Homological Mirror Symmetry, Deformation Quantization and Non-Commutative Geometry

摘要

Mathematical foundation of mirror symmetry belongs both to geometry and algebra. Present paper deals with some aspects of the latter. Our methods and conjectures have an algebraic (categorical) nature. On the other hand, based o n the ideas of (Kol), (KoSol), (KoSol2), we stress the role of non-commutative geometry of mirror symmetry, making an attempt to connect it with deformation quantization. The latter can be thought of generalization of the theory of differential operators (or D-modules) to arbitrary symplectic manifolds.