There is 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.
However, for integral *k* there is 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 *k*th
power of the
canonical gerbe over *G*. Its nerve gives a topological group that is an
extension of *G* by the Eilenberg-MacLane space
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*).

Click on this to see the transparencies of my talk:

- From Loop Groups to 2-Groups - in PDF or Postscript.

The talk is based on this paper:

- John Baez, Alissa Crans, Danny Stevenson and Urs Schreiber, From Loop Groups to 2-Groups

- John Baez and Aaron Lauda, Higher-Dimensional Algebra V: 2-Groups
- John Baez and Alissa Crans, Higher-Dimensional Algebra VI: Lie 2-Algebras
- Toby Bartels, Higher Gauge Theory: 2-Bundles

Also, Urs Schreiber and I will soon be coming out with a related paper. You can see the latest draft here:

- John Baez and Urs Schreiber, Higher Gauge Theory: 2-Connections on 2-Bundles.

In 2004, I gave a related talk at a conference in honor of Larry Breen's birthday.

© 2005 John Baez

baez@math.removethis.ucr.andthis.edu