# Gradings of Lie algebras, magical spin geometries and matrix factorizations

## TL;DR

This paper constructs explicit matrix factorizations related to Spin(14) and exceptional Lie algebras, revealing deep connections with octonion algebras, the Freudenthal-Tits magic square, and algebraic geometry.

## Contribution

It introduces new matrix factorizations for Spin(14) invariants and extends these to the entire magic square, linking Lie algebra gradings with geometric and algebraic structures.

## Key findings

- Constructed a rank fourtenn matrix factorization of the Spin(14) invariant polynomial.
- Extended the matrix factorizations to the entire Freudenthal-Tits magic square.
- Proposed a conjecture about a spherical vector bundle on a double-octic threefold.

## Abstract

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.