Categorical Structures on Bundle Gerbes and Higher Geometric Prequantisation
Abstract
We present a construction of a 2Hilbert space of sections of a bundle gerbe, a suitable candidate for a prequantum 2Hilbert space in higher geometric quantisation. We introduce a direct sum on the morphism categories in the 2category 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 rigcategory, which acts on generic morphism categories of bundle gerbes. We continue by presenting a categorification of the hermitean 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 twovariable adjunction for the aforementioned rigmodule category structure on morphism categories. Its right internal hom is the module action, composed by taking the dual of 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 a given bundle gerbe. The resulting categories of sections carry a rigmodule structure over the category of finitedimensional Hilbert spaces. A suitable definition of 2Hilbert spaces is given, modifying previous definitions by the use of twovariable adjunctions. We prove that the category of sections of a bundle gerbe fits into this framework, thus obtaining a 2Hilbert 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 bundle gerbes naturally reduce to their counterparts on hermitean line bundles with connections. In several places in this thesis, we provide examples, making 2Hilbert spaces of sections and dimensional reduction very explicit.
 Publication:

arXiv eprints
 Pub Date:
 September 2017
 arXiv:
 arXiv:1709.06174
 Bibcode:
 2017arXiv170906174B
 Keywords:

 Mathematical Physics;
 High Energy Physics  Theory;
 Mathematics  Category Theory;
 Mathematics  Differential Geometry
 EPrint:
 PhD thesis, 139 pages