Fukaya categories of hyperkahler reductions: general request for information

Question

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 Hyperkahler analogues of Kahler quotients). 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.

Answer 1

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??

Answer 2

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 $HF(L,L)\cong H_*(L)$.