## Document Type

Article - post-print

## Publication Date

2005

## 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 g_k each having 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 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 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 kth power of the canonical gerbe over G. Its nerve gives a topological group that is an extension of G by K(Z,2). When k = +-1, this topological group can also be obtained by killing the third homotopy group of G. Thus, when G = Spin(n), it is none other than String(n).

## Original Publication Citation

Baez, J.; Crans, A.; Schreiber, U.; and Stevenson, D. “From Loop Groups to 2-Groups.” Homology, Homotopy, and Applications, Vol. 9 (2007), No. 2: 101 – 135.

## Digital Commons @ LMU & LLS Citation

Baez, John C.; Stevenson, Danny; Crans, Alissa S.; and Schreiber, Urs, "From Loop Groups to 2-Groups" (2005). *Mathematics, Statistics and Data Science Faculty Works*. 65.

https://digitalcommons.lmu.edu/math_fac/65

## Comments

This is the post-print version of the article.