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 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 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 P(k)G whose Lie 2-algebra is equivalent to g k. The objects of P(k)G 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 kth power of the canonical gerbe over G. Its nerve gives a topological group |P(k)G| that is an extension of G by K( Z, 2). When k = +/- 1, |P(k)G| can also be obtained by killing the third homotopy group of G. Thus, when G = Spin( n), | P(k)G| is none other than String( n).