The geometry of 2-vector bundles

A vector bundle is a space where each point is replace by an entire vector space in a nice and continuous manner. Vector bundles and their geometry are important in connection with quantum field theories, however, it appears that finer structures are also of importance. In this connection, Baas (NTNU), Richter (Hamburg), Rognes (UiO) and I have a construction for "2-vector bundles", where vector spaces are exchanged for 2-vector spaces. Just as a vector is a tuple of numbers, a 2-vector is a tuple of vector spaces, and matrices of numbers are exchanged for matrices of vector spaces. The surprise is that this gives rise to a theory connected to quantum field theories and to "the prime factorisation of the sphere spectrum".

A shortcoming in connection with the application to quantum field theory is that the geometry of 2-vector bundles is not developed. This is not a simple question, for example, one does not have a good idea of what curvature should mean, one has no index theory and there is a complete lack of a theory for determinants for 2-vector spaces; and problems in this regard are interesting enough in themselves. Some of the questions will require a bit of differential geometry.