For a fixed prequantum line bundle $L$ over a hyperK\"ahler manifold $X$, we find a natural $\operatorname{ Sp}(1)$-action on $\Omega^*(X, L)$ intertwining a twistor family of $\operatorname{ Spin}^{ \operatorname{ c}}$-Dirac Laplacians on the spaces of $L$-valued $(0, *)$-forms on $X$, noting that $L$ is holomorphic for only one complex structure in the twistor family. This establishes a geometric quantization of $X$ via Gukov-Witten brane quantization and leads to a proposal of a mathematical definition of $\operatorname{ Hom}(\overline{ \mathcal{ B}}_{ \operatorname{ cc}}, \mathcal{ B}_{ \operatorname{ cc}})$ for the canonical coisotropic A-brane $\mathcal{ B}_{ \operatorname{ cc}}$ on $X$ and its conjugate brane $\overline{ \mathcal{ B}}_{ \operatorname{ cc}}$.
Categories: math.SG math-ph math.DG math.MP