Categorical quantization on Kähler manifolds

Generalizing deformation quantizations with separation of variables of a Kähler manifold $M$, we adopt Fedosov’s gluing argument to construct a category $\mathsf{D}\mathsf{Q}$, enriched over sheaves of $\mathbb{C}[[\mathrm{\hslash }]]$-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 $\mathsf{D}\mathsf{Q}$, generalizing Chan-Leung-Li’s notion [4] of quantizable functions. Upon evaluation of quantizable morphisms at $\mathrm{\hslash }=\frac{\sqrt{-1}}{k}$, we obtain an enriched category ${\mathsf{D}\mathsf{Q}}_{\mathrm{qu},k}$. We show that, when $M$ is prequantizable, ${\mathsf{D}\mathsf{Q}}_{\mathrm{qu},k}$ is equivalent to the category $\mathsf{G}\mathsf{Q}$ of holomorphic vector bundles over $M$ with morphisms being holomorphic differential operators, via a functor obtained from Bargmann-Fock actions.