# nLab fivebrane 6-group

Contents

cohomology

## Spin geometry

spin geometry

Dynkin labelsp. orth. groupspin grouppin groupsemi-spin group
SO(2)Spin(2)Pin(2)
B1SO(3)Spin(3)Pin(3)
D2SO(4)Spin(4)Pin(4)
B2SO(5)Spin(5)Pin(5)
D3SO(6)Spin(6)
B3SO(7)Spin(7)
D4SO(8)Spin(8)SO(8)
B4SO(9)Spin(9)
D5SO(10)Spin(10)
B5SO(11)Spin(11)
D6SO(12)Spin(12)
$\vdots$$\vdots$
D8SO(16)Spin(16)SemiSpin(16)
$\vdots$$\vdots$
D16SO(32)Spin(32)SemiSpin(32)

string geometry

# Contents

## 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

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.

(…)

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

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