group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
$\cdots \to$ ninebrane 10-group $\to$ Fivebrane group $\to$ string group $\to$ spin group $\to$ special orthogonal group $\to$ orthogonal group.
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 $\mathrm{O}(n)$:
The homotopy groups of $O(n)$ are for $k \in \mathbb{N}$ and for sufficiently large $n$
By co-killing these groups step by step one gets
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.
$n$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Whitehead tower of orthogonal group | orientation | spin | string | fivebrane | ninebrane | |||||||||||||
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 | $\mathbb{Z}_{480}$ | $\mathbb{Z}_2$ |
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
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