Gradings of Lie algebras, magical spin geometries and matrix factorizations

Abstract

We describe a remarkable rank 14 14 matrix factorization of the octic S p i n 14 \mathrm {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 \mathbb {Z} -grading of e 8 \mathfrak {e}_8 . Intriguingly, the whole story can in fact 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 S p i n 14 \mathrm {Spin}_{14} , we construct a special rank seven vector bundle on a double-octic threefold, that we conjecture to be spherical.