∞-Lie theory

cohomology

# Contents

## Idea

The fivebrane 6-group $\mathrm{Fivebrane}\left(n\right)$ 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

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

$\frac{1}{6}{p}_{2}:B\mathrm{String}\left(n\right)\to {B}^{7}ℝ/ℤ$\frac{1}{6} \mathbf{p}_2 : \mathbf{B} String(n) \to \mathbf{B}^7 \mathbb{R}/\mathbb{Z}

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

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

$\begin{array}{ccc}B\mathrm{Fivebrane}\left(n\right)& \to & *\\ ↓& & ↓\\ B\mathrm{String}\left(n\right)& \stackrel{\frac{1}{6}{p}_{2}}{\to }& {B}^{7}ℝ/ℤ\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\array{ \mathbf{B} Fivebrane(n) &\to& {*} \\ \downarrow && \downarrow \\ \mathbf{B}String(n) &\stackrel{\frac{1}{6}\mathbf{p}_2}{\to}& \mathbf{B}^7 \mathbb{R}/\mathbb{Z} } \,.

## 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 $B\mathrm{String}\left(n\right)$ that is built in degree $k\le 7$ from smooth $k$-simplices in the Lie group $\mathrm{Spin}\left(n\right)$. This morphism assigns to a 7-simplex $\varphi :{\Delta }_{\mathrm{Diff}}^{7}\to \mathrm{Spin}\left(n\right)$ the integral

${\int }_{{\Delta }_{\mathrm{Diff}}^{7}}{\varphi }^{*}{\mu }_{7}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\in ℝ/ℤ$\int_{\Delta^7_{Diff}} \phi^* \mu_7 \;\;\in \mathbb{R}/\mathbb{Z}

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

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

${\mathrm{cosk}}_{7}\left(\begin{array}{c}{Q}_{7}\subset \mathrm{hom}\left({\Delta }_{\mathrm{Diff}}^{7},G\right)×\left(U\left(1\right){\right)}^{8\cdot 7\cdot 6\cdot 5\cdot 4}\\ ↓↓↓↓↓↓↓↓\\ ⋮\\ ↓↓↓↓↓↓\\ {Q}_{4}\subset \mathrm{hom}\left({\Delta }_{\mathrm{Diff}}^{4},G\right)×\left(U\left(1\right){\right)}^{20}\\ ↓↓↓↓↓\\ {Q}_{3}\subset \mathrm{hom}\left({\Delta }_{\mathrm{Diff}}^{3},G\right)×\left(U\left(1\right){\right)}^{4}\\ ↓↓↓↓\\ \mathrm{hom}\left({\Delta }_{\mathrm{Diff}}^{2},G\right)×U\left(1\right)\\ ↓↓↓\\ \mathrm{hom}\left({\Delta }_{\mathrm{Diff}}^{1},G\right)\\ ↓↓\\ *\end{array}\right)$\mathbf{cosk}_7 \left( \array{ Q_7 \subset hom(\Delta^7_{Diff}, G) \times (U(1))^{8 \cdot 7 \cdot 6 \cdot 5 \cdot 4} \\ \downarrow \downarrow \downarrow\downarrow \downarrow \downarrow \downarrow \downarrow \\ \vdots \\ \downarrow \downarrow \downarrow\downarrow \downarrow \downarrow \\ Q_4 \subset hom(\Delta^4_{Diff}, G) \times (U(1))^{20} \\ \downarrow \downarrow \downarrow\downarrow \downarrow \\ Q_3 \subset hom(\Delta^3_{Diff}, G) \times (U(1))^4 \\ \downarrow \downarrow \downarrow\downarrow \\ hom(\Delta^2_{Diff}, G) \times U(1) \\ \downarrow \downarrow \downarrow \\ hom(\Delta^1_{Diff}, G) \\ \downarrow \downarrow \\ * } \right)

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

(…)

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

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

• 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.

Revised on June 14, 2012 13:17:04 by Urs Schreiber (131.130.238.252)