Abstract:
Given a holomorphic symplectic manifold X, I will associate to X a virtual de Rham dg category and a dg category of canonical D-branes of type B wrapped on spin complex Lagrangians in X along with its deformation quantisation.
For any suitable collection of complex Lagrangians, I will upgrade the deformation quantisation, with supports in the collection, to a formal deformation whose central fibre is the category of D-branes and whose generic fibre is the deformation quantisation mentioned above. I will then show that the latter is quasi-isomorphic to the (base-change of) virtual de Rham category and explain the formality of the de Rham category, thus making the formal deformation “generically formal”.
Time permitting, I will introduce the Kaledin class obstructing formality of a dg category and explain how the proper Calabi-Yau structure on the formal deformation leads to “generic formality => formality”, thus showing the formality of the dg category of D-branes.
This story can be thought of as a B-side analogue of Ivan Smith’s conjecture on the formality of the Solomon-Verbitsky Fukaya category under Kapustin's duality between type A and type B D-branes on X.
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