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 $\mathrm{String}(n)$. 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 $\mathrm{String}(n)$.

37 pages, sign mistakes corrected by David Roberts

References in corpus (1)

• Bundle gerbes for Chern-Simons and Wess-Zumino-Witten theories

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• Smooth 2-Group Extensions and Symmetries of Bundle Gerbes
• Division Algebras, Supersymmetry and Higher Gauge Theory
• Principal $\infty$-Bundles and Smooth String Group Models
• Categorified Symplectic Geometry and the String Lie 2-Algebra
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• Integrating central extensions of Lie algebras via Lie 2-groups
• Survey of mathematical foundations of QFT and perturbative string theory