nLab 2-spin structure

Idea

A 2-spin structure is a special tangential structure in the Whitehead tower of the orthogonal group, bridging the more important fivebrane structure and ninebrane structure. Its name comes from the fact, that it corresponds to a spin structure under eight-fold Bott periodicity.

Lift

Let $x_10\in H^10(B2\text{-}Orient,\mathbb{Z}_2)$ be the universal characteristic class from the short exact sequence $B2\text{-}Spin\rightarrow B2\text{-}Orient\rightarrow K(\mathbb{Z}_2,10)$. It classifies lifts, meaning that a 2-orientation $f\colon M\rightarrow B2\text{-}Orient$ lifts to a 2-spin structure $f\colon M\rightarrow B2\text{-}Spin$ if and only if:

$$f^*x_10=0\in H^10(M,\mathbb{Z}_2).$$

(Sati 14, Def. 2.5)

In this case the set of all 2-spin structures is a torsor for $H^9(M,\mathbb{Z}_2)$. (Sati 14, Prop. 4.1 (ii))

Twisted 2-Spin structure

If $i\colon M\hookrightarrow X$ is a submanifold (seen as a brane in spacetime) with a 2-orientation $f\colon M\rightarrow B2\text{-}Orient$, then for a singular cohomology class $\alpha\in H^10(X,\mathbb{Z}_2)$ with:

$$f^*x_10 =i^*\alpha \in H^10(M,\mathbb{Z}_2)$$

a homotopy between their classifying maps $M\rightarrow K(\mathbb{Z}_2,10)$ is a twisted 2-spin structure. ($\alpha=0$ gives back an ordinary 2-spin structure.)

(Sati 14, Def. 5.2)

(Sati 14) asks for explicit examples of twisted 2-spin structures.

References

• Hisham Sati, Ninebrane structures (arXiv:1405.7686)