# nLab Fivebrane group

Contents

cohomology

### Theorems

$\cdots \to$ ninebrane 10-group $\to$ Fivebrane group $\to$ string group $\to$ spin group $\to$ special orthogonal group $\to$ orthogonal group.

# Contents

## Definition

The Fivebrane group $Fivebrane(n)$ is defined to be, as a topological group, the 7-connected cover of the String group $String(n)$, for any $n \in \mathbb{N}$.

Notice that $String(n)$ itself if the 3-connected cover of $Spin(n)$, which is itself is the simply connected cover of the special orthogonal group $SO(n)$, which in turn is the connected component (of the identity) of the orthogonal group $O(n)$. Hence $Fivebrane(n)$ is one element in the Whitehead tower of $\mathrm{O}(n)$:

$\cdots \to Fivebrane(n) \to String(n) \to Spin(n) \to SO(n) \to \mathrm{O}(n) \,.$

The homotopy groups of $O(n)$ are for $k \in \mathbb{N}$ and for sufficiently large $n$

$\array{ \pi_{8k+0}(O) & = \mathbb{Z}_2 \\ \pi_{8k+1}(O) & = \mathbb{Z}_2 \\ \pi_{8k+2}(O) & = 0 \\ \pi_{8k+3}(O) & = \mathbb{Z} \\ \pi_{8k+4}(O) & = 0 \\ \pi_{8k+5}(O) & = 0 \\ \pi_{8k+6}(O) & = 0 \\ \pi_{8k+7}(O) & = \mathbb{Z} } \,.$

By co-killing these groups step by step one gets

$\array{ cokill\; this &&&& to \;get \\ \\ \pi_{0}(O) & = \mathbb{Z}_2 &&& SO \\ \pi_{1}(O) & = \mathbb{Z}_2 &&& Spin \\ \pi_{2}(O) & = 0 \\ \pi_{3}(O) & = \mathbb{Z} &&& String \\ \pi_{4}(O) & = 0 \\ \pi_{5}(O) & = 0 \\ \pi_{6}(O) & = 0 \\ \pi_{7}(O) & = \mathbb{Z} &&& Fivebrane } \,.$

## Further information…

Note that if $3 \leq n \leq 6$, one needs to take extra care, as $String(n)$ is not 6-connected in this range (see orthogonal group for a table of the relevant homotopy groups). There are nontrivial intermediate steps in the Whitehead tower

…should eventually go here. For the time being have a look at Fivebrane structure.

$n$012345678910111213141516
Whitehead tower of orthogonal grouporientationspinstringfivebraneninebrane
homotopy groups of stable orthogonal group$\pi_n(O)$$\mathbb{Z}_2$$\mathbb{Z}_2$0$\mathbb{Z}$000$\mathbb{Z}$$\mathbb{Z}_2$$\mathbb{Z}_2$0$\mathbb{Z}$000$\mathbb{Z}$$\mathbb{Z}_2$
stable homotopy groups of spheres$\pi_n(\mathbb{S})$$\mathbb{Z}$$\mathbb{Z}_2$$\mathbb{Z}_2$$\mathbb{Z}_{24}$00$\mathbb{Z}_2$$\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$$\mathbb{Z}_2 \oplus \mathbb{Z}_2$
image of J-homomorphism$im(\pi_n(J))$0$\mathbb{Z}_2$0$\mathbb{Z}_{24}$000$\mathbb{Z}_{240}$$\mathbb{Z}_2$$\mathbb{Z}_2$0$\mathbb{Z}_{504}$000$\mathbb{Z}_{480}$$\mathbb{Z}_2$

The term fivebrane group and the role of this topological group in quantum anomaly cancellaton conditions in dual heterotic string theory was found by Hisham Sati and appeared in

The term shortly after was picked up in

The refinement to fivebrane principal infinity-connections, hence differential fivebrane structures was then discussed in

Discussion in a comprehensive context is in section 5 of