2017/09/18 by Severin Bunk, Bunk, Severin
Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Bundle #Categorical variable #Category Theory (math.CT) #Computer science #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Homotopy and Cohomology in Algebraic Topology #Materials science #Mathematical Physics (math-ph) #hep-th #math-ph #math.CT #math.DG #math.MP
paper · pdf · doi:10.48550/arxiv.1709.06174
PhD thesis, 139 pages
arxiv created 2017/09/18 · openalex publication_date 2017/09/18 · arxiv updated 2017/09/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 introduce a direct sum on the morphism categories in the 2-category 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, 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 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 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 rig-module structure over the category of finite-dimensional Hilbert spaces. 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 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 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.