From Loop Groups to 2-Groups

TL;DR

This paper explores the deep connections between Lie 2-algebras, Kac-Moody extensions, and the String group, constructing a new infinite-dimensional Lie 2-group that models these relationships for compact simple Lie groups.

Contribution

It constructs an explicit infinite-dimensional Lie 2-group for integral levels, linking Lie 2-algebras, Kac-Moody extensions, and the String group in a novel way.

Findings

Constructed a Lie 2-group with objects as paths in G

Linked the 2-group to the kth power of the canonical gerbe

Identified the 2-group with String(n) for G=Spin(n)

Abstract

We describe an interesting relation between Lie 2-algebras, the Kac-Moody central extensions of loop groups, and the group $\mathrm{String}(n)$. 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 $\mathrm{Lie}(G)$ 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 $G = \mathrm{Spin}(n)$, it is none other than $\mathrm{String}(n)$.