Sp(1)-symmetric hyperkähler quantisation

We provide a new general scheme for the geometric quantisation of Sp(1)- symmetric hyperkahler manifolds, considering Hilbert spaces of holomorphic sections with respect to the complex structures in the hyperkahler 2-sphere. Under properness of an associated moment map, or other finiteness assumptions, we construct unitary (super) representations of groups acting by Riemannian isometries preserving the 2-sphere, and we study their decomposition in irreducible components. We apply this scheme to hyperkahler vector spaces, the Taub–NUT metric on R4, moduli spaces of framed SU(r)-instantons on R4, and in part to the Atiyah–Hitchin manifold of magnetic monopoles in R3.