nLab 2-spin group

Idea

A 2-Spin group is a topological group appearing in the Whitehead tower of the orthogonal group as an intermediate step between the more important fivebrane group and ninebrane group:

$$\ldots \rightarrow Ninebrane(n) \rightarrow 2\text{-}Spin(n) \rightarrow 2\text{-}Orient(n) \rightarrow Fivebrane(n) \rightarrow String(n) \rightarrow Spin(n) \rightarrow SO(n) \rightarrow O(n).$$

2-Spin groups are denoted that way, because in limit the construction removes the same homotopy group as for the spin group, which describes spin structures, under the eight-fold Bott periodicity:

$$2\text{-}Spin \coloneqq O\langle 10\rangle =O\langle 11\rangle;$$

$$Spin \coloneqq O\langle 2\rangle =O\langle 3\rangle.$$

(In similar fashion, ninebrane groups would be the 2-string groups.)

References

• Hisham Sati, Ninebrane structures (arXiv:1405.7686)