The claim that the $K$-theory of the derived Fukaya category of a symplectic Calabi-Yau manifold (exact with suitable convexity at infinity, or compact) factorizes through homology $H_{n}(X)$ modulo torsion seems well known to symplectic topology experts, although a precise reference seems rather difficult to find. We now sketch a high brow viewpoint explained to the author by P. Seidel and S. Rezchikov, and will later explain in more detail a more pedestrian approach in the exact setting. The claim is a formal consequence of the existence of maps

$$K_{0}(D^{b}Fuk(X))\xrightarrow{ch_{0}}HH_{0}(D^{b}Fuk(X))$$

$$HH_{0}(D^{b}Fuk(X))\xrightarrow{OC}QH^{n}(X;\Lambda_{nov})\simeq H_{n}(X)\otimes\Lambda_{nov}.$$

Here $HH_{0}(D^{b}Fuk(X))$ is the Hochschild homology in degree zero. The first map sends the K-theory class of an unobstructed Lagrangian brane $L$ (compact, graded, oriented, with spin and bounding cochain structure) to the unit $1_{L}\in HF^{0}(L,L)\to HH_{0}(D^{b}Fuk(X))$; the well definition of this map is an essentially algebraic fact. Suppose $L,L^{\prime}$ are isomorphic in $D^{b}Fuk(X)$, then there are closed morphisms $\alpha\in CF^{0}(L,L^{\prime})$ and $\beta\in CF^{0}(L^{\prime},L)$ whose derived category compositions are equal to $1_{L}\in HF^{0}(L,L)$ and $1_{L^{\prime}}\in HF^{0}(L^{\prime},L^{\prime})$ in cohomology. The Hochschild differential of $\beta\otimes\alpha$ exhibits $1_{L}-1_{L^{\prime}}$ as a coboundary in the Hochschild chain complex, so $1_{L}=1_{L^{\prime}}$ in $HH_{0}(D^{b}Fuk(X))$. Some additional calculation shows the compatibility with distinguished triangles.

The second map is a special case of the open-closed string map, and in general requires working over the Novikov field. One then needs the claim that $1_{L}$ is sent to the homology class $[L]\in H_{n}(X)$, without quantum correction. The intuitive meaning of the open-closed string map is to consider holomorphic discs with boundary on $L$ with an unconstrained boundary marked point, and find the cycle in $X$ traced out by an interior marked point. The claim amounts to saying that the only contribution comes from constant maps. Unfortunately, the author is unable to locate a general reference. Granted this claim, we would get by composition a map from $K_{0}(D^{b}Fuk(X))$ to $H_{n}(X;\Lambda_{nov})$ which sends the K-theory class of $L$ to the homology class $[L]$.