Differential graded categories in holomorphic symplectic geometry

Let $(\mathrm{X},\sigma )$ be a holomorphic symplectic manifold. We study the differential graded category of canonical Lagrangian $\mathrm{D}$-branes ${\mathcal{D}}_{\mathrm{L}\mathrm{a}\mathrm{g}}(\mathrm{X},\sigma )$ along with its deformation quantisation, spanned by quantised orientations, $\mathcal{D}\mathcal{Q}(\mathrm{X},\sigma )$, and the virtual de Rham category ${\mathcal{D}\mathcal{R}}^{\mathrm{v}\mathrm{i}\mathrm{r}}(\mathrm{X},\sigma )$. We prove the formality of these dg categories when localised at a countable collection of orientable compact Kähler Lagrangian submanifolds with pairwise clean intersections. Along the way, we define Kaledin classes of minimal ${\mathrm{A}}_{\mathrm{\infty }}$-categories and show that they are the obstructions to formality. In addition, we obtain a formality criterion for flat weakly proper Calabi-Yau dg categories.