2-vector bundles
arXiv:2106.12198
Abstract
We develop a ready-to-use comprehensive theory for (super) 2-vector bundles over smooth manifolds. It is based on the bicategory of (super) algebras, bimodules, and intertwiners as a model for 2-vector spaces. We discuss symmetric monoidal structures and the corresponding notions of dualizability, and we derive a classification in terms of Cech cohomology with values in a crossed module. One important feature of our 2-vector bundles is that they contain bundle gerbes as well as ordinary algebra bundles as full sub-bicategories, and hence provide a unifying framework for these so far distinct objects. We provide several examples of isomorphisms between bundle gerbes and algebra bundles, coming from representation theory, twisted K-theory, and spin geometry.
52 pages. In v2 we have moved a detailed discussion of bimodule bundles and their composition into a separate paper (arXiv:2204.03900), addressing there a problem that was present in v1 concerning the composition of bimodule bundles. Apart from that only smaller corrections