Quantization of Kähler Manifolds via Brane Quantization

Abstract

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 $\mathrm{Hom}({\mathcal{B}}_{\mathrm{cc}},{\mathcal{B}}_{\mathrm{cc}})$ and $\mathrm{Hom}(\mathcal{B},{\mathcal{B}}_{\mathrm{cc}})$ should recover holomorphic deformation quantization of $X$ and geometric quantization of $M$ respectively, where ${\mathcal{B}}_{\mathrm{cc}}$ is a canonical coisotropic A-brane on $X$ and $\mathcal{B}$ is a Lagrangian A-brane supported on $M$. Assuming $M$ is spin and Kähler with a prequantum line bundle $L$, Chan-Leung-Li [10] constructed a subsheaf ${\mathcal{O}}_{\mathrm{qu}}^{(k)}$ of smooth functions on $M$ with a non-formal star product and a left ${\mathcal{O}}_{\mathrm{qu}}^{(k)}$-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 $\mathrm{Hom}({\mathcal{B}}_{\mathrm{cc}},{\mathcal{B}}_{\mathrm{cc}})$ on $\mathrm{Hom}(\mathcal{B},{\mathcal{B}}_{\mathrm{cc}})$. By Fedosov's gluing arguments, we also construct a ${\mathcal{O}}_{\mathrm{qu}}^{(k)}$-${\overline{\mathcal{O}}}_{\mathrm{qu}}^{(k)}$-bimodule structure on the sheaf of smooth sections of $L^{\otimes 2k}$ to realize the actions of $\mathrm{Hom}({\mathcal{B}}_{\mathrm{cc}},{\mathcal{B}}_{\mathrm{cc}})$ and $\mathrm{Hom}({\overline{\mathcal{B}}}_{\mathrm{cc}},{\overline{\mathcal{B}}}_{\mathrm{cc}})$ on $\mathrm{Hom}({\overline{\mathcal{B}}}_{\mathrm{cc}},{\mathcal{B}}_{\mathrm{cc}})$, which is related to the analytic geometric Langlands program.

arXiv comment29 pages