nLab 2-orientation

Idea

A 2-orientation 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 orientation under eight-fold Bott periodicity.

Lift

Let $x_9\in H^9(BFivebrane,\mathbb{Z}_2)$ be the universal characteristic class from the short exact sequence $B2\text{-}Orient\rightarrow BFivebrane\rightarrow K(\mathbb{Z}_2,9)$. It classifies lifts, meaning that a fivebrane structure $f\colon M\rightarrow BFivebrane$ lifts to a 2-orientation $f\colon M \rightarrow B2\text{-}Orient$ if and only if:

$$f^*x_9 =0\in H^9(M,\mathbb{Z}_2).$$

(Sati 14, Def. 2.4, Eq. (2.7))

In this case the set of all 2-orientations is a torsor for $H^8(M,\mathbb{Z}_2)$. (Sati 14, Prop. 4.1 (i))

Twisted 2-orientation

If $i\colon M\hookrightarrow X$ is a submanifold (seen as a brane in spacetime) with a fivebrane structure $f\colon M\rightarrow BFivebrane$, then for a singular cohomology class $\alpha\in H^9(X,\mathbb{Z}_2)$ with:

$$f^*x_9 =i^*\alpha \in H^9(M,\mathbb{Z}_2)$$

a homotopy between their classifying maps $M\rightarrow K(\mathbb{Z}_2,9)$ is a twisted 2-orientation. ($\alpha=0$ gives back an ordinary 2-orientation.)

(Sati 14, Def. 5.1, Eq. (2.10))

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

References

• Hisham Sati, Ninebrane structures (arXiv:1405.7686)