  We explain that when quantising phase spaces with varying symplectic
structures, the bundle of quantum Hilbert spaces over the parameter space has a
natural unitary connection. We then focus on symplectic vector spaces and their
fermionic counterparts. After reviewing how the quantum Hilbert space depends
on physical parameters such as the Hamiltonian and unphysical parameters such
as choices of polarisations, we study the connection, curvature and phases of
the Hilbert space bundle when the phase space structure itself varies. We apply
the results to the two-sphere family of symplectic structures on a
hyper-K\"ahler vector space and to their fermionic analogue, and conclude with
possible generalisations.
