The following paper is now available on the arXiv: John Baez, Alissa Crans, Urs Schreiber and Danny Stevenson
From Loop Groups to 2-Groups http://arxiv.org/abs/math.QA/0504123
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. Indeed, a "strict" Lie 2-group is just a categorical group or crossed module in the category of smooth manifolds. 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 and proportional to k. 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 strict 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 the Eilenberg-Mac Lane space 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). 12-Apr-2005 10:59:38 -0300,6643;000000000000-00000005