Connections and curvings on lifting bundle gerbes

TL;DR

This paper constructs a connection and curving on a lifting bundle gerbe, linking geometric structures on principal bundles to their obstruction classes via Deligne cohomology.

Contribution

It provides a explicit construction of connections and curvings on lifting bundle gerbes using bundle structures, clarifying their cohomological obstructions.

Findings

Connection and curving constructed explicitly

Obstruction class identified with Deligne cohomology

Framework applicable to lifting problems in principal bundles

Abstract

We construct a connection and a curving on a bundle gerbe associated with lifting a structure group of a principal bundle to a central extension. The construction is based on certain structures on the bundle, i.e. connections and splittings. The Deligne cohomology class of the lifting bundle gerbe with the connection and with the curving coincides with the obstruction class of the lifting problem with these structures.