nLab octonionic line bundle

Idea

Recall first that a (topological) line bundle over a (topological) field (or more generally a (topological) ring) $\mathbb{K}$ is, by definition, a rank=1 $\mathbb{K}$-vector bundle and hence a $\mathbb{K}$-fiber bundle with structure group the group of units $GL(1,\mathbb{K}) \simeq \mathbb{K}^\times$. It is then typically a theorem that (over paracompact topological spaces) such fiber bundles all arise, up to isomorphism, as pullbacks of the tautological line bundle on infinite projective space $\mathbb{K}P^\infty$, so that their isomorphism classes are identified with the homotopy classes of continuous maps to this classifying space $\mathbb{K}P^\infty$.

Now, the octonions $\mathbb{O}$ are not an (associative) ring and so a would-be group of units “$\mathbb{O}^\times$” which could serve as the structure group of $\mathbb{O}$-fiber bundles does not exist. This means that the usual definition of $\mathbb{K}$-line bundles does not generalize to $\mathbb{K} \coloneqq \mathbb{O}$.

However, the notion of a tautological line bundle over the octonionic projective line $\mathbb{O}P^1$ still does make sense, and this (rank=8 real vector bundle) is commonly called the tautological octonionic line bundle or the “canonical octonionic line bundle”.

Incidentally, a would-be tautological octonionic line bundle on the octonionic projective plane $\mathbb{O}P^2$ (the “Cayley plane”) does not exist (the non-associativity makes the would-be octonionic lines be quaternionic in some sense, see Dray & Manogue 2015, p. 121), and for $n \geq 3$ even $\mathbb{O}P^n$ does not exist, much less its tautological line bundle. The only tautological octonionic line bundle is hence that over $\mathbb{O}P^1$.

Therefore, by the above analogy, it is then justified (though not common in the literature) to define general octonionic line bundles to be, up to isomorphism, the pullback bundles of this tautological one along continuous maps to $\mathbb{O}P^1$.

Related concepts

• real vector bundle, complex vector bundle, quaternionic vector bundle

References

Discussion of the tautological octonionic line bundle over $\mathbb{O}P^1$:

• John Baez, §3.1 in: The Octonions, Bull. Amer. Math. Soc. 39 (2002) 145-205. [web, pdf, doi:10.1090/S0273-0979-01-00934-X]

• Tevian Dray, Corinne Manogue, §12.2 of: The Geometry of Octonions, World Scientific (2015) [doi:10.1142/8456, web]