Contents

Idea

The fivebrane 6-group $\mathrm{Fivebrane}(n)$ is a smooth version of the topological space that appears in the second step of the Whitehead tower of the orthogonal group.

It is a lift of this through the geometric realization functor $\Pi :$ ∞LieGrpd $\to$ ∞Grpd.

One step below the fivebrane 6-group in the Whitehead tower is the string Lie 2-group.

For the time being see the discussions at

smooth Whitehead tower

and the Motivation section at

infinity-Chern-Weil theory

for more background.

Definition

In the (∞,1)-topos $H=$ ∞LieGrpd we have a smooh refinement of the second fractional Pontryagin class

defined on the delooping of the string Lie 2-group.

The delooping $B\mathrm{Fivebrane}(n)$ of the fivebrane 6-group is the principal ∞-bundle classified by this in $H$, that is the homotopy fiber

Construction

Along the lines of the description at Lie integration and string 2-group, in a canonical model for $H$ the morphism $\frac{1}{6}{p}_2$ is given by a morphism out of a resolution $BQ$ of $B\mathrm{String}(n)$ that is built in degree $k\le 7$ from smooth $k$-simplices in the Lie group $\mathrm{Spin}(n)$. This morphism assigns to a 7-simplex $\varphi :{\Delta }_{\mathrm{Diff}}^7\to \mathrm{Spin}(n)$ the integral

of the degree 7 Lie algebra cocycle ${\mu }_7$ of the special orthogonal Lie algebra $\mathrm{𝔰𝔬}(n)$ which is normalized such that its pullback to $\mathrm{String}(n)$ (..explain…) is the deRham image of the generator in integral cohomology there.

More in detail, a resolution of $B\mathrm{String}(n)$ is given by the coskeleton

where the subobjects are those consisting of 3-simplices in $G$ with 2-faces labeled in $U(1)$ such that the integral of ${\mu }_3$ over the 3-simplex in $ℝ/\mathbb{Z}$ is the signed product of these labels.

(…)

Related concepts

fivebrane 6-group $\to$ string 2-group $\to$ spin group $\to$ special orthogonal group $\to$ orthogonal group

References

The topological fivebrane group with its interpretation in dual heterotic string theory was discussed in

• Hisham Sati, Urs Schreiber, Jim Stasheff, Fivebrane structures Reviews in mathematical physics, 10 (2009) 1197 (web)

and the smooth fivebrane 6-group was indicated. The latter is discussed in more detail in section 4.1 of

• Urs Schreiber, differential cohomology in a cohesive topos

• Jesse Wolfson says he has shown the existence of a presentation of the $\mathrm{Fivebrane}$ smooth 6-group by a locally Kan and degreewise finite-dimensional simplicial smooth manifold.