nLab fivebrane 6-group

Idea

The fivebrane 6-group $Fivebrane(n)$ 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 $\mathbf{H} =$ ∞LieGrpd we have a smooth refinement of the second fractional Pontryagin class

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

defined on the delooping of the string Lie 2-group. Strictly speaking, we need $n\gt 6$, since for low $n$, $String(n)$ is not 6-connected. See orthogonal group for a table of the relevant homotopy groups.

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

$$\array{ \mathbf{B} Fivebrane(n) &\to& {*} \\ \downarrow && \downarrow \\ \mathbf{B}String(n) &\stackrel{\frac{1}{6}\mathbf{p}_2}{\to}& \mathbf{B}^7 \mathbb{R}/\mathbb{Z} } \,.$$

Construction

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

$$\int_{\Delta^7_{Diff}} \phi^* \mu_7 \;\;\in \mathbb{R}/\mathbb{Z}$$

of the degree 7 Lie algebra cocycle $\mu_7$ of the special orthogonal Lie algebra $\mathfrak{so}(n)$ which is normalized such that its pullback to $String(n)$ (..explain…) is the deRham image of the generator in integral cohomology there.

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

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

where the subobjects are those consisting of 3-simplices in $G$ with 2-faces labeled in $U(1)$ such that the integral of $\mu_3$ over the 3-simplex in $\mathbb{R}/\mathbb{Z}$ is the signed product of these labels.

(…)

Related concepts

	$n$	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16
Whitehead tower of orthogonal group		orientation	spin group		string group				fivebrane group	2-orient group		2-spin group	ninebrane group
higher versions		special orthogonal group	spin group		string 2-group				fivebrane 6-group				ninebrane 10-group
homotopy groups of stable orthogonal group	$\pi_n(O)$	$\mathbb{Z}_2$	$\mathbb{Z}_2$	0	$\mathbb{Z}$	0	0	0	$\mathbb{Z}$	$\mathbb{Z}_2$	$\mathbb{Z}_2$	0	$\mathbb{Z}$	0	0	0	$\mathbb{Z}$	$\mathbb{Z}_2$
stable homotopy groups of spheres	$\pi_n(\mathbb{S})$	$\mathbb{Z}$	$\mathbb{Z}_2$	$\mathbb{Z}_2$	$\mathbb{Z}_{24}$	0	0	$\mathbb{Z}_2$	$\mathbb{Z}_{240}$	$\mathbb{Z}_2 \oplus \mathbb{Z}_2$	$\mathbb{Z}_2 \oplus \mathbb{Z}_2 \oplus \mathbb{Z}_2$	$\mathbb{Z}_6$	$\mathbb{Z}_{504}$	0	$\mathbb{Z}_3$	$\mathbb{Z}_2 \oplus \mathbb{Z}_2$	$\mathbb{Z}_{480} \oplus \mathbb{Z}_2$	$\mathbb{Z}_2 \oplus \mathbb{Z}_2$
image of J-homomorphism	$im(\pi_n(J))$	0	$\mathbb{Z}_2$	0	$\mathbb{Z}_{24}$	0	0	0	$\mathbb{Z}_{240}$	$\mathbb{Z}_2$	$\mathbb{Z}_2$	0	$\mathbb{Z}_{504}$	0	0	0	$\mathbb{Z}_{480}$	$\mathbb{Z}_2$

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 $Fivebrane$ smooth 6-group by a locally Kan and degreewise finite-dimensional simplicial smooth manifold.