### New preprint: From Loop Groups to 2-Groups (and the String Group)

#### Posted by Urs Schreiber

I am happy to be able to announce a new preprint:

J. Baez, A. Crans, U. Schreiber & D. Stevenson

Abstract:

We describe an interesting relation between Lie 2-algebras, the Kac–Moody central extensions of loop groups, and the group $\mathrm{String}(n)$. 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 ${\U0001d524}_{k}$ each having $\U0001d524$ 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 ${\U0001d524}_{k}$ as its Lie 2-algebra, except when $k=0$. Here, however, we construct for integral $k$ an infinite-dimensional Lie 2-group ${\mathcal{P}}_{k}G$ whose Lie 2-algebra is

equivalentto ${\U0001d524}_{k}$. The objects of ${\mathcal{P}}_{k}G$ are based paths in $G$, while the automorphisms of any object form the level-$k$ Kac–Moody central extension of the loop group $\Omega G$. This 2-group is closely related to the $k$th power of the canonical gerbe over $G$. Its nerve gives a topological group $\mid {\mathcal{P}}_{k}G\mid $ that is an extension of $G$ by $K(\mathbb{Z}\mathrm{,2})$. When $k=\pm 1$, $\mid {\mathcal{P}}_{k}G\mid $ can also be obtained by killing the third homotopy group of $G$. Thus, when $G=\mathrm{Spin}(n)$, $\mid {\mathcal{P}}_{k}G\mid $ is none other than $\mathrm{String}(n)$.

[**Update:** I am aware of that problem with the incorrectly-displayed TeX code above. I am hoping to find the solution to that problem soon.]

There are two central theorems here:

1) The weak Lie 2-algebras ${\U0001d524}_{k}$ defined in HDA VI are *equivalent* to infinite-dimensional strict Fréchet Lie 2-algebras ${\mathcal{P}}_{k}\U0001d524$. These are related to the Kac-Moody central extension ${\Omega}_{k}\U0001d524$ of the loop algebra $\Omega \U0001d524$ and come from infinite-dimensional Fréchet Lie 2-groups ${\mathcal{P}}_{k}G$.

2) The so-called ‘nerve’ $\mid {\mathcal{P}}_{k}G\mid $ of ${\mathcal{P}}_{k}G$ is, for $G=\mathrm{Spin}(n)$, the topological group $\mathrm{String}(n)$.

The conclusion of the paper is the following:

We have seen that the Lie 2-algebra ${\U0001d524}_{k}$ is equivalent to an infinite-dimensional Lie 2-algebra ${\mathcal{P}}_{k}\U0001d524$, and that when $k$ is an integer, ${\mathcal{P}}_{k}\U0001d524$ comes from an infinite-dimensional Lie 2-group ${\mathcal{P}}_{k}G$. Just as the Lie 2-algebra ${\U0001d524}_{k}$ is built from the simple Lie algebra $\U0001d524$ and a shifted version of $\U0001d532(1)$:

(1)$$0\u27f6\mathrm{b}\U0001d532(1)\u27f6{g}_{k}\u27f6\U0001d524\u27f60\phantom{\rule{thinmathspace}{0ex}},$$the Lie 2-group ${\mathcal{P}}_{k}G$ is built from $G$ and another Lie 2-group:

(2)$$1\u27f6{\mathcal{L}}_{k}G\u27f6{\mathcal{P}}_{k}G\u27f6G\u27f61$$whose geometric realization is a shifted version of $\mathrm{U}(1)$:

(3)$$1\u27f6B\mathrm{U}(1)\u27f6\mid {\mathcal{P}}_{k}G\mid \u27f6G\u27f61\phantom{\rule{thinmathspace}{0ex}}.$$None of these exact sequences split; in every case an interesting cocycle plays a role in defining the middle term. In the first case, the Jacobiator of ${\U0001d524}_{k}$ is $k\nu :{\Lambda}^{3}\U0001d524\to \mathbb{R}$. In the second case, composition of morphisms is defined using multiplication in the level-$k$ Kac–Moody central extension of $\mathcal{O}G$, which relies on the Kac–Moody cocycle $k\omega :{\Lambda}^{2}\mathcal{O}\U0001d524\to R$. In the third case, $\mid {\mathcal{P}}_{k}G\mid $ is the total space of a twisted $B\mathrm{U}(1)$-bundle over $G$ whose Dixmier–Douady class is $k[\nu /2\pi ]\in {H}^{3}(G)$. Of course, all these cocycles are different manifestations of the fact that every simply-connected compact simple Lie algebra has ${H}^{3}(G)=\mathbb{Z}$.

We conclude with some remarks of a more speculative nature. There is a theory of ‘2-bundles’ in which a Lie 2-group plays the role of structure group [3, 4]. Connections on 2-bundles describe parallel transport of 1-dimensional extended objects, e.g. strings. Given the importance of the Kac–Moody extensions of loop groups in string theory, it is natural to guess that connections on 2-bundles with structure group ${\mathcal{P}}_{k}G$ will play a role in this theory.

The case when $G=\mathrm{Spin}(n)$ and $k=1$ is particularly interesting, since then $\mid {\mathcal{P}}_{k}G\mid =\mathrm{String}(n)$. In this case we suspect that $2$-bundles on a spin manifold $M$ with structure $2$-group $\mathcal{P}G$ can be thought as substitutes for principal $\mathrm{String}(n)$-bundles on $M$. It is interesting to think about ‘string structures’ [16] on $M$ from this perspective: given a principal $G$-bundle $P$ on $M$ (thought of as a $2$-bundle with only identity morphisms) one can consider the obstruction problem of trying to lift the structure $2$-group from $G$ to ${\mathcal{P}}_{k}G$. There should be a single topological obstruction in ${H}^{4}(M;\mathbb{Z})$ to finding a lift, namely the characteristic class ${p}_{1}/2$. When this characteristic class vanishes, every principal $G$-bundle on $M$ should have a lift to a $2$-bundle $\mathcal{P}$ on $M$ with structure $2$-group ${\mathcal{P}}_{k}G$. It is tempting to conjecture that the geometry of these $2$-bundles is closely related to the enriched elliptic objects of Stolz and Teichner [20].

## Re: New preprint: From Loop Groups to 2-Groups (and the String Group)

I guess it is time to learn about these things. Maybe I can start with an embarrassingly elementary question: what is this String group good for? I understand (to some extent) the reason why a physicist wants to kill the first homotopy group of SO(n) — it is because the physicist wants to be able to define the parallel transport of fermion fields. When it is possible, lifting from SO(n) to Spin(n) involves (roughly) fixing a Z_2-valued ambiguity for every 1-cycle of spacetime, which one describes as the choice of periodic or antiperiodic boundary conditions for the fermions. Is there an analogous “problem” that one encounters in string theory and gets solved by lifting to String(n)? Is there a Z-valued ambiguity for something associated with 3-cycles of spacetime?