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From Loop Groups to 2-Groups | | John C. Baez
; Alissa S. Crans
; Danny Stevenson
; Urs Schreiber
; | | Date: |
7 Apr 2005 | | Subject: | Quantum Algebra; Differential Geometry MSC-class: 81R10 (Primary), 22E67, 17B67, 18D10 (Secondary) | math.QA hep-th math.DG | | 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 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. Here, however, we construct for integral k 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 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). | | Source: | arXiv, math.QA/0504123 | | Services: | Forum | Review | PDF | Favorites | |
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