Deformation Quantization of Vector Bundles on Lagrangian Subvarieties

We consider a smooth subvariety Y in a smooth algebraic variety X with an algebraic symplectic form. Assume that there exists a deformation quantization Oh of the structure sheaf OX which agrees with the symplectic form. When Y is Lagrangian, for a vector bundle E on Y, we establish necessary and sufficient conditions for the existence of the deformation quantization of E, i.e., an Oh-module Eh such that Eh/hEh=E.If the necessary conditions hold, we describe the set of equivalence classes of such quantizations. In the more general situation when Y is coisotropic, we reformulate the deformation problem into the lifting problem of torsors. We expect a deformation quantization of a line bundle on a coisotropic subvariety is equivalent to a solution of curved Maurer-Cartan equation of a curved L-infinity-algebra.