Fluxes, bundle gerbes and 2-Hilbert spaces

We elaborate on the construction of a prequantum 2-Hilbert space from a bundle gerbe over a 2-plectic manifold, providing the first steps in a program of higher geometric quantisation of closed strings in flux compactifications and of M5-branes in C-fields. We review in detail the construction of the 2-category of bundle gerbes, and introduce the higher geometrical structures necessary to turn their categories of sections into 2-Hilbert spaces. We work out several explicit examples of 2-Hilbert spaces in the context of closed strings and M5-branes on flat space. We also work out the prequantum 2-Hilbert space associated to an M-theory lift of closed strings described by an asymmetric cyclic orbifold of the SU(2) WZW model, providing an example of sections of a torsion gerbe on a curved background. We describe the dimensional reduction of M-theory to string theory in these settings as a map from 2-isomorphism classes of sections of bundle gerbes to sections of corresponding line bundles, which is compatible with the respective monoidal structures and module actions.