From loop groups to 2-groups

Abstract

We describe an interesting relation between Lie 2-algebras, the Kac-Moody central extensions of loop groups, and the group String($n$). 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 $\mathfrak{g}$ 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 $\mathfrak{g}_\mathcal{k}$ as its Lie 2-algebra, except when $k = 0$. Here, however, we construct for integral $k$ an infinite-dimensional Lie 2-group $\mathcal{P}_k G$ whose Lie 2-algebra is equivalent to $\mathfrak{g}_\mathcal{k}$. The objects of $\mathfrak{g}_\mathcal{k}$ are based paths in $G$-, while the automorphisms of any object form the level-$k$ Kac-Moody central extension of the loop group $\Omega{G}$. This 2- group is closely related to the $k$th power of the canonical gerbe over $G$. Its nerve gives a topological group $|\mathcal{P}_k G|$ that is an extension of $G$ by $K(\mathbb{Z}, 2)$. When $k = \pm 1, |\mathcal{P}_k G|$ can also be obtained by killing the third homotopy group of $G$. Thus, when $G = \rm{Spin}(n), |PkG|$ is none other than String($n$).