There is 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.
However, for integral k there is 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 the Eilenberg-MacLane 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).