Geometric quantisation on a hyper-Kähler vector space

In the context of geometric quantisation, one starts with the data of a symplectic manifold together with a pre-quantum line bundle, and obtains a quantum Hilbert space by means of the auxiliary structure of a polarisation, i.e. typically a Lagrangian foliation or a Kähler structure. One common and widely studied problem is that of quantising Hamiltonian flows which do not preserve it. In the simple case of Kähler quantisation of a symplectic vector space, one may consider the space of all Kähler structures, and organise the corresponding Hilbert spaces to form an infinite-rank vector bundle on it, so the quantisation of symplectic transformations may be understood in terms of their action on this bundle.
In this talk I shall consider a variation of this construction for the case of a hyper-Kähler vector space V, in which case the symplectic form and the pre-quantum line bundle are allowed to change together with the complex structure. In a joint work with Andersen and Rembado, we construct a bundle of Hilbert spaces on the sphere of Kähler structures, on which a flat connection is naturally defined. The group of isometries of V which preserve this sphere globally also acts on the bundle, and the construction determines a quantisation of this group. Time permitting, I shall discuss our main application of this construction—the quantisation of the circle action on the moduli space of reductive Higgs bundles over an elliptic curve.