Central extensions of groups of sections

TL;DR

This paper investigates when certain Lie algebra extensions, derived from invariant bilinear forms on the Lie algebra of a structure group, can be integrated into Lie group extensions, especially for non-connected and infinite-dimensional groups.

Contribution

It provides a complete characterization of integrable extensions for groups with finitely many components and offers general conditions for integrability in the context of Lie group bundles.

Findings

Complete characterization for finitely many components

Sufficient conditions for integrability based on group K data

Extension analysis applicable to infinite-dimensional groups

Abstract

If q : P -> M is a principal K-bundle over the compact manifold M, then any invariant symmetric V-valued bilinear form on the Lie algebra k of K defines a Lie algebra extension of the gauge algebra by a space of bundle-valued 1-forms modulo exact forms. In the present paper we analyze the integrability of this extension to a Lie group extension for non-connected, possibly infinite-dimensional Lie groups K. If K has finitely many connected components we give a complete characterization of the integrable extensions. Our results on gauge groups are obtained by specialization of more general results on extensions of Lie groups of smooth sections of Lie group bundles. In this more general context we provide sufficient conditions for integrability in terms of data related only to the group K.