Differential graded categories in holomorphic symplectic geometry

Abstract

Let $(\mathrm{X},\sigma)$ be a holomorphic symplectic manifold. We study the differential graded category of canonical Lagrangian $\mathrm{D}$-branes $\mathcal{D}_\mathrm{Lag}(\mathrm{X},\sigma)$ along with its deformation quantisation, spanned by quantised orientations, $\mathcal{DQ}(\mathrm{X},\sigma)$, and the virtual de Rham category $\mathcal{DR}^{\mathrm{vir}}(\mathrm{X},\sigma)$. We prove the formality of these dg categories when localised at a countable collection of orientable compact K\"{a}hler Lagrangian submanifolds with pairwise clean intersections. Along the way, we define Kaledin classes of minimal $\mathrm{A}_\infty$-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.