Abstract
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 programme 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 with an M-theory lift of closed strings described by an asymmetric cyclic orbifold of the mathsf {S}mathsf {U}(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.
Highlights
The higher analogue of Hilbert spaces of physical states in quantum theory should be a suitable notion of 2-Hilbert spaces, which we describe in some generality in Sect. 7 in the form that we need in this paper
We highlight some of the open questions which are not yet addressed by our formalism. One of these concerns is an already notoriously difficult problem in ordinary geometric quantisation: the second step in that procedure involves the extra choice of a polarisation, which corresponds to locally representing the symplectic manifold M as a cotangent bundle T ∗U
This article is a companion to the longer paper [13] which, among other things, developed new categorical structures on morphisms of bundle gerbes, initiated steps towards higher geometry by introducing 2-bundle metrics, and showed how to obtain a 2-Hilbert space from the category of sections of a bundle gerbe
Summary
Geometric quantisation starts from a symplectic manifold (M, ω), i.e. a manifold M of dimension 2n with a non-degenerate closed 2-form ω; this is part of the data of most classical physical systems Such a manifold is called prequantisable if there exists a hermitean line bundle with connection (L , ∇ L ) on M whose magnetic flux is F L = 2π ω; by Dirac charge quantisation this is equivalent to the statement that the de Rham class of ω lies in the image of the map H2(M, Z) → Hd2R(M). We obtain a geometric structure which realises a closed 3-form on M with integer periods, provided that M is 2-connected This is analogous to how the tautological line bundle in Sect.
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