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.
Categories: math.SG math-ph math.DG math.MP