Generalized Riemann–Hilbert correspondence, wall-crossing and resurgence

Abstract

Together with Yan Soibelman, we propose a new version of homological mirror symmetry, a hypothetical equivalence between two categories associated with a complex algebraic symplectic variety. In the case of cotangent bundles, it gives the classical Riemann–Hilbert correspondence, which is the equivalence between the category of holonomic $D$-modules with regular singularities on a complex algebraic variety and the category of perverse constructible sheaves. In general, the de Rham side is replaced by the algebraic deformation quantization, whereas the Betti side is replaced by the Fukaya category. New theory is applicable not only to $D$-modules, but also to q-difference and elliptic difference equations. The key role is played by the theory of wall-crossing, hypothetically leading to resurgence properties of WKB expansions.