Gradings of Lie algebras, magical spin geometries and matrix factorizations

Abstract

We describe a remarkable rank 14 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 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 Spin 14 , we construct a special rank seven vector bundle on a double-octic threefold, that we conjecture to be spherical.

Graded Lie-Rinehart Algebras

“…On the other hand, gradings appear elsewhere in the theory of Lie algebras, for example in the Cartan decomposition of a finite-dimensional complex semisimple Lie algebra (see for instance [1,6,10,15,16,19,23]). Also, graded modules have attracted the attention of many researchers in the last years (see [2,4,5,8,28,31]).…”