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