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 gk each having 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 gk as its Lie 2-algebra, except when k = 0. Here, however, we construct for integral k an infinite-dimensional Lie 2-group PkG whose Lie 2-algebra is equivalent to gk. The objects of PkG are based paths in G, while the automorphisms of any object form the level-k Kac– Moody central extension of the loop group G. This 2-group is closely related to the kth power of the canonical gerbe over G. Its nerve gives a topological group |PkG| that is an extension of G by K(Z,2). When k = ±1, |PkG| can also be obtained by killing the third homotopy group of G. Thus, when G = Spin(n), |PkG| is none other than String(n).