Orientations and Connective Structures on 2-vector Bundles

TL;DR

This paper investigates obstructions to determinant gerbe constructions on 2-vector bundles, introduces oriented 2-vector bundles to remove these obstructions, and generalizes connective structures to this setting.

Contribution

It defines oriented 2-vector bundles to eliminate obstructions and extends connective structures to 2-vector bundles, enabling determinant gerbe definitions.

Findings

Identification of a specific obstruction in determinant gerbe construction

Introduction of oriented 2-vector bundles to remove the obstruction

Generalization of connective structures to 2-vector bundles

Abstract

In work by Ausoni, Dundas and Rognes a half magnetic monopole is discovered and describes an obstruction to creating a determinant K(ku) \to ku*. In fact it is an obstruction to creating a determinant gerbe map from K(ku) to K(Z,3). We describe this obstruction precisely using monoidal categories and define the notion of oriented 2-vector bundles, which removes this obstruction so that we can define a determinant gerbe. We also generalize Brylinskis notion of a connective structure to 2-vector bundles, in a way compatible with the determinant gerbe.