Kontsevich–Soibelman conjecture for global torus fibrations

Let $B$ be a closed $\mathbb{Z}$-affine manifold, let $M=T^*B/T^*_{\mathbb{Z}}B$ be the associated symplectic torus fibration, and let $\mathcal{O}_B$ be its natural sheaf of $\Lambda_q$-algebras. Let $\mathcal{F}(M)$ be the Fukaya category, with gradings and signs defined using the canonical trivialization of $\mathcal{K}_M^2$ and relative $\operatorname{Pin}$ structures. Kontsevich–Soibelman conjecture. All of $\mathcal{F}(M)$ embeds cohomologically fully and faithfully into $\mathcal{D}(\mathcal{O}_B)$, the derived dg category of $\mathcal{O}_B$-module sheaves.

The preceding theorem gives this embedding for the full subcategory of Lagrangian sections. The conjecture extends that result to all Fukaya-category objects and is presented as an expected global relationship between symplectic geometry and sheaf theory.