Noncommutative Fukaya category?

Question

After reading through part of Victor Ginzburg's notes on Calabi-Yau algebras, I have a question about a principle in mirror symmetry. Let $(X,X')$ be a mirror pair of Calabi-Yau varieties then mirror symmetry predicts a bijection between
$$M_\mathbb{C}(X) \leftrightarrow M_K(X'),$$ where $M_{\mathbb{C}}(X)$ are smooth CY-deformations of $X$ and $M_{K}(X')$ is the 'stringy' Kahler resolutions of $X'$. The homological mirror symmetry conjecture of Maxim Kontsevich predicts an equivalence of categories $D^b(coh(X_c))\simeq D(Fuk(X'_{c'}))$ where $c\leftrightarrow c'$ under the bijection on the moduli spaces. The problem is that there are singular Calabi-Yau varieties without any smooth deformations or any smooth crepant resolutions. So the homological mirror symmetry conjecture doesn't seem to have a lot of substance in these cases. An insight of Michael Van den Bergh is that every Calabi-Yau variety should have a noncommutative deformation or noncommutative crepant resolution. Then under the bijection on moduli spaces it is then possible for a noncommutative deformation to map to commutative Kahler resolution and vice verse. To extend homological mirror conjecture to include these noncommutative spaces it seems plausible to define the category of coherent sheaves on a noncommutative space using finitely generated projective modules. So here is my question:

The definition of the Fukaya category on a symplectic manifold uses techniques that only seem to be available in the geometric context, so is there a plausible definition of the `Fukaya category' of a noncommutative space in order to make the noncommutative homological mirror symmetry conjecture hold?

Answer 1

Since no one else has tried to answer, I'll take a shot. It seems to me that there are threads of ideas in this story that in the very distant future might be woven together to give a possible answer.

To begin, we should note that there seems to be a general idea, discussed in this mathoverflow question, Deformation quantization and quantum cohomology (or Fukaya category) -- are they related?. That one could define the Fukaya category as modules over a deformation quantization of $C^{\infty}(X)$ corresponding to the symplectic form $\omega$.

The basic idea is that in two naive respects this category of modules behaves a lot like the Fukaya category. Firstly, the Hochschild cohomology of the deformation quantization is almost by definition the Poisson cohomology of the symplectic form $\omega$, which in turn is known to be isomorphic to $H^*(X)((t))$. As an equation:

$$HH^*(A_\omega,A_\omega) \cong H^*(X)((t)) $$

Second, one can define a reasonable notion of modules with support on a Lagrangian submanifold and for any Lagrangian L, produce canonical holonomic modules supported there. One can compute that $$Ext(M_L,M_L) \cong H^*(L)((t))$$ There is some hope that one can put in the instanton corrections in a formal algebraic way and a fair amount of work has been done in this direction.

This story works best so far for the Fukaya category of $T^*X$ where the deformation quantization is roughly the algebra of differential operators. This is related to more work than I could competently summarize. I'll just mention, work of Nadler and Zaslow, Tsygan and Tamarkin. This approach is used by Kapustin and Witten to incorporate co-isotropic branes into the Fukaya category in their famous study of the Geometric Langlands. There, they are after some enlargement of Nadler's infinitesimal Fukaya category of $T^*(X)$. Note however that this not the same Fukaya category(the wrapped Fukaya category) that one studies in the context of mirror symmetry, but perhaps things will work better in the compact case if that is ever put on firm ground.

This was all a prelude to say that deformation quantization places you firmly in the land of non-commutative geometry anyways. Things like differential operators for non-commutative rings can make sense http://www.springerlink.com/content/r0rqguawu1960qxy/. I've never really looked at Van Den Bergh's work, but perhaps the passage from the sheaf of algebraic functions to the sheaf $C^\infty(X)$ is another stumbling point. One of Maxim's Kontsevich's ideas (see his Lefschetz lecture notes http://www.ihes.fr/~maxim/TEXTS/Kontsevich-Lefschetz-Notes.pdf) is that for any saturated dg-algebra there should maybe exist some nuclear algebra which bears the same formal relationship as the algebra of algebraic functions and smooth functions.