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, $\mathcal{D}\mathcal{Q}(\mathrm{X},\sigma )$, 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 Kähler 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.

50pp., comments welcome