A Riemann-Hilbert correspondence for holomorphic Lagrangian intersections
Given a pair of holomorphic Lagrangian submanifolds of a holomorphic symplectic manifold (plus some additional data), we may consider the following two objects. On the one hand, Bussi (based on work with Brav, Dupont, Joyce, and Szendroi) assign to such data a perverse sheaf supported on the Lagrangian intersection together with a canonical monodromy automorphism. On the other hand, the theory of deformation quantization (as developed in this context by d’Agnolo, Kashiwara, Polesello, Schapira, and others) assigns to such data a dg-category and a pair of objects whose Hom complex is also known to form a perverse sheaf, albeit one defined over a field of Laurent series C((h))). In my work in progress with Pavel Safronov, we explain how to enhance the deformation quantization category so that the Hom complexes are naturally identified with the perverse sheaf of Bussi (with coefficients in the complex numbers, together with the monodromy operator), shedding light on Joyce's conjectural description of the holomoprhic Fukaya category. As an application, we will present a comparison between the sheaf-theoretic Floer homology of Abouzaid and Manolescu and the skein module of a 3-manfold.