Abstract
We develop a new framework for the study of generalized Killing spinors, where every generalized Killing spinor equation, possibly with constraints, can be formulated equivalently as a system of partial differential equations for a polyform satisfying algebraic relations in the Kähler–Atiyah bundle constructed by quantizing the exterior algebra bundle of the underlying manifold. At the core of this framework lies the characterization, which we develop in detail, of the image of the spinor squaring map of an irreducible Clifford module Sigma of real type as a real algebraic variety in the Kähler–Atiyah algebra, which gives necessary and sufficient conditions for a polyform to be the square of a real spinor. We apply these results to Lorentzian four-manifolds, obtaining a new description of a real spinor on such a manifold through a certain distribution of parabolic 2-planes in its cotangent bundle. We use this result to give global characterizations of real Killing spinors on Lorentzian four-manifolds and of four-dimensional supersymmetric configurations of heterotic supergravity. In particular, we find new families of Einstein and non-Einstein four-dimensional Lorentzian metrics admitting real Killing spinors, some of which are deformations of the metric of text {AdS}_4 space-time.
Highlights
Engineering, Bucharest, Romania references cited below) require the study of more general linear first-order partial differential equations for spinor fields
In order to do this, we assume that S is endowed with a fixed connection D : (S) → (T ∗ M ⊗ S) (which in practice will depend on various geometric structures on (M, g) relevant to the specific problem under consideration) and consider the equation: D =0 (1)
They occur in these physics theories through the notion of “supersymmetric configuration”, whose definition involves spinors parallel under a connection D on S which is parameterized by geometric structures typically defined on fiber bundles, gerbes or Courant algebroids associated to (M, g) [28,51,81]
Summary
One approach to the study of supergravity Killing spinor equations is the so-called “method of bilinears” [42,84,85], which was successfully applied in various cases to simplify the local partial differential equations characterizing certain supersymmetric configurations and solutions. We solve question (2) in the affirmative by reformulating constrained generalized Killing spinor equations on a spacetime (M, g) of such signatures ( p, q) as an equivalent system of algebraic and partial differential equations for the square polyform. The sign-equivalence class of a nowhere-vanishing spinor ∈ (S) determines and is determined by a parabolic pair of one-forms (u, l) considered up to transformations of the form (u, l) → (−u, l) and l → l + cu with c ∈ R We use this result to characterize spin Lorentzian four-manifolds (M, g) with H 1(M, g) =.
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