From loop groups to 2-groups

By John C. Baez, Alissa S. Crans, Danny Stevenson, Urs Schreiber. 2005

167 citations recorded by the source corpus.

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 gk 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 gk as its Lie 2-algebra, except when k = 0. Here, however, we construct for integral k an infinite-dimensional Lie 2-group PkG whose Lie 2-algebra is equivalent to gk. The objects of PkG are based paths in G, while the automorphisms of any object form the level-k Kac– Moody central extension of the loop group G. This 2-group is closely related to the kth power of the canonical gerbe over G. Its nerve gives a topological group |PkG| that is an extension of G by K(Z,2). When k = ±1, |PkG| can also be obtained by killing the third homotopy group of G. Thus, when G = Spin(n), |PkG| is none other than String(n).

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/3899c845fef685864884e602dfdbb8313271b3be

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Citations in the corpus

Twisted Differential String and Fivebrane Structures (2009, methodology): “The finite dimensional but weak Lie 2-algebra gμ3 is equivalent [9] to the strict but infinite-dimensional Lie 2-algebra (Ω̂g→ Pg).”
Higher gauge theory (2005, background): “To go beyond this and develop a theory of nonabelian higher gauge fields, we must let the topology dictate the algebra.”
An invitation to higher gauge theory (2010, background): “Moreover, it fits into an exact sequence 1 → Lk(G) → ST RINGk(G)−→G → 1 where the middle term, the level-k string 2-group of G, has very interesting properties [9].”
Central extensions of smooth 2–groups and a finite-dimensional string 2–group (2009, result): “This idea is not new, and goes back to the work of Baez-Lauda [BL04], Henriques [Hen06], and BaezCrans-Schreiber-Stevenson [BSCS07].”
The Classifying Space of a Topological 2-Group (2008, unknown): “A T ] 2 7”
Integrating $L_\infty $-algebras (2006, methodology): “Given a pretopology in our sense, one recovers a pretopology in Grothendieck’s sense by declaring {Ui → X} to be a cover if ∐ i Ui → X is a cover.”