Seeking for a mathematical definition of morphism spaces between coisotropic A-branes has been a long-standing problem for understanding mirror symmetry. In 2009, a paper of Gukov-Witten showed that this problem is also closely related to deformation quantization and geometric quantization.

In this talk, I shall briefly survey proposals to this problem in the past literature, including the recent works of Gaiotto-Witten, Bischoff-Gualtieri and Qin. Then I shall explain my recent joint work with NaiChung Conan Leung that for a fixed prequantum line bundle L over a hyperKahler manifold X, there is a natural but hidden Sp(1)-symmetry intertwining a twistor family of Spin^c-Dirac operators on the spaces of L-valued (0, *)-forms on X. It leads to a proposed definition of the morphism space of a brane-conjugate brane system for a space-filling coisotropic A-brane on a symplectic manifold, and it establishes geometric quantization via brane quantization on a hyperKahler manifold. Finally, I shall also compare and contrast our result and a related work of Andersen-Malusà-Rembado on Sp(1)-symmetric hyperKahler quantization.