Parallel spinors for $\mathrm{G}_2^*$ and isotropic structures

Abstract: We obtain a correspondence between irreducible real parallel spinors on pseudo-Riemannian manifolds $(M,g)$ of signature $(4,3)$ and solutions of an associated differential system for three-forms that satisfy a homogeneous algebraic equation of order two in the K\"ahler-Atiyah bundle of $(M,g)$. Applying this general framework, we obtain an intrinsic algebraic characterization of $\mathrm{G}_2^*$-structures as well as the first explicit description of isotropic irreducible spinors in signature $(4,3)$ that are parallel under a general connection on the spinor bundle. This description is given in terms of a coherent system of mutually orthogonal and isotropic one forms and follows from the characterization of the stabilizer of an isotropic spinor as the stabilizer of a highly degenerate three-form that we construct explicitly. Using this result, we show that isotropic spinors parallel under a metric connection with torsion exist when the connection preserves the aforementioned coherent system. This allows us to construct a natural class of metrics of signature $(4,3)$ on $\mathbb{R}^7$ that admit spinors parallel under a metric connection with torsion.