Lie groups of bundle automorphisms and their extensions

TL;DR

This paper investigates abelian extensions of the Lie algebra of automorphisms of principal bundles, classifies their cocycles, and discusses conditions for their integration into Lie group extensions, with special focus on trivial bundles.

Contribution

It classifies fundamental cocycles for gauge algebra extensions and describes conditions for extending these to automorphism groups and integrating into Lie group extensions.

Findings

All central extensions of gauge algebra from three cocycle types.

Extensions extend to automorphism groups under certain conditions.

Descriptions of the second cohomology group classifying twists.

Abstract

We describe natural abelian extensions of the Lie algebra $\aut(P)$ of infinitesimal automorphisms of a principal bundle over a compact manifold $M$ and discuss their integrability to corresponding Lie group extensions. Already the case of a trivial bundle $P = M \times K$ is quite interesting. In this case, we show that essentially all central extensions of the gauge algebra $C^\infty(M,\fk)$ can be obtained from three fundamental types of cocycles with values in one of the spaces $\fz := C^\infty(M,V)$, $\Omega^1(M,V)$ and $\Omega^1(M,V)/\dd C^\infty(M,V)$. These cocycles extend to $\aut(P)$, and, under the assumption that $TM$ is trivial, we also describe the space $H^2({\cal V}(M),\fz)$ classifying the twists of these extensions. We then show that all fundamental types have natural generalizations to non-trivial bundles and explain under which conditions they extend to $\aut(P)$ and integrate to global Lie group extensions.