Connections on central extensions, lifting gerbes, and finite-dimensional obstruction vanishing

TL;DR

This paper investigates the classification of connections on central extensions of principal bundles, introduces a new connective structure on lifting gerbes for reductive groups, and proves a vanishing result for certain obstruction classes.

Contribution

It provides a classification of all connections on central extensions and introduces a novel connective structure on lifting gerbes for reductive groups.

Findings

Admissible connections correspond to parallel trivializations of the lifting gerbe.

A new connective structure on lifting gerbes is introduced for reductive structure groups.

Proves a vanishing result for Neeb's obstruction classes in finite-dimensional Lie groups.

Abstract

Given a central extension of Lie groups, we study the classification problem of lifting the structure group together with a given connection. For reductive structure groups we introduce a new connective structure on the lifting gerbe associated to this problem. Our main result classifies all connections on the central extension of a given principal bundle. In particular, we find that admissible connections are in one-to-one correspondence with parallel trivializations of the lifting gerbe. Moreover, we prove a vanishing result for Neeb's obstruction classes for finite-dimensional Lie groups.