Categorical structures on bundle gerbes and higher geometric prequantisation

Abstract

We present a construction of a 2-Hilbert space of sections of a bundle gerbe, a suitable candidate for a prequantum 2-Hilbert space in higher geometric quantisation. We start by briefly recalling the construction of the 2-category of bundle gerbes, with minor alterations that allow us to endow morphisms with additive structures. The morphisms in the resulting 2-categories are investigated in detail. We introduce a direct sum on morphism categories of bundle gerbes and show that these categories are cartesian monoidal and abelian. Endomorphisms of the trivial bundle gerbe, or higher functions, carry the structure of a rig-category, a categorised ring, and we show that generic morphism categories of bundle gerbes form module categories over this rig-category. We continue by presenting a categorification of the hermitean bundle metric on a hermitean line bundle. This is achieved by introducing a functorial dual that extends the dual of vector bundles to morphisms of bundle gerbes, and constructing a two-variable adjunction for the aforementioned rig-module category structure on morphism categories. Its right internal hom is the module action, composed by taking the dual of the acting higher functions, while the left internal hom is interpreted as a bundle gerbe metric. Sections of bundle gerbes are defined as morphisms from the trivial bundle gerbe to the bundle gerbe under consideration. We show that the resulting categories of sections carry a rig-module structure over the category of nite-dimensional Hilbert spaces with its canonical direct sum and tensor product. A suitable definition of 2-Hilbert spaces is given, modifying previous definitions by the use of two-variable adjunctions. We prove that the category of sections of a bundle gerbe, with its additive and module structures, fits into this framework, thus obtaining a 2-Hilbert space of sections. In particular, this can be constructed for prequantum bundle gerbes in problems of higher geometric quantisation. We define a dimensional reduction functor and show that the categorical structures introduced on the 2-category of bundle gerbes naturally reduce to their counterparts on hermitean line bundles with connections. In several places in this thesis, we provide examples, making 2-Hilbert spaces of sections and dimensional reduction very explicit.