### Talk in Bonn on String 2-Group

#### Posted by Urs Schreiber

Next Tuesday, May 2, 2006 there is a talk

On the String 2-Group ($\to $)

at the Bonn Math Institute ( 17:00-18:00, seminar room E (room 3, Meckenheimer Allee 160)).

**Abstract.**

We try to convey the main idea for

$\u2022$ what the String-group ${\mathrm{String}}_{G}$ is

$\u2022$ and how it is the nerve of a 2-group
${\mathrm{Str}}_{G}$

as well as

$\u2022$ what a ${\mathrm{Str}}_{G}$-2-bundle is

$\u2022$ and how it is ‘the same’ as a
${\mathrm{String}}_{G}$-bundle.

The first point is due to [BCSS,Henriques], which will be reviewed in section 2. The second point has been discussed in [Jurčo] using the language of bundle gerbes. In section 3 we review this, using a 2-functorial language which is natural with respect to the 2-group nature of ${\mathrm{Str}}_{G}$.

**Motivation**

The main motivation for the following discussion has its origin in theoretical physics.

Elementary particles with spin are described by sections of spin bundles. From the physical point of view, the necessity of a spin structure on spacetime may be deduced from a certain global anomaly for the path integral of a single, pointlike, fermion. The path integral (albeit a somewhat heuristic device) can be regarded as a single valued function on the space of configurations of the particle only if the (first and) second Stiefel-Whitney class of spacetime vanishes. In other words, if spacetime admits a spin struture.

It is possible to generalize this argument to the case where
the fermion is line-like. (In theoretical physics such a
hypothetical object is called a superstring.) It was
found that in this case there is another obstruction,
which this time is measured by
the first Pontryagin class of spacetime [Killingback].
This is interpreted as saying that the *loop space* over spacetime
admits a spin structure. In fact, this condition implies the famous
(to high energy physicists) Green-Schwarz anomaly cancellation,
which has been one of the main reasons why physicists
considered superstrings a promising idea to pursue.

As for the pointlike fermion, this situation may be reformulated in terms of lifts of bundles. There is a topological group called $\mathrm{String}(n)$, which is a 3-connected cover of $\mathrm{Spin}(n)$. The Pontryagin class is the obstruction controlling the lift of $\mathrm{Spin}(n)$-bundles to $\mathrm{String}(n)$-bundles [StolzTeichner,Stevenson].

$\phantom{\rule{thinmathspace}{0ex}}$

For several reasons one may suspect that this situation is naturally described in terms of categorical algebra. Indeed, it can be shown that $\mathrm{String}(n)$ is nothing but the geometric realization of the nerve of a certain category with group structure, called $\mathrm{Str}(n)$ - a (Fréchet Lie) 2-group. This is the content of section 2.

Moreover, the obstruction to lifting a $\mathrm{Spin}(n)$-bundle to a $\mathrm{String}(n)$-bundle is the same as that for lifting it to a 2-bundle (gerbe) with structure 2-group $\mathrm{Str}ofn$. This is the content of section 3 [Jurčo].

## Re: Talk in Bonn on String 2-Group

Hey,

Regarding how String(n)-bundles are “the same” as a Str_G-2-bundle. Maybe someone can explain a simpler version to me: Is it the case that circle-bundles are “the same” as 2-bundles for the 2-group with no 1-arrows and Z 2-arrows? What does it mean?

D.