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 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).

An invitation to higher gauge theory

“…There is a way to take any topological 2-group and squash it down to a topological group [9,19]. Applying this trick to ST RIN G k (G) when k = 1, we obtain a topological group whose homotopy groups match those of Gexcept for the third homotopy group, which has been made trivial.…”

“…Weak Lie 2-algebras are more tricky [14]. Baez et al [9] dodged this problem by showing that the string Lie 2-algebra is equivalent (in some precise sense) to a strict Lie 2-algebra, which however is infinite-dimensional. They then constructed the infinite-dimensional strict Lie 2-group corresponding to this strict Lie 2-algebra.…”

Division algebras and supersymmetry III

“…We choose, however, to focus on α. This simplifies our later work, and because Lie 2-algebras based on j have already been the subject of much scrutiny [3,10,31,47], it should be possible to combine what we do here with the work of other authors to arrive at a more complete picture.…”

“…There are more sophisticated approaches to integrating the string Lie 2-algbera, like those due to Baez, Crans, Schreiber and Stevenson [10] or Schommer-Pries [47], and a general technique to integrate any Lie n-algebra due to Henriques [31], which Schreiber [50] has in turn generalized to handle Lie n-superalgebras and more. All three techniques involve generalizing the notion of Lie 2-group (or Lie n-group, for Henriques and Schreiber) away from the world of finite-dimensional manifolds, and the latter three generalize the notion of 2-group to one in which products are defined only up to equivalence.…”

n-Groupoids and Stacky Groupoids

“…More generally, to any compact simply connected group G one can associate its string group String G . It has various models, given by Stolz and Teichner [33,34] using an infinite-dimensional extension of G, by Brylinski [9] using a U (1)-gerbe with the connection over G, and recently by Baez et al [4] using Lie 2-groups and Lie 2-algebras. Henriques [21] constructs the string group as a higher group that we study in this paper and as an integration object of a certain Lie 2-algebra with an integration procedure which is also studied in [18,39,42].…”