A 2-group construction from an extension of the 3-loop group \(\varOmega ^3G\)
Let \(G\) be a Lie group. The paper under review studies the \(3\)-loop group \(\Omega^3 G\) of based maps from the \(3\)-sphere to \(G\). It turns out that the related Mickelsson-Faddeev cocycle depends crucially on the defining Lie algebra valued \(1\)-forms over the domain and, in particular, the corresponding Lie group extension is not central. The Mickelsson-Faddeev extension of \(\Omega^3 G\) admits an action of a certain extension of the group of all smooth maps from a \(3\)-ball to \(G\) which are flattened on the boundary. This is used to construct a strict \(2\)-group associated to the Mickelsson-Faddeev extension of \(\Omega^3 G\).
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