Hidden \({\text {Sp}}(1)\)-Symmetry and Brane Quantization on HyperKähler Manifolds

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