What is Known about the $K$-Theory of Fukaya Categories?

Question

Some Background: In Kontsevich and Soibelman's theory of motivic DT-invariants, one is interested in something like the ``number'' of objects in a 3-Calabi-Yau category $\mathcal{C}$ having a fixed class $\gamma$ in $K_0(\mathcal{C})$. One of the motivating examples is in counting the number of certain Lagrangian submanifolds (with some extra data) in a Calabi-Yau 3-fold $X$ with a fixed class in $H_3(X)$. In other words, we want to count objects in the Fukaya category $\mathcal{D}^bLag(X)$ of $X$ subject to some constraint.

Question 1: Is it known (that under certain circumstances) that $K_0(\mathcal{D}^bLag(X)$) is isomorphic to $H_3(X)$?

I guess while we are at it, what about the higher $K$-groups, or the $K$-theory spectrum. My suspicion is not much is known, since I believe that very little is known about the higher $K$-theory of triangulated categories in general. Any comments or references would be much appreciated.

Answer 1

Let's say $X$ is an exact symplectic manifold, and $\pi:X\rightarrow\mathbb{C}$ is a Lefschetz fibration whose critical values are $z_1,\cdot\cdot\cdot,z_k\in\mathbb{C}$. Let $\gamma_1,\cdot\cdot\cdot,\gamma_k$ be vanishing paths emanating from $z_1,\cdot\cdot\cdot,z_k$ which do not intersect with each other, go to infinity in the direction of $\mathbb{R}_+$, and parallel to each other at infinity. Denote by $\Delta_1,\cdot\cdot\cdot,\Delta_k$ the corresponding Lefschetz thimbles, and assume that they are graded. Then a result of Seidel says that $\{\Delta_1,\cdot\cdot\cdot,\Delta_k\}$ is a full exceptional collection in the triangulated category $D^\pi\mathscr{F}(\pi)$. In particular we have

$K_0\big(D^\pi\mathscr{F}(\pi)\big)\cong\mathbb{Z}^k$.

Another general result due to Abouzaid considers the case when $X=T^\ast Q$ is a cotangent bundle of some closed manifold $Q$. In this case, the wrapped Fukaya category $\mathscr{W}(X)$ is well defined and is generated by a cotangent fiber, from which one concludes

$K_0\big(D^\pi\mathscr{W}(X)\big)\cong\mathbb{Z}$.

Answer 2

For $X$ Weinstein, it's a result of Oleg Lazarev that the map $H_n(X) \to SH(X)$ factors as

$$H_n(X) \twoheadrightarrow K_0(Fuk(X)) \to HH_\bullet(Fuk(X)) = SH_\bullet(X)$$

Here the map to $K_0(X)$ is defined by sending each n-handle in the Weinstein presentation to the corresponding cocore, which is an element of the (wrapped) Fukaya category. Surjectivity follows from the (relatively recent) result that cocores generate, and Oleg shows (by a use of the wrapping exact triangle) that the n-1 handles impose relations. The map $K_0(X) \to HH_\bullet(X)$ is the trace map.

(A nice consequence of this result is that if the map $H_n(X) \to K_0(Fuk(X))$ is not an isomorphism, one can conclude the Weinstein conjecture for the contact boundary of $X$.)

In fact one should expect similar maps to exist for all of $H_\bullet(X)$, though now I don't know any predictions about their properties. The reason I say the maps should exist is that for the microlocal sheaf category on an arboreal skeleton L, it follows from the local calculation made in my work with Takeda that there are canonical maps

$$H_\bullet(L) \to K_\bullet(msh(L)) \to HH_\bullet(msh(L))$$

A strong enough version of the sheaf/fukaya correspondence should identify the top degree part of these with those described by Oleg.