A 2-group construction from an extension of the 3-loop group $\Omega^3G$

TL;DR

This paper constructs a 2-group from a 3-loop group extension of a Lie group, extending previous one-dimensional work to three dimensions with a non-central extension, advancing the understanding of higher loop groups.

Contribution

It introduces a new 2-group construction based on a non-central extension of the 3-loop group, generalizing earlier one-dimensional models to three dimensions.

Findings

Defined the 3-loop group as a subgroup of smooth maps from a 3-ball to G

Constructed a 2-group using an automorphic action on the Mickelsson-Faddeev extension

Extended the framework of higher loop groups to three dimensions with non-central extensions

Abstract

We define a 3-loop group $\Omega^3G$ as a subgroup of smooth maps from a 3-ball to a Lie group $G$, and then construct a 2-group based on an automorphic action on the Mickelsson-Faddeev extension of $\Omega^3G$. In this we follow the strategy of Murray et al., who earlier described a similar construction in one dimension. The three-dimensional situation presented here is further complicated by the fact that the 3-loop group extension is not central.