Gradings of Lie algebras, magical spin geometries and matrix factorizations

We describe a remarkable rank fourtenn matrix factorization of the octic Spin(14)-invariant polynomial on either of its half-spin representations. We observe that this representation can be, in a suitable sense, identified with a tensor product of two octonion algebras. Moreover the matrix factorisation can be deduced from a particular Z-grading of the exceptional Lie algebra $\mathfrak{e}_8$. Intriguingly, the whole story can be extended to the whole Freudenthal-Tits magic square and yields matrix factorizations on other spin representations, as well as for the degree seven invariant on the space of three-forms in several variables. As an application of our results on Spin(14), we construct a special rank seven vector bundle on a double-octic threefold, that we conjecture to be spherical.