Hidden Sp(1)-Symmetry and Brane Quantization on HyperKähler manifolds

For a fixed prequantum line bundle $L$ over a hyperKähler manifold $X$, we find a natural $\mathrm{Sp}(1)$-action on ${\mathrm{\Omega }}^{\ast }(X,L)$ intertwining a twistor family of ${\mathrm{Spin}}^{\mathrm{c}}$-Dirac Laplacians on the spaces of $L$-valued $(0,\ast )$-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 $\mathrm{Hom}({\bar{\mathcal{B}}}_{\mathrm{cc}},{\mathcal{B}}_{\mathrm{cc}})$ for the canonical coisotropic A-brane ${\mathcal{B}}_{\mathrm{cc}}$ on $X$ and its conjugate brane ${\bar{\mathcal{B}}}_{\mathrm{cc}}$.