Homology, Homotopy and Applications

Volume 9 (2007)

Number 2

From loop groups to 2-groups

Pages: 101 – 135

DOI: http://dx.doi.org/10.4310/HHA.2007.v9.n2.a4

Authors

John C. Baez (Department of Mathematics, University of California at Riverside)

Alissa S. Crans (Department of Mathematics, Loyola Marymount University, Los Angeles, California, U.S.A.)

Urs Schreiber (Organizationseinheit Mathematik, Schwerpunkt Algebra und Zahlentheorie, Universität Hamburg, Germany)

Danny Stevenson (Department of Mathematics, University of California at Riverside)

Abstract

We describe an interesting relation between Lie 2-algebras, the Kac-Moody central extensions of loop groups, and the group ${\rm String}(n)$. 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 $\mathfrak{g}_k$ each having $\mathfrak{g}$ 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 $\mathfrak{g}_k$ as its Lie 2-algebra, except when $k = 0$. Here, however, we construct for integral $k$ an infinite-dimensional Lie 2-group ${\cal P}_kG$ whose Lie 2-algebra is equivalent to $\mathfrak{g}_k$. The objects of ${\cal P}_kG$ are based paths in $G$, while the automorphisms of any object form the level-$k$ Kac-Moody central extension of the loop group $\Omega G$. This 2-group is closely related to the $k$th power of the canonical gerbe over $G$. Its nerve gives a topological group $|{\cal P}_kG|$ that is an extension of $G$ by $K(\mathbb{Z},2)$. When $k = \pm 1$, $|{\cal P}_kG|$ can also be obtained by killing the third homotopy group of $G$. Thus, when $G = {\rm Spin}(n)$, $|{\cal P}_kG|$ is none other than ${\rm String}(n)$.

Keywords

Gerbe; Kac-Moody extension; Lie 2-algebra; loop group; string group; 2-group

2010 Mathematics Subject Classification

22E67

Full Text (PDF format)