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I'd really like to hear any references or information people have about the Fukaya categories of hyperkahler reductions of vector spaces (for more informations on the varieties, see Nick Proudfoot's thesis). These are very nice symplectic manifolds (hyperkahler and exact, in particular), so I feel like their Fukaya categories should themselves be nice, but I've never found a good reference on them.

The marquee example of this is Nakajima quiver varieties. I would be very interested to hear anything about the Fukaya categories of these.

A question of particular interest to me the Floer homology of images of invariant Lagrangian subspaces in the quotient.

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This sounds like a lovely topic for one or more thesis projects. The relevant definitions are in Seidel's book, as are powerful tools for describing Fukaya categories. The Wehrheim-Woodward functoriality theorem may well come in handy too.

The only thing of any substance that occurs to me is that, if I'm not mistaken, the (exact) Fukaya category of $\mathrm{Hilb}^n(\mathbb{C}^2)$ is empty. Translation on the plane induces an automorphism of Hilb which should be Hamiltonian (because that's so away from the digaonal, and so everywhere by density - no?). That point needs to be checked. Yet such maps will displace any exact Lagrangian $L$ from itself, so $HF(L,L)=0$, contradicting the fact that the $HF(L,L)\cong H_*(L)$.

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I usually work in the balanced Hilbert scheme (where the center of mass is required to be the origin), where the translation argument doesn't work. On the other hand, I can't tell you any compact smooth Lagrangians in there (unlike some other examples, there aren't any obvious holomorphic ones). –  Ben Webster Dec 14 '09 at 0:56
    
I agree with you that it would be a good thesis problem; I'm just not sure I know anyone with the right mix of expertises to supervise such a thesis. –  Ben Webster Dec 14 '09 at 1:07
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At the risk of writing things that are obvious to those listening in: this is Nadler-land, no?

If $X$ is a smooth complex variety with reductive group $G$ acting, and $\mu_{\mathbb C}: T^*X\rightarrow {\mathfrak g}^*$ is the complex moment map, then $\mu_{\mathbb C}^{-1}(0)/G = T^*(X/G)$ provided one interprets all quotients as stacks.

If $T^*X$ is hyperkahler and we do the hyperkahler quotient for the maximal compact of $G$, picking a nontrivial real moment value $\mu_{\mathbb R}^{-1}(\zeta)$ at which to reduce amounts (by Kirwan) to imposing a GIT stability on $\mu_{\mathbb C}^{-1}(0)$---i.e. to picking a nice open subset of the cotangent stack $T^*(X/G)$ that is actually a variety. A stack version of Nadler's "microlocal branes" theorem would describe the (suitable, undoubtedly homotopical/derived) exact Fukaya category as the constructible derived category of $X/G$.

Since I'm completely ignorant of how the Nadler-Zaslow/Nadler story actually works, I'd like to then imagine that such an equivalence microlocalizes properly to give an equivalence over the hyperkahler reduction (i.e. the nice open set) as well? Admittedly, by microlocalizing to the stable locus one should avoid all the derived unpleasantness (this should be analogous to what happens in Bezrukavnikov-Braverman's proof of "generic" geometric Langlands for $GL_n$ in characteristic $p$, where by localizing to the generic locus, ${\mathcal D}$-module really means ${\mathcal D}$-module, not "module over the enveloping algebroid of the tangent complex" or something like that).

Admittedly, I don't have a clue how to deal with the issue that the base $X$ in the important examples is typically affine...maybe if one forces some kind of boundary conditions also in the $X$-direction one could make the Fukaya category nontrivial in Tim's example of the Hilbert scheme??

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One of the dangers of not putting enough context is a question is getting an excellent explanation of your own motivation. Which is to say, I basically agree with you, but part of what I was looking for was how to really look at this story from the Fukaya side, since one has some topological intuition in there that I don't know how to make precise on the microlocal side of things. Either that, or how to do precisely the stuff that you skipped over. –  Ben Webster Dec 17 '09 at 22:02
    
Which is also to say "can I do computations in that microlocalization"? Ginzburg and Chriss tell me how to compute Ext-algebras in constructible sheaves using convolution algebras. Is there any way to "microlocalize" this fact? –  Ben Webster Dec 17 '09 at 22:11
    
So, the risk/probability of writing things obvious to those present was one. :-) Here's a closely related question for the symplectic experts: are there known nontrivial examples of a Fukaya category in the presence of a B-field for which such a microrestriction functor is an equivalence? [Do people even think about equivariant Fukaya categories?] –  Thomas Nevins Dec 18 '09 at 15:32
    
Rats, I wasn't thinking clearly when I wrote that...what I should have said was, are there nontrivial examples in which, in the presence of a B-field, the Fukaya categories of $\mu_{\mathbb R}^{-1}(0) \cap \mu_{\mathbb C}^{-1}(0)/K$ (where $K$ is the maximal compact of the group $G$) and $\mu_{\mathbb R}^{-1}(\zeta) \cap \mu_{\mathbb C}^{-1}(0)/K$ coincide? [For example, if you put in a generic B-field, would you expect the Fukaya categories of the $n$th symmetric product of ${\mathbb C}^2$ and its Hilbert scheme of $n$ points to be equivalent? –  Thomas Nevins Dec 18 '09 at 22:46
    
Equivariant transversality can be painful, and I don't think anyone has defined equivariant Fukaya categories beyond involutions. Actually, it's hard to underestimate the range of non-trivial examples we understand... But we do know that $(\mathbb{C}^2)^n$ contains no exact Lagrangians, $S_n$-equivariant or otherwise, so I'd be inclined to replace $\mathbb{C}^2$ by some other affine surface. –  Tim Perutz Dec 22 '09 at 23:24
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