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 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 ΩG.This 2group 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).

Locations

• Homology Homotopy and Applications - View - PDF