nLab
Fivebrane group

$\cdots \to$ Fivebrane group $\to$ string group $\to$ spin group $\to$ special orthogonal group $\to$ orthogonal group.

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Definition

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

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

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

$$\begin{array}{cc}{\pi }_{8k+0}(O)& ={\mathbb{Z}}_2\\ {\pi }_{8k+1}(O)& ={\mathbb{Z}}_2\\ {\pi }_{8k+2}(O)& =0\\ {\pi }_{8k+3}(O)& =\mathbb{Z}\\ {\pi }_{8k+4}(O)& =0\\ {\pi }_{8k+5}(O)& =0\\ {\pi }_{8k+6}(O)& =0\\ {\pi }_{8k+7}(O)& =\mathbb{Z}\end{array}.$$\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

$$\begin{array}{ccccc}\mathrm{cokill}\mathrm{this}& & & & \mathrm{to}\mathrm{get}\\ \\ {\pi }_0(O)& ={\mathbb{Z}}_2& & & \mathrm{SO}\\ {\pi }_1(O)& ={\mathbb{Z}}_2& & & \mathrm{Spin}\\ {\pi }_2(O)& =0\\ {\pi }_3(O)& =\mathbb{Z}& & & \mathrm{String}\\ {\pi }_4(O)& =0\\ {\pi }_5(O)& =0\\ {\pi }_6(O)& =0\\ {\pi }_7(O)& =\mathbb{Z}& & & \mathrm{Fivebrane}\end{array}.$$\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…

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

References

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

• Hisham Sati, Urs Schreiber, Jim Stasheff, Fivebrane structures , Reviews in Math. Phys. (arXiv:0805.0564)

The term shortly after was picked up in

• Christopher Douglas, André Henriques, Michael Hill?, Homological obstructions to string orientations (arXiv:0810.2131)