Parallel spinors for \(\text {G}_2^*\) 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 Kähler-Atiyah bundle of (M, g). Applying this general framework, we obtain an intrinsic algebraic characterization of \(\text {G}_2^*\)-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.