid of an integrable Poisson manifold. and then forming ITS Fukaya category? This symplectic groupoid is naturally attached to the Poisson manifold, so any invariant of it is a Poisson invariant. If M has the zero Poisson structure, the symplectic groupoid is T*M. If M is the dual g* of a Lie algebra, the groupoid is the cotangent bundle of the simply connected Lie group G. If M is symplectic, the groupoid is the fundamental groupoid of M (just $M x M^{opp}$ if M is simply connected).


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