Spinors of real type as polyforms and the generalized Killing equation

Abstract We develop a new framework for the study of generalized Killing spinors, where every spinor equation, possibly with constraints, can be formulated equivalently as system partial differential equations polyform satisfying algebraic relations in Kähler–Atiyah bundle constructed by quantizing exterior algebra underlying manifold. At core this lies characterization, which we detail, image squaring map an irreducible Clifford module $$\Sigma $$ Σ real type variety algebra, gives necessary and sufficient conditions to square spinor. apply these results Lorentzian four-manifolds, obtaining description on such manifold through certain distribution parabolic 2-planes its cotangent bundle. use result give global characterizations spinors four-manifolds four-dimensional supersymmetric configurations heterotic supergravity. In particular, find families Einstein non-Einstein metrics admitting some are deformations metric $$\text {AdS}_4$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">AdS4 space-time.