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We develop a new framework for the study of generalized Killing spinors,where generalized Killing spinor equations, possibly with constraints, can beformulated equivalently as systems of partial differential equations for apolyform satisfying algebraic relations in the K\"ahler-Atiyah bundleconstructed by quantizing the exterior algebra bundle of the underlyingmanifold. At the core of this framework lies the characterization, which wedevelop in detail, of the image of the spinor squaring map of an irreducibleClifford module $\Sigma$ of real type as a real algebraic variety in theK\"ahler-Atiyah algebra, which gives necessary and sufficient conditions for apolyform to be the square of a real spinor. We apply these results toLorentzian four-manifolds, obtaining a new description of a real spinor on sucha manifold through a certain distribution of parabolic 2-planes in itscotangent bundle. We use this result to give global characterizations of realKilling spinors on Lorentzian four-manifolds and of four-dimensionalsupersymmetric configurations of heterotic supergravity. In particular, we findnew families of Einstein and non-Einstein four-dimensional Lorentzian metricsadmitting real Killing spinors, some of which are deformations of the metric ofAdS$_4$ space-time.

Spinors of real type as polyforms and the generalized Killing equation