nLab Fivebrane group

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

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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.

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				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 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)

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

• Domenico Fiorenza, Urs Schreiber, Jim Stasheff, Cech Cocycles for Differential characteristic Classes, Advances in Theoretical and Mathematical Phyiscs, Volume 16 Issue 1 (2012) (arXiv:1011.4735)

• Hisham Sati, Urs Schreiber, Jim Stasheff, Twisted Differential String and Fivebrane Structures, Communications in Mathematical Physics October 2012, Volume 315, Issue 1, pp 169-213 (arXiv:0910.4001)

Discussion in a comprehensive context is in section 5 of

• Urs Schreiber, differential cohomology in a cohesive topos (arXiv:1310.7930)