Categorical quantization on K\"ahler manifolds

Generalizing deformation quantizations with separation of variables of a K\"ahler manifold M, we adopt Fedosov's gluing argument to construct a category DQ, enriched over sheaves of C[[]]-modules on M, as a quantization of the category of Hermitian holomorphic vector bundles over M with morphisms being smooth sections of hom-bundles. We then define quantizable morphisms among objects in DQ, generalizing Chan-Leung-Li's notion [4] of quantizable functions. Upon evaluation of quantizable morphisms at = √-1k, we obtain an enriched category DQqu, k. We show that, when M is prequantizable, DQqu, k is equivalent to the category GQ of holomorphic vector bundles over M with morphisms being holomorphic differential operators, via a functor obtained from Bargmann-Fock actions.