From Loop Groups to 2-Groups

We describe an interesting relation between Lie 2-algebras, the Kac-Moody central extensions of loop groups, and the group $\mathrm{S}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{n}\mathrm{g}(n)$. A Lie 2-algebra is a categorified version of a Lie algebra where the Jacobi identity holds up to a natural isomorphism called the “Jacobiator”. Similarly, a Lie 2-group is a categorified version of a Lie group. If $G$ is a simply-connected compact simple Lie group, there is a 1-parameter family of Lie 2-algebras ${\mathfrak{g}}_k$ each having $\mathrm{L}\mathrm{i}\mathrm{e}(G)$ as its Lie algebra of objects, but with a Jacobiator built from the canonical 3-form on $G$. There appears to be no Lie 2-group having ${\mathfrak{g}}_k$ as its Lie 2-algebra, except when $k = 0$. Here, however, we construct for integral k an infinite-dimensional Lie 2-group whose Lie 2-algebra is equivalent to ${\mathfrak{g}}_k$. The objects of this 2-group are based paths in $G$, while the automorphisms of any object form the level-$k$ Kac-Moody central extension of the loop group of $G$. This 2-group is closely related to the $k$th power of the canonical gerbe over $G$. Its nerve gives a topological group that is an extension of $G$ by $K(\mathbb{Z},2)$. When $k=\pm 1$, this topological group can also be obtained by killing the third homotopy group of $G$. Thus, when $G = $\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}$(n)$,itisnoneotherthan$$\mathrm{S}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{n}\mathrm{g}$(n)$.