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 ${\rm String}(n)$. 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 $\mathfrak{g}_k$ as its Lie 2-algebra, except when $k = 0$. Here, however, we construct for integral $k$ an infinite-dimensional Lie 2-group ${\cal P}_kG$ whose Lie 2-algebra is equivalent to $\mathfrak{g}_k$. The objects of ${\cal P}_kG$ 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 $|{\cal P}_kG|$ that is an extension of $G$ by $K(\mathbb{Z},2)$. When $k = \pm 1$, $|{\cal P}_kG|$ can also be obtained by killing the third homotopy group of $G$. Thus, when $G = {\rm Spin}(n)$, $|{\cal P}_kG|$ is none other than ${\rm String}(n)$.

Keywords

Gerbe, Kac-Moody extension, Lie 2-algebra, loop group, string group, 2-group

2010 Mathematics Subject Classification

22E67

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2023 Nov. 22. A number of sign errors have been found in this paper. For an explanation of how to correct those errors, see the authors’ $\href{https://dx.doi.org/10.4310/HHA.2023.v25.n2.e18}{\textrm{Erratum}}$ in volume 25, no. 2 of this journal.

Published 1 January 2007