  In their physical proposal for quantization [20], Gukov-Witten suggested
that, given a symplectic manifold $M$ with a complexification $X$, the A-model
morphism spaces $\operatorname{Hom}(\mathcal{B}_{\operatorname{cc}},
\mathcal{B}_{\operatorname{cc}})$ and $\operatorname{Hom}(\mathcal{B},
\mathcal{B}_{\operatorname{cc}})$ should recover holomorphic deformation
quantization of $X$ and geometric quantization of $M$ respectively, where
$\mathcal{B}_{\operatorname{cc}}$ is a canonical coisotropic A-brane on $X$ and
$\mathcal{B}$ is a Lagrangian A-brane supported on $M$.
  Assuming $M$ is spin and K\"ahler with a prequantum line bundle $L$,
Chan-Leung-Li [10] constructed a subsheaf
$\mathcal{O}_{\operatorname{qu}}^{(k)}$ of smooth functions on $M$ with a
non-formal star product and a left
$\mathcal{O}_{\operatorname{qu}}^{(k)}$-module structure on the sheaf of
holomorphic sections of $L^{\otimes k} \otimes \sqrt{K}$.
  In this paper, we give a careful treatment of the relation between
(holomorphic) deformation quantizations of $M$ and $X$. As a result,
Chan-Leung-Li's work [10] provides a mathematical realization of the action of
$\operatorname{Hom}(\mathcal{B}_{\operatorname{cc}},
\mathcal{B}_{\operatorname{cc}})$ on $\operatorname{Hom}(\mathcal{B},
\mathcal{B}_{\operatorname{cc}})$. By Fedosov's gluing arguments, we also
construct a
$\mathcal{O}_{\operatorname{qu}}^{(k)}$-$\overline{\mathcal{O}}_{\operatorname{qu}}^{(k)}$-bimodule
structure on the sheaf of smooth sections of $L^{\otimes 2k}$ to realize the
actions of $\operatorname{Hom}(\mathcal{B}_{\operatorname{cc}},
\mathcal{B}_{\operatorname{cc}})$ and
$\operatorname{Hom}(\overline{\mathcal{B}}_{\operatorname{cc}},
\overline{\mathcal{B}}_{\operatorname{cc}})$ on
$\operatorname{Hom}(\overline{\mathcal{B}}_{\operatorname{cc}},
\mathcal{B}_{\operatorname{cc}})$, which is related to the analytic geometric
Langlands program.
