2-vector bundles

TL;DR

This paper develops a comprehensive theory of (super) 2-vector bundles over smooth manifolds, unifying bundle gerbes and algebra bundles within a bicategorical framework, with applications in geometry and topology.

Contribution

It introduces a bicategory-based model for 2-vector bundles, classifies them via Cech cohomology, and unifies bundle gerbes and algebra bundles in a single framework.

Findings

Contains bundle gerbes as sub-bicategories

Provides classification via Cech cohomology with crossed modules

Includes examples from representation theory, twisted K-theory, and spin geometry

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.