Differential graded categories in holomorphic symplectic geometry

TL;DR

This paper explores the structure and formality of differential graded categories associated with holomorphic symplectic manifolds, introducing new invariants and criteria for their formality.

Contribution

It establishes the formality of certain dg categories in holomorphic symplectic geometry and introduces Kaledin classes as obstructions to formality.

Findings

Proves formality of dg categories localized at specific Lagrangian submanifolds.

Defines Kaledin classes as obstructions to formality.

Provides a criterion for the formality of flat weakly proper Calabi-Yau dg categories.

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.