Sp(1)-symmetric hyper-Kähler quantisation

We provide a new general scheme for the geometric quantisation of Sp(1)-symmetric hyper-Kähler manifolds, considering Hilbert spaces of holomorphic sections over the associated twistor space. Under properness of an associated moment map, or other finiteness assumptions, we construct unitary quantum (super) representations of central extensions of certain subgroups of Riemannian isometries preserving the hyper-Kähler 2-sphere, and we study their decomposition in irreducible components. This quantisation scheme applies to hyper-Kähler vector spaces, the Taub-NUT metric on R×S³, moduli spaces of framed SU(r)-instantons on R⁴, and moduli spaces of monopoles on R³.