nLab
fivebrane 6-group

Urs Schreiber: this is not in the literature

Idea

The smooth $\mathrm{Fivebrane}(n)$ 6-group is to the string 2-group as the topological fivebrane group is to the topological string group.

The motivation and background described at string 2-group applies to the fivebrane 6-group with string structure everywhere replaced by fivebrane structure.

Definition

In a smooth (∞,1)-topos $H$ with $B\mathrm{String}(n)$ denoting the delooping of the string 2-group, there is canonically (up to equivalence) a morphism

$$\frac{1}{6}{p}_{2}B\mathrm{String}(n)\to B^{7}R/Z$$\frac{1}{6} p_2 \mathbf{B} String(n) \to \mathbf{B}^7 R/Z

being a smooth incarnation of the second fractional Pontryagin class?. The delooping $B\mathrm{Fivebrane}(n)$ of the fivebrane 6-group is the homotopy fiber

$$\begin{array}{ccc}B\mathrm{Fivebrane}(n)& \to & *\\ \downarrow & & \downarrow \\ B\mathrm{String}(n)& \stackrel{\frac{1}{6}{p}_2}{\to }& B^{7}R/Z\end{array}$$\array{ \mathbf{B} Fivebrane(n) &\to& {*} \\ \downarrow && \downarrow \\ \mathbf{B}String(n) &\stackrel{\frac{1}{6}p_2}{\to}& \mathbf{B}^7 R/Z }

of $\frac{1}{2}{p}_2$ in $H$.

Along the lines of the description at 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

$${\int }_{{\Delta }_{\mathrm{Diff}}^7}{\varphi }^{*}{\mu }_7\in R/Z$$\int_{\Delta^7_{Diff}} \phi^* \mu_7 \;\;\in R/Z

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

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