Quantization of Lagrangians in algebraic geometry

Let $X$ be a smooth algebraic variety with an algebraic symplectic form and $Y$ a smooth Lagrangian subvariety. For a fixed deformation quantization $A$ of $X$ and a line bundle $L$ on $Y$ we could ask whether $L$ deforms to a module over $A$. We answer this question under a mild technical assumption.
Joint work with D. Kaledin, V. Ginzburg and J. Pecharich.