Spinors of real type as polyforms and the generalized Killing equation

We develop a new framework for the study of generalized Killing spinors, where generalized Killing spinor equations, possibly with constraints, can be formulated equivalently as systems of partial differential equations for a polyform satisfying algebraic relations in the K\"ahler-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 K\"ahler-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 AdS$_4$ space-time.

1. Introduction 1.1. Background and context. Let (M, g) be a pseudo-Riemannian manifold of signature (p, q), equipped with a bundle of irreducible real Clifford modules S. If (M, g) admits a spin structure, then S carries a canonical connection ∇ S which lifts the Levi-Civita connection of g. This allows one to define the notions of parallel and Killing spinors, both of which were studied extensively in the literature [1][2][3][4][5]. Developments in supergravity and differential geometry (see references cited below) require the study of more general linear first-order partial differential equations for spinor fields. It is therefore convenient to develop a general framework which subsumes all such spinorial equations as special cases. In order to do this, we assume that S is endowed with a fixed connection D : Γ(S) → Γ(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: (1) for a real spinor ǫ ∈ Γ(S). Solutions to this equation are called generalized Killing spinors with respect to D or simply D-parallel spinors on (M, g). We also consider linear constraints of the form: where Q ∈ Γ(Hom(S, W ⊗ S)), with W a vector bundle defined on M . Solutions ǫ ∈ Γ(M ) of the system of equations (1) and (2) are called constrained generalized Killing spinors on (M, g).
Generalized Killing spinors play a fundamental role in supergravity and string theory [22][23][24]. 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) [25][26][27]. This produces the notion of supergravity Killing spinor equations -particular instances of (systems of) constrained generalized Killing spinor equations which are specific to the physics theory under consideration. Pseudo-Riemannian manifolds endowed with parameterizing geometric structures for which such equations admit non-trivial solutions are called supersymmetric configurations. They are called supersymmetric solutions if they also satisfy the equations of motion of the given supergravity theory. The study of supergravity Killing spinor equations was pioneered by P. K. Tod [23,24] and later developed systematically in several references, including [28][29][30][31][32][33][34][35][36][37][38][39][40][41][42]. The study of supersymmetric solutions of supergravity theories provided an enormous boost to the subject of generalized Killing spinors and to spinorial geometry as a whole, which resulted in a large body of literature both in physics and mathematics, the latter of which is largely dedicated to the case of Euclidean signature in higher dimensional theories. We refer the reader to [11,26,27,[43][44][45] and references therein for more details and exhaustive lists of references.
Supergravity Killing spinor equations pose a number of new challenges when compared to simpler spinorial equations traditionally considered in the mathematics literature. First, supergravity Killing spinor equations must be studied for various theories and in various dimensions and signatures (usually Riemannian and Lorentzian), for real as well as complex spinors. In particular, this means that every single case in the modulo eight classification of real Clifford algebras must be considered, adding a layer of complexity to the problem. Second, such equations involve spinors parallel under non-canonical connections coupled to several other objects such as connections on gerbes, principal bundles or maps from the underlying manifold into a Riemannian manifold of special type. These objects, together with the underlying pseudo-Riemannian metric, must be treated as parameters of the supergravity Killing spinor equations, yielding a highly nontrivial non-linearly coupled system. Moreover, the formulation of supergravity theories relies on the Dirac-Penrose 1 rather than on the Cartan approach to spinors. As a result, spinors appearing in such theories need not be associated to a spin structure or other a priory classical spinorial structure but involve the more general concept of a (real or complex) Lipschitz structure (see [46][47][48][49]). The latter naturally incorporates the 'R-symmetry' group of the theory and is especially well-adapted for geometric formulations of supergravity. Third, applications require the study of the moduli space of supersymmetric solutions of supergravity theories, involving the metric and all other geometric objects entering their formulation. This set-up yields remarkably nontrivial moduli problems for which the automorphism group(oid) of the system is substantially more complicated than the more familiar infinite-dimensional gauge group of automorphisms of a principal bundle or the diffeomorphism group of a compact manifold. Given these aspects, the study of supergravity Killing spinor equations and of moduli spaces of supersymmetric solutions of supergravity theories requires methods 1 When constructing such theories, one views spinors as sections of given bundles of Clifford modules. The existence of such bundles on the given space-time is postulated when writing down the theory, rather than deduced through the associated bundle construction from a specific classical spinorial structure assumed on to exist on that spacetime. and techniques specifically dedicated to their understanding [32, 36-38, 45, 47-54]. Developing such methods in a systematic manner is one of the goals of this article.

1.2.
Main results. One approach to the study of supergravity Killing spinor equations is the socalled "method of bilinears" [23,24,32], which was successfully applied in various cases to simplify the local partial differential equations characterizing certain supersymmetric configurations and solutions. The idea behind this method is to consider the polyform constructed by taking the 'square' of the Killing spinor (instead of the spinor itself) and use the corresponding constrained generalized Killing spinor equations to extract a system of algebraic and partial differential equations for this polyform, thus producing necessary conditions for a constrained generalized Killing spinor to exist on (M, g). These conditions can also be exploited to obtain information on the structure of supersymmetric solutions of the supergravity theory at hand. The main goal of the present work is to develop a framework inspired by these ideas aimed at investigating constrained generalized Killing spinors on pseudo-Riemannian manifolds by constructing a mathematical equivalence between real spinors and their polyform squares.
Whereas the fact that the 'square of a spinor' [55][56][57] (see Definition 3.16 in Section 3) yields a polyform has been known for a long time (and the square of certain spinors with particularly nice stabilizers is well-known in specific -usually Riemannian -cases [55]), a proper mathematical theory to systematically characterize and compute spinor squares in every dimension and signature has been lacking so far. In this context, the fundamental questions to be addressed are 2 : (1) What are the necessary and sufficient conditions for a polyform to be the square of a spinor, in every dimension and signature? (2) Can we (explicitly, if possible) translate constrained generalized Killing spinor equations into equivalent algebraic and partial differential equations for the square polyform? In this work, we solve both questions for irreducible real spinors when the signature (p, q) of the underlying pseudo-Riemannian manifold satisfies p − q ≡ 8 0, 2, i.e. when the corresponding Clifford algebra is simple and of real type. We solve question (1) by fully characterizing the space of polyforms which are the (signed) square of spinors as the set of solutions of a system of algebraic equations which define a real affine variety in the space of polyforms. Every polyform solving this algebraic system can be written as the square of a real spinor which is determined up to a sign factor -and vice-versa. Following [36][37][38], the aforementioned algebraic system can be neatly written using the geometric product. The latter quantizes the wedge product, thereby deforming the exterior algebra to a unital associative algebra which is isomorphic to the Clifford algebra. This algebraic system can be considerably more complicated in indefinite signature than in the Euclidean case. On the other hand, 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. Altogether, this produces an equivalent reformulation of the constrained generalized Killing spinor problem as a more transparent and easier to handle system of partial differential equations for a polyform satisfying certain algebraic equations in the Kähler-Atiyah bundle of (M, g). We believe that the framework developed in this paper is especially useful in pseudo-Riemannian signature and in higher dimensions, where the spin group does not act transitively on the unit sphere in spinor space and hence representation theory cannot be easily exploited to understand the square of a spinor in purely representation theoretic terms. One of our main results (see Theorem 4.26 for details and notation) is: Theorem 1.1. Let (M, g) be a connected, oriented and strongly spin pseudo-Riemannian manifold of signature (p, q) and dimension d = p + q, such that p − q ≡ 8 0, 2. Let W be a vector bundle on M and (S, Γ, B) be a paired real spinor bundle on (M, g) whose admissible pairing has symmetry and adjoint types σ, s ∈ {−1, 1}. Fix a connection D = ∇ S − A on S (where A ∈ Ω 1 (M, End(S))) and a 2 A systematic approach of this type was first used in references [40,41] for generalized Killing spinor equations in certain 8-dimensional flux compactifications of M-theory, using the Kähler-Atiyah bundle approach to such problems developed previously in [36][37][38]. morphism of vector bundles Q ∈ Γ(End(S) ⊗ W). Then there exists a nontrivial generalized Killing spinor ǫ ∈ Γ(S) with respect to the connection D which also satisfies the linear constraint Q(ǫ) = 0 iff there exists a nowhere-vanishing polyform α ∈ Ω(M ) which satisfies the following algebraic and differential equations: for every polyform β ∈ Ω(M ), whereÂ ∈ Ω 1 (M, ∧T * M ) andQ ∈ Γ(∧T * M ⊗ W) are the symbols of A and Q while π, τ are the canonical automorphism and anti-automorphism of the Kähler-Atiyah bundle (∧T * M, ⋄) of (M, g). If ǫ is chiral of chirality µ ∈ {−1, 1}, then we have to add the condition: * (π • τ )(α) = µ α , where * is the Hodge operator of (M, g). Moreover, any such polyform α determines a nowherevanishing real spinor ǫ ∈ Γ(S), which is unique up to a sign and satisfies the constrained generalized Killing spinor equations with respect to D and Q.
When (M, g) is a Lorenzian four-manifold, we say that a pair of nowhere-vanishing one-forms (u, l) defined on M is parabolic if u and l are mutually orthogonal with u null and l spacelike of unit norm. Applying the previous result, we obtain (see Theorem 4.32 and Section 4.6 for detail and terminology): Theorem 1.2. Let (M, g) be a connected and spin Lorentzian four-manifold of "mostly plus" signature such that H 1 (M, Z 2 ) = 0 and S be a real spinor bundle associated to the spin structure of (M, g) (which is unique up to isomorphism). Then there exists a natural bijection between the set of global smooth sections of the projective bundle P(S) and the set of trivializable and co-oriented distributions (Π, H) of parabolic 2-planes in T * M . Moreover, there exist natural bijections between the following two sets: (a) The set Γ(Ṡ)/Z 2 of sign-equivalence classes of nowhere-vanishing real spinors ǫ ∈ Γ(S).
(b) The set of strong equivalence classes of parabolic pairs of one-forms (u, l) ∈ P(M, g).
In particular, the sign-equivalence class of a nowhere-vanishing spinor ǫ ∈ Γ(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) = 0 which admit real Killing spinors and supersymmetric bosonic heterotic configurations associated to "paired principal bundles" (P, c) over such a manifold through systems of partial differential equations for u and l, which we explore in specific cases. Taking (M, g) to be of signature (3, 1), we prove the following results (see Theorems 5.3 and 6.6), where * and d * denote the Hodge operator and codifferential of (M, g) while ∇ g denotes the action of its Levi-Civita on covariant tensors: Theorem 1.3. (M, g) admits a nontrivial real Killing spinor with Killing constant λ 2 ∈ R iff it admits a parabolic pair of one-forms (u, l) which satisfies: for some κ ∈ Ω 1 (M ). In this case, u ♯ ∈ X(M ) is a Killing vector field with geodesic integral curves. Theorem 1.4. A bosonic heterotic configuration (g, ϕ, H, A) of (M, P, c) is supersymmetric iff there exists a parabolic pair of one-forms (u, l) which satisfies (here ρ def. = * H ∈ Ω 1 (M )): for some one-form κ ∈ Ω 1 (M ) and some g P -valued one-form χ A ∈ Ω 1 (M, g P ) which is orthogonal to u. In this case, u ♯ ∈ X(M ) is a Killing vector field.
Let H be the Poincaré half plane with coordinates x ∈ R, y ∈ R >0 . Using the last result above, we show (see Subsection 5.4) that the following one-parameter family of metrics defined on R 2 × H : (where R 2 has Cartesian coordinates x v , x u ) admits real Killing spinors for every F ∈ C ∞ (H). We also show that these metrics are Einstein with Einstein constant Λ = −3λ 2 when F has the form: , where BY and BJ are the spherical Bessel functions and a 1 , . . . , a 4 ∈ R. These give deformations of the AdS 4 spacetime of cosmological constant Λ, which obtains for a 1 = a 2 = a 3 = a 4 = 0.
1.3. Open problems and further directions. Theorem 1.1 refers exclusively to real spinors in signature p − q ≡ 8 0, 2, for which the irreducible Clifford representation map is an isomorphism. It would be interesting to extend this result to the remaining signatures, which encompass two main cases, namely real spinors of complex and quaternionic type (the latter of which can be reducible). This would yield a rich reformulation of the theory of real spinors through polyforms subject to algebraic constraints, which could be used to study generalized Killing spinors in all dimensions and signatures. It would also be interesting to extend Theorem 1.1 to other types of spinorial equations, such as those characterizing harmonic or twistor spinors and generalizations thereof, investigating if it is possible to develop an equivalent theory exclusively in terms of polyforms.
Some important signatures satisfy the condition p − q ≡ 8 0, 2, most notably signatures (2, 0), (1, 1), (3, 1) and (9, 1). The latter two are especially relevant to supergravity theories and one could apply the formalism developed in this article to study moduli spaces of supersymmetric solutions in these cases. Several open problems of analytic, geometric and topological type exist regarding the heterotic system in four and ten Lorentzian dimensions, as the mathematical study of its Riemannian analogue shows [45]. Most problems related to existence, classification, construction of examples and moduli are open and give rise to interesting analytic and geometric questions on Lorentzian four-manifolds and ten-manifolds. In this direction, we hope that Appendix B can serve as a brief introduction to heterotic supergravity in four Lorentzian dimensions for mathematicians who may be interested in such questions. The local structure of ten-dimensional supersymmetric solutions to heterotic supergravity was explored in [58][59][60], where that problem was reduced to a minimal set of partial differential equations on a local Lorentzian manifold of special type.
1.4. Outline of the paper. Section 2 gives the description of rank-one endomorphisms of a vector space which are (anti-)symmetric with respect to a non-degenerate bilinear pairing assumed to be symmetric or skew-symmetric. Section 3 develops the algebraic theory of the square of a spinor culminating in Theorem 3.20, which characterizes it through a system of algebraic conditions in the Kähler-Atiyah algebra of the underlying quadratic vector space. In Section 4, we apply this to real spinors on pseudo-Riemannian manifolds of signature (p, q) satisfying p − q ≡ 8 0, 2, obtaining a complete characterization of generalized constrained Killing spinors as polyforms satisfying algebraic and partial differential equations which we list explicitly. Section 5 applies this theory to real Killing spinors in four Lorentzian dimensions, obtaining a new global characterization of such. In Section 6, we apply the same theory to the study of supersymmetric configurations of heterotic supergravity, whose mathematical formulation is explained briefly in an appendix.
Notations and conventions. Throughout the paper, we use Einstein summation over repeated indices. We let R × denote the group of invertible elements of any commutative ring R. In particular, the multiplicative group of non-zero real numbers is denoted by R × = R. For any positive integer n, the symbol ≡ n denotes the equivalence relation of congruence of integers modulo n, while Z n = Z/nZ denotes the corresponding congruence group. All manifolds considered in the paper are assumed to be smooth, connected and paracompact, while all fiber bundles are smooth. The set of globallydefined smooth sections of any fiber bundle F defined on a manifold M is denoted by Γ(F ). We denote by G 0 the connected component of the identity of any Lie group G. Given a vector bundle S on a manifold M , the dual vector bundle is denoted by S * while the bundle of endomorphisms is denoted by End(S) ≃ S * ⊗ S. The trivial real line bundle on M is denoted by R M . The space of globally-defined smooth sections of S is denoted by Γ(S), while the set of those globally-defined smooth sections of S which do not vanish anywhere on M is denoted by · Γ(S). The complement of the origin in any R-vector space Σ is denoted byΣ while the complement of the image 0 S of the zero section of a vector bundle S defined on a manifold M is denoted byṠ. The inclusion · Γ(S) ⊂ · Γ(S) is generally strict. If A is any subset of the total space of S, we define: Notice the relation · Γ(S) = Γ(Ṡ). All pseudo-Riemannian manifolds (M, g) are assumed to have dimension at least two and signature (p, q) satisfying p − q ≡ 8 0, 2; in particular, all Lorentzian four-manifolds have "mostly plus" signature (3,1). For any pseudo-Riemannian manifold (M, g), we denote by , g the (generally indefinite) metric induced by g on the total exterior bundle We denote by ∇ g the Levi-Civita connection of g and use the same symbol for its action on tensors. The equivalence class of an element ξ of an R-vector space Σ under the sign action of Z 2 on Σ is denoted byξ ∈ Σ/Z 2 and called the sign equivalence class of ξ.
Acknowledgements. The work of C. I. L. was supported by grant IBS-R003-S1. The work of C. S. S. is supported by the Humboldt Research Fellowship ESP 1186058 HFST-P from the Alexander von Humboldt Foundation. The research of V.C. and C.S.S. was partially funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy -EXC 2121 Quantum Universe -390833306.

Representing real vectors as endomorphisms in a paired vector space
Let Σ be an R-vector space of positive even dimension N ≥ 2, equipped with a non-degenerate bilinear pairing B : Σ × Σ → R, which we assume to be either symmetric or skew-symmetric. In this situation, the pair (Σ, B) is called a paired vector space. We say that B (or (Σ, B)) has symmetry type σ ∈ {−1, 1} if: B(ξ 1 , ξ 2 ) = σB(ξ 2 , ξ 1 ) ∀ ξ 1 , ξ 2 ∈ Σ . Thus B is symmetric if it has symmetry type +1 and skew-symmetric if it has symmetry type −1. Let (End(Σ), •) be the unital associative R-algebra of linear endomorphisms of Σ, where • denotes composition of linear maps. Given E ∈ End(Σ), let E t ∈ End(Σ) denote the adjoint of E taken with respect to B, which is uniquely determined by the condition: The map E → E t is a unital anti-automorphism of the R-algebra (End(Σ), •).

2.1.
Tame endomorphisms and the squaring maps.
Thus E is tame iff it vanishes or is of unit rank. Let: be the real determinantal variety of tame endomorphisms of Σ and: be its open subset consisting of endomorphisms of rank one. We view T as a real affine variety of dimension 2N − 1 in the vector space End(Σ) ≃ R N 2 andṪ as a semi-algebraic variety. Elements of T can be written as: for some ξ ∈ Σ and β ∈ Σ * , where Σ * = Hom(Σ, R) denotes the vector space dual to Σ. Notice that tr(E) = β(ξ). When E ∈ T is non-zero, the vector ξ and the linear functional β appearing in the relation above are non-zero and determined by E up to transformations of the form (ξ, β) → (λξ, λ −1 β) with λ ∈ R × . In particular,Ṫ is a manifold diffeomorphic with the quotient where R × acts with weights +1 and −1 on the two copies of R N \ {0}.

Admissible endomorphisms.
The maps E ± need not be surjective. To characterize their images, we introduce the notion of admissible endomorphism. Let (Σ, B) be a paired vector space of type σ. Let: denote the real cone of B-admissible endomorphisms of Σ.
Remark 2.5. Tame endomorphisms are not related to admissible endomorphisms in any simple way. A tame endomorphism need not be admissible, since it need not be (anti-)symmetric with respect to B. An admissible endomorphism need not be tame, since it can have rank larger than one (as shown by a quick inspection of explicit examples in four dimensions). Let: denote the real cone of those endomorphisms of E which are both tame and B-admissible and consider the open setŻ def.
Lemma 2.6. We have: Hence an endomorphism E ∈ End(Σ) belongs toŻ iff there exists a non-zero vector ξ ∈Σ and a sign factor κ ∈ {−1, 1} such that: Moreover, κ is uniquely determined by E through this equation while ξ is determined up to sign.

Combining the two inclusions above gives
Definition 2.7. The signature κ E ∈ {−1, 1} of an element E ∈Ż with respect to B is the sign factor κ determined by E as in Lemma 2.6. When E = 0, we set κ E = 0.
In view of the above, define: = Im(E ± ) . Then: Let Z 2 ≃ {−1, 1} act on Σ and on Z ⊂ End(E) by sign multiplication. Then E + and E − induce the same map between the quotients (which is a bijection by Lemma 2.6). We denote this map by: Definition 2.9. The bijection (8) is called the class squaring map of (Σ, B).

2.3.
The manifoldŻ and the projective squaring map. Given any endomorphism A ∈ End(Σ), define a (possibly degenerate) bilinear pairing B A on Σ through: Notice that B A is symmetric iff A t = σA and skew-symmetric iff A t = −σA.
Proof. Let | · | 2 0 denotes the norm induced by any scalar product on Σ. Then the diffeomorphism: ) satisfies the desired properties.
The mapsĖ ± :Σ → End(Σ) \ {0} induce the same map:    (9) is non-degenerate iff the endomorphism A ∈ End(Σ) is invertible. The following result is immediate: Proposition 2.14. Let B ′ be a non-degenerate symmetric pairing on Σ. Then there exists a unique endomorphism A ∈ GL(E) (called the operator of B ′ with respect to B) such that B ′ = B A . Moreover, A is invertible and satisfies A t = σA. Furthermore, the transpose E T of any endomorphism E ∈ End(Σ) with respect to B ′ is given by: and in particular we have A T = A t = σA.
Let A ∈ End(Σ) be a taming of B and denote by (−, −) def.
= B A the corresponding scalar product. Relation (11) shows that the matrixB of B with respect to a ( , )-orthonormal basis {e 1 , . . . , e N } of Σ is the inverse of the matrixÂ of A in the the same basis. Distinguish the cases: 2. When B is skew-symmetric, we have: where J is a (−, −)-compatible complex structure on Σ. This gives A −1 = −J and hence A = J, which is antisymmetric with respect to both (−, −) and B. Setting N = 2n, we can choose {e 1 , . . . , e n } to be a basis of Σ over C which is orthonormal with respect to the Hermitian scalar product defined by (−, −) and J and take e n+i = Je i for all i = 1, . . . , n. Then the basis {e 1 , . . . , e N } over R is (−, −)-orthonormal while being a Darboux basis for B and we have: where I n is the identity matrix of size n. = E • A ∈ End(Σ) is B ′ -admissible iff the following relations hold: Proof. Let T denote transposition of endomorphisms with respect to B ′ . By definition, E A is B ′ -admissible if: Since A is invertible, the second of these conditions amounts to the second relation in (12). On the other hand, we have: where we used Proposition 2.14. Hence the first condition in (13) is equivalent with σE t A = EA, which in turn amounts to E t = σE since A is invertible. = E tr(E) . Then P 2 = P (which implies rk(P ) = tr(P )) and tr(P ) = 1, whence rk(E) = rk(P ) = tr(P ) = 1.
where µ ∈ {−1, +1}. When B is symmetric, the set K 0 ⊂ Σ is the isotropic cone of B and K µ are the positive and negative unit "pseudo-spheres" defined by B. When B is skew-symmetric, we have K 0 = Σ and K µ = ∅. Lemma 2.6 and Proposition 2.18 imply: Corollary 2.19. Assume that B is symmetric (i.e. σ = +1). For any µ ∈ {−1, 1}, the set E + (K µ ) ∪ E − (K µ ) is the real algebraic submanifold of End(Σ) given by: Proposition 2.20. If B is a scalar product, then every non-zero B-admissible endomorphism E ∈ C \ {0} is tame, whence Z = C. In this case, the signature of E with respect to B is given by: (14) κ E = sign(tr(E)) .
Proof. Let E ∈ C. By Proposition 2.18, the first statement follows if we can show that tr(E) = 0 when E = 0. Since E is admissible, it is symmetric with respect to the scalar product B and hence diagonalizable with eigenvalues λ 1 , . . . , λ N ∈ R. Taking the trace of equation E 2 = tr(E) E gives: Since the right-hand side is a sum of squares, it vanishes iff λ 1 = . . . = λ N = 0, i.e. iff E = 0. This proves the first statement. To prove the second statement, recall from Lemma 2.6 that any non-zero tame admissible endomorphism E has the form E = κ E ξ ⊗ ξ * for some ξ ∈Σ. Taking the trace of this relation gives: which implies (14) since B(ξ, ξ) > 0.
A quick inspection of examples shows that there exist nontrivial admissible endomorphisms which are not tame (and thus satisfy tr(E) = 0) as soon as there exists a totally isotropic subspace of Σ of dimension at least two. In these cases we need to impose further conditions on the elements of C in order to guarantee tameness. To describe such conditions, we consider the more general equation: which is automatically satisfied by every E ∈ Im(E + ) ∪ Im(E − ).
Proposition 2.21. The following statements are equivalent for any endomorphism E ∈ C which satisfies the condition E t = σE: (a) E is B-admissible and rk(E) = 1.
(b) We have E 2 = tr(E)E and there exists an endomorphism A ∈ End(Σ) satisfying: (c) E = 0 and the relation: is holds for every endomorphism A ∈ End(Σ).
Proof. We first prove the implication (b) ⇒ (a). By Proposition 2.18, it suffices to consider the case tr(E) = 0. Assume A ∈ End(Σ) satisfies (15). Define: where ǫ ∈ R >0 is a positive constant. For ǫ > 0 small enough, A ǫ is invertible and the endomorphism E ǫ def.
= E • A ǫ has non-vanishing trace given by tr(E ǫ ) = ǫ. The first relation in (15) gives: Hence P def.
The implication (a) ⇒ (c) follows directly from Lemma 2.6, which shows that E ∈ Im(E κ ) for some sign factor κ. For the implication (c) ⇒ (b), notice first that setting A = Id in (16) gives E 2 = tr(E). Non-degeneracy of the bilinear form induced by the trace on the space End(Σ) now shows that we can choose A in equation (16) such that tr(E • A) = 0. In this case, there exists a non-zero vector ξ ∈ Σ such that: where ξ ∨ = σξ * • A denotes the linear functional dual to ξ with respect B ′ while: is the signature of E • A with respect to B ′ and: is the signature of E with respect to B.
To prove the implication (a) ⇒ (b), assume that E is B-admissible and of rank one. Then Proposition 2.21 shows that we have Since B ′ is a scalar product, Proposition 2.20 implies that E A is tame and hence tr(E A ) = 0 since E A is non-zero. Thus tr(E • A) = 0. Combining everything, this shows that (b) holds.
To prove the implication (b) ⇒ (a), assume that E satisfies E t = σE as well as (15). By Proposition 2.17, this implies that E A is B ′ -admissible. Since tr(E A ) = tr(E • A) = 0, Proposition 2.18 implies rk(E A ) = 1. Hence E A is nonzero, tame and admissible with respect to the scalar product B ′ . We Proposition 2.20) and ξ ∈ Σ is a non-zero vector. Here ξ ∨ ∈ Σ * is the dual of ξ with respect to B ′ , which is given by: = σ κ ′ ) because A is invertible. Thus E belongs to Im(E κ ) and hence is B-admissible and of rank one.
Remark 2.23. When A is a taming of B, Proposition 2.22 shows that conditions (15) and the condition E t = σE automatically imply E 2 = tr(E)A and hence tameness of E. In this case, Proposition 2.21 shows that E also satisfies E • B • E = tr(E • B)E for any B ∈ End(Σ). Corollary 2.24. Let A be a taming of B. Then the image of the projective squaring map PE is the real algebraic submanifold of P(End(Σ)) given by: Any vector ξ ∈ Σ expands as: For any S ∈ End(Σ), letŜ denote the matrix of S in the basis ∆.
Example 2.25. Let B be a scalar product on Σ having ∆ as an orthonormal basis. Then: and the relations E 2 ξ = tr(E ξ )E ξ and E t ξ = E ξ follow from this form. Let E ∈ End(Σ) satisfy E 2 = tr(E)E and E t = E. The second of these conditions implies: Condition E 2 = tr(E)E amounts to b 2 = k 1 k 2 , implying that k 1 and k 2 have the same sign unless at least one of them vanishes (in which case b must also vanish). Since E is B-symmetric (and hence diagonalizable), its trace tr(E) = k 1 + k 2 vanishes iff E = 0. Assume E = 0 and set: Then k 1 = κξ 2 1 , k 2 = κξ 2 2 and b = κξ 1 ξ 2 , showing that E = E ξ for some ξ ∈ Σ \ {0}. Hence conditions E 2 = tr(E)E and E = E t characterize endomorphisms of the form E ξ .  We have:Ê and the relations E 2 ξ = tr(E ξ )E ξ and E t ξ = E ξ follow directly from this form, where t denotes the adjoint taken with respect to B. Let E ∈ End(Σ) satisfy E t = E. Then: Direct computation shows that the conditions E 2 = tr(E)E and E • A • E = tr(E • A)E are equivalent to each other in this two-dimensional example and amount to the relation b 2 = −k 1 k 2 , which implies that E vanishes iff k 1 = k 2 . Let us assume that E = 0 and set: Then it is easy to see that k 1 = κξ 2 1 , k 2 = −κξ 2 2 and b = κξ 1 ξ 2 , which implies E = E ξ . In this example endomorphisms E that can be written in the form E ξ are characterized by the condition E t = E, together with either of the two equivalent conditions Notice that tr(E) = k 1 + k 2 can vanish in this case. However, in this low-dimensional example, the conditions E • E = tr(E)E and E t = E suffice to characterize endomorphisms of the form E ξ , including those which satisfy tr(E) = 0. The complex structure A of Σ with matrix given by: (e 1 , e 1 ) = (e 2 , e 2 ) = 1 , (e 1 , e 2 ) = (e 2 , e 1 ) = 0 .
We have: which implies E 2 ξ = 0 and E t ξ = −E ξ , where t denotes transposition with respect to B. Let E ∈ End(Σ) be an endomorphism satisfying E t = −E. This condition implies: Notice that tr(E) = 0. Direct computation shows that the conditions E 2 = 0 and E • A • E = tr(E • A)E are equivalent to each other in this two-dimensional example and amount to the relation k 2 = bc, which in particular shows that E vanishes iff b = −c. Assume that E = 0 and set: Then it is easy to see that b = κξ 2 1 , c = κξ 2 2 and k = κξ 1 ξ 2 , which shows that E = E ξ . Hence endomorphism which can be written in this form are characterized by the condition E t = −E together with either of the conditions which are equivalent to each other in this low-dimensional example.

2.7.
Including linear constraints. The following result will be used in Sections 3 and 4.
Proof. Take ξ ∈ Σ and assume Q(ξ) = 0. Then: Then the same calculation as before gives: Example 2.29. Let (Σ, B) be a two-dimensional Euclidean vector space with orthonormal basis ∆ as in Example 2.25. Let Q ∈ End(Σ) have matrix: in this basis. Given ξ ∈ Σ, Example 2.25 gives: 3. From real spinors to polyforms where h * denotes the metric dual to h. Let Cl(V * , h * ) be the real Clifford algebra of this dual quadratic space, viewed as a Z 2 -graded associative algebra with decomposition: In our conventions, the Clifford algebra satisfies (notice the sign !): Let π denote the standard automorphism of Cl(V * , h * ), which acts as minus the identity on V * ⊂ Cl(V * , h * ) and τ denote the standard anti-automorphism, which acts as the identity on V * ⊂ Cl(V * , h * ). These two commute and their composition is an anti-automorphism denoted byτ = π • τ = τ • π. Let Cl × (V * , h * ) denote the group of units Cl(V * , h * ). Its twisted adjoint action is the morphism of groups Ad : Cl × (V * , h * ) → Aut(Cl(V * , h * )) defined through: We denote by L (V * , h * ) ⊂ Cl(V * , h * ) the Clifford group, which is defined as follows: This fits into the short exact sequence: Recall that the pin and spin groups of (V * , h * ) are the subgroups of L (V * , h * ) defined through: where N : and Pin(V * , h * ) are disconnected; the first have two connected components while the last has four. The connected components of the identity in Spin(V * , h * ) and Pin(V * , h * ) coincide, being given by: Let Σ be a finite-dimensional R-vector space and γ : Cl(V * , h * ) → End(Σ) a Clifford representation. Then Spin(V * , h * ) acts on Σ through the restriction of γ and (20) induces the short exact sequence: which in turn gives the exact sequence: Here SO 0 (V * , h * ) denotes the connected component of the identity of the special orthogonal group SO(V * , h * ). In signatures p−q ≡ 8 0, 2 (the "real/normal simple case" of [38]), the algebra Cl(V * , h * ) is simple and isomorphic (as a unital associative R-algebra) to the algebra of square real matrices of size N = 2 d 2 . In such signatures Cl(V * , h * ) admits a unique irreducible real left module Σ, which has dimension N . This irreducible representation is faithful and surjective, hence in such signatures the representation map γ : Cl(V * , h * ) ≃ − → End(Σ) is an isomorphism of unital R-algebras. We will equip Σ with a non-degenerate bilinear pairing which is compatible with Clifford multiplication. Ideally, such compatibility should translate into invariance under the natural action of the pin group. However, this condition cannot be satisfied when if pq = 0. Instead, we consider the weaker notion of admissible bilinear pairing introduced in [57,61] (see [36][37][38] for applications to supergravity), which encodes the best compatibility condition with Clifford multiplication that can be imposed on a bilinear pairing on Σ in arbitrary dimension and signature. The following result of [56] summarizes the main properties of admissible bilinear pairings. Theorem 13.17] Suppose that h has signature p − q ≡ 8 0, 2. Then the irreducible real Clifford module Σ admits two non-degenerate bilinear pairings B + : Σ×Σ → R and B − : Σ×Σ → R (each determined up to multiplication by a non-zero real number) such that: for all x ∈ Cl(V * , h * ) and ξ 1 , ξ 2 ∈ Σ. The symmetry properties of B + and B − are as follows in terms of the modulo 4 reduction of k def.
In addition, if B s (with s ∈ {−1, 1}) is symmetric, then it is of split signature unless pq = 0, in which case B s is definite.
Definition 3.2. The sign factor s appearing in the previous theorem is called the adjoint type of B s , hence B + is of positive adjoint type (s = +1) and B − is of negative adjoint type (s = −1).
Relations (22) can be written as: where t denotes the B s -adjoint. The symmetry type of an admissible bilinear form B will be denoted by Notice that σ depends both on s and on the mod 4 reduction of d 2 .
for an appropriate non-zero real constant C. For specific applications, we will choose to work with B + or with B − depending on which admissible pairing yields the computationally simplest polyform associated to a given spinor ξ ∈ Σ. When pq = 0, we will take B s to be positive-definite (which we can always achieve by rescaling it with a non-zero constant of appropriate sign). See [38] for a useful discussion of properties of admissible pairings in various dimensions and signatures.
Remark 3.5. Directly from their definition, the pairings B s satisfy: This relation yields: for all x = θ 1 · θ 2 with h * -orthogonal θ 1 , θ 2 ∈ V * . This implies that B s is invariant under the action of Spin(V * , h * ) and hence also under Spin 0 (V * , h * ). If h is positive-definite, then B + is Pin(V * , h * )-invariant, since it satisfies: For completeness, let us give an explicit construction of B + and B − . Pick an h * -orthonormal basis e i i=1,...,d of V * and let: be the finite multiplicative subgroup of Cl(V * , h * ) generated by the elements ±e i . Averaging over K( e i ), we construct an auxiliary positive-definite inner product (−, −) on Σ which is invariant under the action of this group. This product satisfies: Fix an orientation on V * + (which induces a unique orientation on V * − compatible with the orientation of V * induced from that of V ) and denote by ν + and ν − the corresponding pseudo-Riemannian volume forms. We have ν = ν + ∧ ν − . For p (and hence q) odd, define: whereas for p (and hence q) even, set: Then B ± are admissible pairings in the sense of Theorem 3.1. Direct computation using equations (25) and (26) gives the following result, which fixes the constant C appearing in Remark 3.4.
Proposition 3.6. The admissible pairings B + and B − constructed above are related as follows: Thus we can normalize B ± such that the constant in (24) is given by Proof. We have to show the relation:

Consider orthonormal basis
A simple computation using relation (23) shows that (28) holds To identify spinors with polyforms, we will use an explicit realization of Cl(V * , h * ) as a deformation of the exterior ∧V * . This model (which goes back to the work of Chevalley and Riesz [62][63][64]) has an interpretation as a deformation quantization of the odd symplectic vector space obtained by parity change from the quadratic space (V, h) (see [65,66]). It can be constructed using the symbol map and its inverse, the quantization map. Consider first the linear map f : V * → End(∧V * ) given by: We have: By the universal property of Clifford algebras, it follows that f extends uniquely to a morphism f : Definition 3.8. The symbol (or Chevalley-Riesz) map is the linear map l : Cl(V * , h * ) → ∧V * defined through: The symbol map is an isomorphism of filtered vector spaces. We have: As expected, l is not a morphism of algebras. The inverse: of l (called the quantization map) allows one to view Cl(V * , h * ) as a deformation of the exterior algebra (∧V * , ∧) (see [65,66]). Using l and Ψ, we transport the algebra product of Cl(V * , h * ) to an h-dependent unital associative product defined on ∧V * , which deforms the wedge product.
Let T (V * ) denote the tensor algebra of the (parity change of) V * , viewed as a Z-graded associative superalgebra whose Z 2 -grading is the reduction of the natural Z-grading; thus elements of V have integer degree one and they are odd. Let: denote the Z-graded Lie superalgebra of all superderivations. The minus one integer degree component Der −1 (T (V * )) is linearly isomorphic with the space Hom(V * , R) = V acting by contractions: while the zero integer degree component Der 0 (T (V * )) = End(V * ) = gl(V * ) acts through: We have an isomorphism of super-Lie algebras: The action of this super Lie algebra preserves the ideal used to define the exterior algebra as a quotient of T (V * ) and hence descends to a morphism of super Lie algebras L Λ : Contractions also preserve the ideal used to define the Clifford algebra as a quotient of T (V * ). On the other hand, endomorphisms Together with contractions, they induce a morphism of super Lie algebras L Cl : The following result states that L Λ and L Cl are compatible with l and Ψ.
3 Notice that the geometric product is not compatible with the grading of ∧V * given by form rank, but only with its mod 2 reduction, because the quantization map does not preserve Z-gradings. Hence the Kähler-Atiyah algebra is not isomorphic with Cl(V * , h * ) in the category of Clifford algebras defined in [48]. As such, it provides a different viewpoint on spin geometry, which is particularly useful for our purpose (see [36][37][38][39]).
This proposition shows that quantization is equivariant with respect to affine orthogonal transformations of (V * , h * ). In signatures p − q ≡ 8 0, 2, composing Ψ with the irreducible representation γ : Cl(V * , h * ) → End(Σ) (which in such signatures is a unital isomorphism of algebras) gives an isomorphism of unital associative R-algebras 4 : Since Ψ γ is an isomorphism of algebras and (∧V * , ⋄) is generated by V * , the identity together with Remark 3.11. We sometimes denote the action of a polyform α ∈ ∧V * as an endomorphism on Σ by a dot (this corresponds to Clifford multiplication through the isomorphism Ψ γ ): The trace on End(Σ) transfers to the Kähler-Atiyah algebra through the map Ψ γ (see [38]): Definition 3.12. The Kähler-Atiyah trace is the linear functional: We will see in a moment that S does not depend on γ or h. Since Ψ γ is a unital morphism of algebras, we have: Lemma 3.13. For any 0 < k ≤ d, we have: Proof. Let e i i=1,...,d be an orthonormal basis of (V * , h * ). For i = j we have e i ⋄ e j = −e j ⋄ e i and hence (e i ) −1 ⋄ e j ⋄ e i = −e j . Let 0 < k ≤ d and 1 ≤ i 1 < · · · < i k ≤ d. If k is even, then: and hence S(e i1 ⋄ · · · ⋄ e i k ) = 0. Here we used cyclicity of the Kähler-Atiyah trace and the fact that e i k anticommutes with e i1 , . . . , e i k−1 . If k is odd, let j ∈ {1, . . . , d} be such that j ∈ {i 1 , . . . , i k } (such a j exists since k < d). We have: and we conclude.
Let α (k) ∈ ∧ k V * denote the degree k component of α ∈ ∧V * . Lemma 3.13 implies: Proposition 3.14. The Kähler-Atiyah trace is given by: In particular, S does not depend on the irreducible representation γ of Cl(V * , h * ) or on h. Lemma 3.15. Let α ∈ ∧V * and B be an admissible bilinear pairing of (Σ, γ) of adjoint type s ∈ {−1, 1}. Then the following equation holds: where Ψ γ (α) t is the B-adjoint of Ψ γ (α) and we defined the s-transpose of α through: Proof. Follows immediately from (23) and relations (29).

Spinor squaring maps.
Definition 3.16. Let Σ = (Σ, γ, B) be a paired simple Clifford module for (V * , h * ). The signed spinor squaring maps of Σ are the quadratic maps: are the signed squaring maps of the paired vector space (Σ, B) which were defined in Section 2. Given a spinor ξ ∈ Σ, the polyforms E + Σ (ξ) and E − Σ (ξ) = −E + Σ (ξ) are called the positive and negative squares of ξ relative to the admissible pairing B.
. Consider the following subsets of ∧V * : Σ, B)) . Since Ψ γ is a linear isomorphism, Section 2 implies that E ± Σ are two-to-one except at 0 ∈ Σ and: Moreover, E ± Σ induce the same bijective map: Notice that Z is a cone in ∧V * , which is the union of the opposite half cones Z ± .
We will sometimes denote by α ξ def.
This does not hold in other signatures, for which the construction of spinor squaring maps is more delicate (see [38]).
The following result is a direct consequence of Proposition 3.10.
We are ready to give the algebraic characterization of spinors in terms of polyforms.
Theorem 3.20. Let Σ = (Σ, γ, B) be a paired simple Clifford module for (V * , h * ) of symmetry type σ and adjoint type s. Then the following statements are equivalent for a polyform α ∈ ∧V * : (a) α is a signed square of some spinor ξ ∈ Σ, i.e. it lies in the set Z(Σ). (b) α satisfies the following relations: The following relations hold: for any polyform β ∈ ∧V * . In particular, the set Z(Σ) depends only on σ, s and (V * , h * ).
In view of this result, we will also denote Z(Σ) by Z σ,s (V * , h * ).
Remark 3.23. We have: where {e i } i=1,...,d is the contragradient orthonormal basis of (V, h). For simplicity, set: Then the degree one component in (36) reads: and its dual vector field (α (1) ) ♯ = κ Proof. It is easy to see that the set: gives an orthogonal basis of End(Σ) with respect to the nondegenerate and symmetric bilinear pairing induced by the trace: In particular, the endomorphism E def.
The following shows that the choice of admissible pairing used to construct the spinor square map is a matter of taste, see also Remark 3.4.
Proposition 3.25. Let ξ ∈ Σ and denote by α ± ξ ∈ Z + the positive polyform squares of ξ relative to the admissible pairings B + and B − of (Σ, γ), which we assume to be normalized such that they are related through (27). Then the following relation holds: where c : ∧ V * → ∧V * is the linear map which acts as multiplication by k! (d−k)! in each degree k.
3.4. Linear constraints. The following result will be used in Sections 3 and 4.

3.5.
Real chiral spinors. Theorem 3.20 can be refined for chiral spinors of real type, which exist in signature p − q ≡ 8 0. In this case, the Clifford volume form ν ∈ Cl(V * , h * ) squares to 1 and lies in the center of Cl ev (V * , h * ), giving a decomposition as a direct sum of simple associative algebras: We decompose Σ accordingly: The subspaces Σ (±) ⊂ Σ are preserved by the restriction of γ to Cl ev (V * , h * ), which therefore decomposes as a sum of two irreducible representations: distinguished by the value which they take on the volume form ν ∈ Cl ev (V * , h * ): , Proposition 3.26 shows that this amounts to the condition: For any µ ∈ {−1, 1} and κ ∈ {−1, 1}, define: κ (Σ) (which are two to one except at the origin). In turn, the latter induce bijectionsÊ for a fixed polyform β ∈ ∧V * which satisfies S(α ⋄ β) = 0. (c) The following conditions are satisfied: for every polyform β ∈ ∧V * . In this case, the real chiral spinor of chirality µ which corresponds to α through the either of the maps E In particular, Z (µ) (Σ) depends only on σ, s and (V * , h * ) and will also be denoted by Z then we have α (k) = 0 and α (d−k) = 0.
Proof. Follows immediately from Corollary 3.21 and the second relation in (38).
3.6. Low-dimensional examples. Let us describe Z and Z (µ) for some low-dimensional cases.
Notice that the nullity condition on α (1) is equivalent with (anti-)selfduality.
3.6.3. Signature (3, 1). This case is relevant for supergravity applications and will arise in Sections 5 and 6. Let (V * , h * ) be a Minkowski space of "mostly plus" signature (3, 1). Its irreducible Clifford module (Σ, γ) is four-dimensional and both admissible pairings B ± are skew-symmetric. We work with the admissible pairing B = B − of negative adjoint type.
Let P(V * , h * ) denote the set of parabolic pairs of one-forms. The binary relations defined above are equivalence relations on this set; moreover, strong equivalence implies equivalence, which in turn implies weak equivalence.
Recall that a 2-plane Π in V * is called parabolic (with respect to h * ) if the restriction h * Π of h * to Π has one-dimensional kernel. This happens iff Π is tangent to the light cone of the Minkowski space (V * , h * ) along a null line. This line coincides with K h (Π) def.
= ker(h * Π ) and is called the null line of Π. If Π ⊂ V * is a parabolic 2-plane, then any element of Π which does not belong to K h (Π) is spacelike. The two connected components of the complement Π \ K h (Π) are the spacelike half-planes of Π. An orientation of the null line K h (Π) is called a time orientation of Π, while an orientation of the quotient line Π/K h (Π) is called a co-orientation of Π. Notice that a co-orientation of Π amounts to a choice H of one of the spacelike half-spaces of Π. A co-oriented parabolic 2-plane in V * is a pair (Π, H), where Π is a parabolic two-plane in V * and H is a co-orientation of Π. The set of spacelike unit norm elements of Π has two connected components, each of which is an affine line parallel to K h (Π). These two affine lines are related by the inversion of Π with respect to the origin. Notice that a co-orientation H of Π amounts to a choice L of one of these two affine lines. Namely, we associate to L that spacelike half-plane H L of Π which contains L. Given u ∈ K h (Π) \ {0}, a unit norm spacelike element l ∈ Π is determined up to transformations of the form l → ζl + cu, where ζ ∈ {−1, 1} and c ∈ R.
Remark 3.31. Parabolic 2-planes correspond to degenerate complete flags in (V * , h * ) (see Appendix A). Notice that a parabolic 2-plane Π determines a short exact sequence of vector spaces: with K = K h (Π) and N = Π/K and induces a scalar product on the quotient line N . Conversely, giving a "parabolic" metric on a 2-plane Π amounts to giving a short exact sequence of this form together with a scalar product on N . A time orientation of Π is orientation of K while a coorientation is an orientation of N . Since the determinant line of Π is given by det(Π) = ∧ 2 Π = K ⊗L, a time orientation and a co-orientation taken together determine an orientation of Π.
A basis of a parabolic 2-plane Π ⊂ V * is called parabolic if its two elements form a parabolic pair. By Sylvester's theorem, any parabolic plane Π admits parabolic bases.
Proposition 3.32. The map (u, l) → Span R (u, l) induces a bijection between the set of weak equivalence classes of parabolic pairs of one-forms and the set of all parabolic 2-planes in (V * , h * ).
Proof. If (u, v) is a parabolic pair, then Span R (u, v) is a parabolic 2-plane, which depends only on the weak equivalence class of (u, v). Conversely, it is easy to see that any two parabolic bases of a parabolic 2-plane Π in V * are weakly-equivalent as parabolic pairs. Proposition 3.32 implies: Corollary 3.33. The map (u, l) → (Span R (u, l), H l ) induces a bijection between the set P(V * , h * )/ ≡ of equivalence classes of parabolic pairs of one-forms and the set of all co-oriented parabolic 2-planes in (V * , h * ), where H l is the unique spacelike half-plane of the parabolic 2-plane Span R (u, l) which contains the vector l.
Theorem 3.34. A polyform α ∈ ∧V * is a signed square of a nonzero spinor (i.e. it belongs to the set Z −,− (V * , h * )) iff it has the form: for a parabolic pair of one-forms (u, l) ∈ P(V * , h * ). In this case, u is uniquely determined by α while l is determined by α up to transformations of the form: where c ∈ R is arbitrary. Moreover, (u, l) is determined by the sign equivalence class of α up to strong equivalence of parabolic pairs. This gives a natural bijection between the sets Z −,− (V * , h * )/Z 2 and P(V * , h * )/ ∼ .
= α (2) ∈ ∧ 2 V * . For β = 1, the first condition in (49) gives (u + ω) ⋄ (u + ω) = 0, which reduces to the following relations upon expanding the geometric product: Here , h is the metric induced by h on ∧V * . The second condition in (50) amounts to ω = u ∧ l for some l ∈ V * determined up to the transformations (48). Using this in (50) gives the condition: which is invariant under the transformations (48). For β = u, the first equation in (49) amounts to h * (u, u) = 0, whence h * (u, l) = 0 by (51). It remains to show that h * (l, l) = 1. Since u is non-zero and null, there exists a non-zero null one- For β = v, the first condition in (49) reduces to: Direct computation shows that this equation amounts to h * (l, l) = 1 and we conclude. Given l ∈ V * of unit norm and orthogonal to u, there exists a unique c ∈ R such that P v (l + c u) = 0. This "choice of gauge" could be useful for spinors on time-oriented Lorentzian four-manifolds.  = Span(u, l) = Ru ⊕ L. Then Π is a parabolic 2-plane in V * depending only Ru and L and we have K h (Π) = Ru. Rescaling α by a non-zero real number corresponds to rescaling u = α (1) by the same. Hence Rα determines the line Ru = K h (Π) and relation (47) shows that Rα determines and is determined by the co-oriented parabolic 2-plane (Π, H L ). This proves the first statement. Now recall that the sign-equivalence class of a non-zero spinor ξ determines and is determined by the sign equivalence class of its square polyform α through the mapÊ Σ . By (47), the sign change α → −α corresponds to u → −u and l → l. Thus the sign equivalence class of u is uniquely determined by that of α and hence by that of ξ. This establishes the bijection between the three sets in the second statement.
Remark 3.37. A parabolic pair (u, l) and a polyform square α are recovered from the triplet (Π, H,û) by taking u to be any representative of the sign equivalence classû and l to be any vector lying on the unit norm affine line L contained in H and setting α = u + u ∧ l.
Let us count the degrees of freedom encoded in α = u + u ∧ l. Apriori, the null one-form u has three degrees of freedom while the space-like one-form l has four, which are reduced to two by the requirement that l has unit norm and is orthogonal to u. Since l is defined only up to l → l + c u (c ∈ R), its degrees of freedom further reduce from two to one. This gives a total of four degrees of freedom, matching those of a real spinor in four-dimensional Lorentzian signature.
We have: which gives: (ν − ⋄ ω) (0) = −k 1 . Furthermore, we compute: These products realize the Lie algebra sl(2, R) upon defining a Lie bracket by the commutator: Since ∧ 2 + V * = sl(2, R), the Killing form B of sl(2, R) gives a symmetric non-degenerate pairing of signature (1, 2) on ∧ 2 + V * , which can be rescaled to coincide with that induced induced by h. Then: Proposition 3.38. A polyform α ∈ ∧V * is a signed square of a real chiral spinor ξ ∈ Σ (−) of negative chirality iff α is a self-dual two-form of zero norm.
Proof. It suffices to consider the case α = 0. By the discussion above, a non-zero polyform α = 0 belongs to the set Z (−) −,+ (V * , h * ) only if α = ω is self-dual and of zero norm (which is equivalent to the first three equations in (52)). Once these conditions are satisfied, the only equation that remains to be solved is the fourth equation in (52). To solve it, we take β = ν − . Since (ν − ⋄ ω) (0) = −4 k 1 (as remarked above), we conclude that (ν − ⋄ ω) (0) = 0 iff ω = 0, whence taking β = ν − is a valid choice. A computation shows that this equation is automatically satisfied and thus we conclude.
Remark 3.39. Subsection 3.6.2 together with Proposition 3.38 show that the square of a chiral spinor in signatures (1, 1) and (2, 2) is given by an (anti-)self-dual form of zero norm in middle degree. The reader can verify, through a computation similar to the one presented in this subsection, that the same statement holds in signature (3,3). It is tempting to conjecture that the square of a chiral spinor in general split signature (p, p) corresponds to an (anti-)self-dual p-form of zero norm, the latter condition being automatically implied when p is odd. Verifying this conjecture would be useful in the study of manifolds of split signature which admit parallel chiral spinors [70].

Constrained Generalized Killing spinors of real type
To study constrained generalized Killing spinors of real type, we will extend the theory of Section 3 to bundles of real irreducible Clifford modules equipped with an arbitrary connection. Throughout this section, let (M, g) denote a connected pseudo-Riemannian manifold of signature (p, q) and even dimension d = p + q ≥ 2, where p − q ≡ 8 0, 2. Since M is connected, the pseudo-Euclidean vector bundle (T M, g) is modeled on a fixed quadratic vector space denoted by (V, h). For any point m ∈ M , we thus have an isomorphism of quadratic spaces (T m M, g m ) ≃ (V, h). Accordingly, the cotangent bundle T * M (endowed with the dual metric g * ) is modeled on the dual quadratic space (V * , h * ). We denote by Cl(M, g) the bundle of real Clifford algebras of the cotangent bundle (T * M, g * ), which is modeled on the real Clifford algebra Cl(V * , h * ). Let π and τ be the canonical automorphism and anti-automorphism of the Clifford bundle, given by fiberwise extension of the corresponding objects defined in Section 3 and setπ = π • τ . We denote by (Λ(M ), ⋄) the exterior bundle Λ(M ) = ⊕ d j=0 ∧ j T * M , equipped with the pointwise extension ⋄ of the geometric product of Section 3 (which depends on the metric g). This bundle of unital associative algebras is called the Kähler-Atiyah bundle of (M, g) (see [36,38]). The map Ψ of Section 3 extends to a unital isomorphism of bundles of algebras: which allows us to view the Kähler-Atiyah bundle as a model for the Clifford bundle. We again denote by π, τ andπ = π • τ the (anti-)automorphisms of the Kähler-Atiyah bundle obtained by transporting the corresponding objects from the Clifford bundle through Ψ. The Kähler-Atiyah trace of Section 3 extends to a morphism of vector bundles: S : Λ(M ) → R M whose induced map on smooth sections satisfies: is the unit function defined on M . By Proposition 3.14, we have: In particular, S does not depend on the metric g. The following encodes a well-know property of the Clifford bundle, which also follows from the definition of ⋄ (cf. [36,38]).
Proposition 4.1. The canonical extension to Λ(M ) of the Levi-Civita connection ∇ g of (M, g) to Λ(M ) (which we again denote by ∇ g ) acts by derivations of the geometric product:   In the signatures p − q ≡ 8 0, 2 considered in this paper, a simple bundle of Clifford modules satisfies rk S = dim V = 2 d 2 , where d is the dimension of M . Reference [48] proves that (M, g) admits a bundle of simple real Clifford modules iff it admits a real Lipschitz structure of type γ. In signatures p − q ≡ 8 0, 2, the latter corresponds to an adjoint-equivariant (a.k.a. "untwisted") Pin(V * , h * )structure Q on (M, g) and (S, Γ) is isomorphic (as a unital bundle of algebras) with the bundle of 5 Since S need not be associated to a spin structure on (M, g), this generalizes the traditional notion of spinor. In signatures p − q ≡ 8 0, 2, S is associated to an untwisted Pin structure (see [48]) so its sections could also be called "pinors".
real Clifford modules associated to Q through the natural representation of Pin(V * , h * ) in Σ. The obstructions to existence of such structures were given in [48]; when p − q ≡ 8 0, 2, they are a slight modification of those given in [73] for ordinary (twisted adjoint-equivariant) Pin(V * , h * )-structures. = ϕ −1 L • Γ. The map Ψ γ of Section 3 extends to a unital isomorphism of bundles of algebras: which allows us to identify bundles (S, Γ) of modules over Cl(T * M, g * ) with bundles of modules (S, Ψ Γ ) over the Kähler-Atiyah algebra. We denote by a dot the external multiplication 6 of (S, Ψ Γ ), whose action on global sections is: Let tr : End(S) → R M be the fiberwise trace morphism, whose map induced on sections we denote by the same symbol. The results of Section 3 imply: where Id U is the identity endomorphism of U .  M is paracompact, the defining algebraic properties of an admissible pairing can be formulated equivalently as follows using global sections (see [38]), when viewing (S, Γ) as a bundle (S, Ψ Γ ) of modules over the Kähler-Atiyah algebra of (M, g): Definition 4.9. We say that (M, g) is strongly spin if it admits a Spin 0 (V * , h * )-structure -which we call a strong spin structure. In this case, a real spinor bundle (S, Γ) on (M, g) is called strong if it associated to a strong spin structure.
When (M, g) is strongly spin, then it is strongly orientable in the sense that its orthonormal coframe bundle admits a reduction to an SO 0 (V * , h * )-bundle.
Remark 4.10. When pq = 0, the special orthogonal and spin groups are connected while the pin group has two connected components. In this case, orientability and strong orientability are equivalent, as are the properties of being spin and strongly spin. When pq = 0, the groups SO(V * , h * ) and Spin(V * , h * ) have two connected components, while Pin(V * , h * ) has four and we have Pin(V * , h * )/Spin 0 (V * , h * ) ≃ Z 2 × Z 2 . In this case, (M, g) is strongly orientable iff it is orientable and in addition the principal Z 2 -bundle associated to its bundle of oriented coframes through the group morphism is strongly spin, the short exact sequence: induces a sequence in Cech cohomology which implies that Spin 0 (V * , h * )-structures form a torsor over H 1 (M, Z 2 ). A particularly simple case arises when H 1 (M, Z 2 ) = 0 (for example, when M is simply-connected). In this situation, M is strongly orientable and any untwisted Pin(V * , h * )structure on (M, g) reduces to a Spin 0 (V * , h * )-structure since H 1 (M, Z 2 × Z 2 ) = H 1 (M, Z 2 ⊕ Z 2 ) = 0. Similarly, any Spin(V * , h * )-structure on (M, g) reduces to a Spin 0 (V * , h * )-structure. Up to isomorphism, in this special case there exists at most one Spin(V * , h * )-structure, one Spin 0 (V * , h * )structure and one real spinor bundle on (M, g), which is automatically strong.
The following gives sufficient conditions for existence of admissible pairings on real spinor bundles: Moreover, the Levi-Civita connection ∇ g of (M, g) lifts to a unique connection on S (denoted ∇ S and called the spinorial connection of S), which acts by module derivations: Proof. The first statement follows from the associated bundle construction since admissible pairings are Spin 0 (V * , h * )-invariant by Proposition 3.7. The second and third statements are standard (see [71,Chapter 3]). The last statement follows since the holonomy of ∇ S is contained in Spin 0 (V * , h * ), whose action on Σ preserves B.
With the assumptions of the proposition, the spinorial connection induces a linear connection (denoted D S ) on the bundle of endomorphisms End(S) = S * ⊗ S. By definition, we have: Proposition 4.12. Suppose that (M, g) is strongly spin and let (Σ, Γ) be a strong real spinor bundle over (M, g). Then D S : Γ(End(S)) → Γ(T * M ⊗ End(S)) acts by derivations: Moreover, Ψ Γ induces a unital isomorphism of algebras (Ω(M ), ⋄) ≃ (Γ(End(S)), •) which is compatible with ∇ g and D S : Proof. That D S acts by algebra derivations of Γ(End(S)) is standard. Proposition 4.11 gives: for all A ∈ Γ(End(S)), ǫ ∈ Γ(S) and X ∈ X(M ), where α def.
Remark 4.14. Writing D = ∇ S − A with A ∈ Ω 1 (End(S)), we have: whereÂ ∈ Ω 1 (M, Λ(M )) is the symbol of A, which we shall also call the symbol of D. A linear constraint datum for (S, Γ) is a pair (W, Q), where W is a real vector bundle over M and Q ∈ Γ(End(S) ⊗ W) ≃ Γ(Hom(S, S ⊗ W)). Given such a datum, the condition: is called the linear constraint on ǫ defined by Q. We say that ǫ is a (real) constrained generalized Killing spinor if it satisfies the system formed by (53) and (54).
Remark 4.16. Supersymmetric solutions of supergravity theories can often be characterized as manifolds admitting certain systems of generalized constrained Killing spinors, see for instance [36,37]. This extends the notion of generalized Killing spinor considered [6,9,10,72].
Suppose that (M, g) is strongly spin and (S, Γ) is a strong real spinor bundle. Then we can write D = ∇ S − A with A ∈ Ω 1 (End(S)), where ∇ S is the spinorial connection on S. In this case, the equations satisfied by a constrained generalized Killing spinor can be written as: and their solutions are called constrained generalized Killing spinors relative to (A, W, Q). When A is given, we sometimes denote D by D A . Using connectedness of M and the parallel transport of D, equation (53) implies that the space of constrained generalized Killing spinors relative to (A, Q, W) is finite-dimensional and that a constrained generalized Killing spinor which is not zero at some point of M is automatically nowhere-vanishing on M ; in this case, we say that ǫ is nontrivial.

4.4.
Spinor squaring maps. Let S = (S, Γ, B) be a paired spinor bundle on (M, g). The admissible pairing B of (S, Γ) allows us to construct extensions to M of the squaring maps E ± : Σ → End(Σ) of Section 2 and of the spinor squaring maps E ± Σ : Σ → ∧V * of Section 3. We denote these by: . Although E ± S preserve fibers, they are not morphisms of vector bundles since they are fiberwise quadratic. By the results of Section 3, these maps are two to one away from the zero section of S (where they branch) and their images -which we denote by Z   = (PE S ) −1 (L α ) of S. A section ǫ of S such that E κ S (ǫ) = α is a section of L Q (α). Since such ǫ must be nowhere-vanishing (because α is), it exists iff L Q (α) is trivial, which happens iff its first Stiefel-Whitney class vanishes. The conclusion follows by setting c Q (α) def.
= w 1 (L Q (α)) ∈ H 1 (M, Z 2 ). Notice that c Q (α) depends only on α and Q, since the Clifford bundle (S, Γ) is associated to Q while all admissible pairings of (S, Γ) are related to each other by automorphisms of S (see Remark 3.4 in Section 3).
which immediately implies the conclusion.
The following proposition shows that c Q (α) can be made to vanish by changing Q.
Remark 4.25. If A is skew-symmetric with respect to B, then (57) simplifies to: In applications to supergravity, A need not be skew-symmetric relative to B.
The polyform α as above is determined by ǫ through the relation: α = E κ S (ǫ) for some κ ∈ {−1, 1}. Moreover, α satisfying the conditions above determines a nowhere-vanishing real spinor ǫ satisfying this relation, which is unique up to sign. Remark 4.27. Suppose that α ∈ Ω(M ) is nowhere-vanishing and satisfies (61) and (62) but we have c Q (α) = 0. Then Proposition 4.22 implies that there exists a unique Spin 0 (V * , h * )-structure Q ′ such that c Q ′ (α) = 0. Thus α is the square of a global section of a paired spinor bundle (S ′ , Γ ′ , B ′ ) associated to Q ′ . Hence a nowhere-vanishing polyform α satisfying (61) and (62) corresponds to the square of a generalized Killing spinor with respect to a uniquely-determined Spin 0 (V * , h * )-structure.
Proof. The algebraic conditions in the Theorem follow from the pointwise extension of Theorem 3.20 and Corollary 3.28. The differential condition follows from Lemma 4.24, which implies that D A ǫ = 0 holds iff (62) does upon noticing that ǫ ∈ Γ(S) vanishes at a point m ∈ M iff its positive polyform square α satisfies α| m = 0. The condition c Q (α) = 0 follows from Proposition 4.18.
In Sections 5 and 6, we apply this theorem to real Killing spinors on Lorentzian four-manifolds and to supersymmetric configurations of heterotic supergravity on principal bundles over such manifolds.   = ker(g * Π ) (where g * Π is the restriction of g * to Π) is called the null line sub-bundle of Π and Π is called co-orientable if the quotient line bundle N h (Π) def.
A co-orientation of Π amounts to the choice of a sub-bundle of half-planes H ⊂ Π such that H m is one of the two spacelike half-planes of Π m for each m ∈ M . In this case, the pair (Π, H) is called a co-oriented distribution of parabolic 2-planes in T * M .  = u + u ∧ l is a signed polyform square of ξ by Section 3.6.3.

4.7.
Real spinors on globally hyperbolic Lorentzian four-manifolds. Let (M, g) be an oriented and spin Lorentzian four-manifold of "mostly plus" signature such that H 1 (M, Z 2 ) = 0. As before, let S = (S, Γ, B) be a paired real spinor bundle on (M, g), where B is skew-symmetric and of negative adjoint type.
Let P v (M, g) denote the set of parabolic pairs of one-forms (u, l) on (M, g) which satisfy P v (l) = 0. The group Z 2 acts on this set by changing the sign of u while leaving l unchanged. Proposition 4.34 and Theorem 4.32 imply: Assume next that (M, g) is globally hyperbolic. By a theorem of A. Bernal and M. Sánchez [74], it follows that (M, g) is isometric to R × N equipped with the warped product metric g = −F dt ⊗ dt + k(t), where N is an oriented three-manifold, F ∈ C ∞ (R × N ) is a strictly positive function and k(t) is a Riemannian metric on N for every t ∈ R. Let V 2 (N, k(t)) be the bundle of ordered orthonormal pairs of one-forms on N . Since N is oriented, any element of V 2 (N, k(t)) determines an element of the principal bundle P SO (3) (N, k(t)) of oriented frames of (N, k(t)), showing that V 2 (N, k(t)) is a principal SO(3)-bundle. Let V 2 be the fiber bundle defined on M = R × N whose fiber at (t, n) ∈ R × M is given by: is the manifold of k(t) n -orthonormal systems of two elements of T * n N . Consider the fiberwise involution i 1 of V 2 defined through: def.
and let Z 2 act on the set C ∞ (R × N ) × × Γ(V 2 ) through the involution: Proposition 4.36. Consider a globally hyperbolic Lorentzian four-manifold: such that N is oriented and spin and H 1 (N, Z 2 ) = 0. Then there exists a bijection between the set Γ(Ṡ)/Z 2 of sign-equivalence classes of nowhere-vanishing real spinors defined on M and the set [C ∞ (R×N ) × ×Γ(V 2 )]/Z 2 . Moreover, there exists a bijection between the sets Γ(P(S)) and Γ(V 2 )/Z 2 , where Z 2 acts on Γ(V 2 ) through the involution i 1 .
Proof. Let v def.
= F 1 2 dt and consider a parabolic pair of one-forms (u, l) on M such that P v (l) = 0. Since l has unit norm and is orthogonal to v, it can be viewed as a family of one-forms (parameterized by t ∈ R) defined on N . We decompose u orthogonally as: = u − P v (u) = u + g * (u, v) v and g * (u, v), g * (u ⊥ , u ⊥ ) are nowhere-vanishing. The spacelike 1-form u ⊥ satisfies: Thus u can be written as: where: The pair (e u , l) determines an orthonormal pair of one-forms (e u (t), l(t)) on (N, k(t)) for all t ∈ R, which gives a section s of the fiber bundle V 2 . It is clear that the parabolic pair (u, l) determines and is determined by the pair (f, s). The conclusion now follows from  )) and using the fact that π 3 (BSpin(3)) = 0. Hence there exists a unique family {e(t)} t∈R of one-forms on N such that (e u (t), l(t), e(t)) is an oriented orthonormal global frame of (T * N, k t ) for all t ∈ R. This produces a parallelization of (M, g) given by (v, e u (t), l(t), e(t)) t∈R . Let: and let Z 2 act on C ∞ (R × N ) × × C ∞ (N, SO(3)) through the involution: where: The previous proposition implies:  (3)) through the involution: (3)) .
We hope that this characterization can be useful in the study of globally hyperbolic Lorentzian four-manifolds admitting spinors satisfying various partial differential equations.

Real Killing spinors on Lorentzian four-manifolds
Definition 5.1. Let (M, g) be a pseudo-Riemannian manifold which is oriented and strongly spin and (S, Γ) be a strong real spinor bundle on (M, g). Let λ ∈ R be a real number. A real Killing spinor of Killing constant λ 2 is a global section ǫ ∈ Γ(S) which satisfies: It is called a parallel spinor if λ = 0.
Real Killing spinors are (unconstrained) generalized Killing spinors relative to the connection D = In this section, we study real Killing spinors when (M, g) is a spin Lorentzian four-manifold of "mostly plus" signature (3, 1) such that H 1 (M, Z 2 ) = 0.
Remark 5.2. When p−q ≡ 8 0, 2, a real Killing spinor can be viewed as a complex Killing spinor which is preserved by a Spin 0 (p, q)-invariant real structure on the complex spinor bundle and which has real (in signature (p, q)) or purely imaginary (in signature (q, p)) Killing constant. When comparing signatures, note that [55] has a sign in the Clifford relation opposite to our convention (19). In the conventions of loc. cit., the real Killing spinors considered below correspond to special cases of imaginary Killing spinors, which were studied in [5,75]. Reference [75] proves that a Lorentzian four-manifold admitting a nontrivial imaginary Killing spinor (which is a real Killing spinor in our convention) with null Dirac current is locally conformal to a Brinkmann space-time 7 . In this section, we give a global characterization of Lorentzian four-manifolds admitting real Killing spinors (see Theorem 5.3).

5.1.
Describing real Killing spinors through differential forms. For the remainder of this section, let (M, g) be a spin Lorentzian four-manifold of "mostly plus" signature which satisfies H 1 (M, Z 2 ) = 0. Let (S, Γ) be a spinor bundle on (M, g). Since H 1 (M, Z 2 ) vanishes, the spinor bundle is automatically strongly spin and unique up to isomorphism. We endow it with an admissible pairing B which is is skew-symmetric and of negative adjoint type.
for some κ ∈ Ω 1 (M ). In this case, u ♯ ∈ X(M ) is a Killing vector field with geodesic integral curves.
Remark 5.4. Our conventions for the wedge product of one-forms are as follows, where S k denotes the permutation group on k letters: where θ 1 , . . . , θ k ∈ Ω 1 (M ). 7 Recall that a Brinkmann space-time is a four-dimensional Lorentzian manifold equipped with a non-vanishing parallel null vector field.
Proof. Recall from Section 4.6 that a spinor ξ associated to (u, l) has a signed polyform square given by α = u + u ∧ l. Theorem 4.26 shows that ξ is a real Killing spinor iff: Expanding the geometric product and isolating degrees, this equation gives: which in turn amounts to (66) for some κ ∈ Ω 1 (M ). The vector field u ♯ is Killing since ∇ g u is an antisymmetric covariant 2-tensor by the first equation in (66). Since u is null and orthogonal to l, the same equation gives ∇ g u ♯ u = 0. Hence ∇ g u ♯ u ♯ = 0, i.e. u ♯ is a geodesic vector field.
Remark 5.5. The first equation in (66) gives: Hence the null vector field u ♯ ∈ X(M ) is not recurrent, i.e. ∇ g does not preserve the rank one distribution spanned by u ♯ . Lorentzian manifolds admitting recurrent vector fields are called almost decent and were studied extensively (see [76][77][78] and references therein).
Taking λ = 0 in Theorem 5.3 gives: Corollary 5.6. (M, g) admits a nontrivial parallel real spinor iff it admits a parabolic pair of one-forms (u, l) which satisfies the following conditions for some one-form κ ∈ Ω 1 (M ): Although Lorentzian manifolds admitting parallel spinors were studied extensively in the literature (see [80,81] and references therein), Corollary 5.6 seems to be new. Recall that u coincides up to sign with the Dirac current of any of the spinors ξ, −ξ determined by the parabolic pair (u, v) (see Remark 3.23). Reference [75] shows that a Lorentzian four-manifold admitting an (imaginary, in the conventions of loc. cit) Killing spinor with null Dirac current is locally conformally Brinkmann.
The following proposition recovers this result in our approach.
Proposition 5.7. Suppose that (M, g) admits a nontrivial real Killing spinor with nonzero Killing constant λ 2 = 0 and let (u, l) be a corresponding parabolic pair of one-forms. Then u is locally conformally parallel iff l is locally equivalent to a closed one-form l ′ by transformations of the form (70). In this case, (M, g) is locally conformal to a Brinkmann space-time.
Proof. The one-form u is locally conformally parallel iff for sufficiently small non-empty open subsets U ⊂ M there exists f ∈ C ∞ (U ) such that the metricĝ = e f g satisfies ∇ĝu = 0 on U . This amounts to: where in the last equality we used the first equation in (66). Taking X = u ♯ and using the fact that u is nowhere-vanishing, null and orthogonal to l gives (df )(u ♯ ) = 0, whence (68) reduces to: which amounts to the condition df = λ(l + cu) for some c ∈ C ∞ (U ). This has local solutions f iff l + cu is closed for some locally-defined function c. In this case, the nowhere-vanishing null one form u is ∇ĝ-parallel and hence (M, g) is locally conformally Brinkmann. 5.2. The Pfaffian system and its consequences. Antisymmetrizing the two equations in (66) gives the Pfaffian system: (69) du = 2λ u ∧ l , dl = κ ∧ u , which implies: Lemma 5.8. Let (u, l) be a parabolic pair of one-forms which satisfies equations (66) for some κ ∈ Ω 1 (M ) and λ ∈ R and let C u ⊂ T M be the rank one distribution spanned by u ♯ . Then l is closed if and only iff κ ∈ Γ(C u ). Moreover, u is closed iff λ = 0.
By the Frobenius theorem, the first equation in (69) implies that the distribution ker(u) = C ⊥ u ⊂ T M integrates to a codimension one foliation of M which is transversally-orientable since u is nowhere-vanishing. Since C u is contained in C ⊥ u , this foliation is degenerate in the sense that the restriction of g to C ⊥ u is a degenerate vector bundle metric. In particular, the three-dimensional vector space C ⊥ u,m is tangent to the causal cone L m ⊂ T m M along the null line C u,m at any point m ∈ M (see Appendix A) and the complement C ⊥ u \ C u consists of spacelike vectors. Since l is orthogonal to u, we have l ♯ ∈ Γ(C ⊥ u ). The vector fields u ♯ and l ♯ span a topologically trivial distribution Π ♯ of parabolic 2-planes contained in C u .
Let S(C ⊥ u ) be any complement of C u in C ⊥ u : . Such a complement is known as a screen bundle of C ⊥ u (see [79] and references therein); in our situation, it can be chosen such that l ♯ ∈ Γ(S(C ⊥ u )), in which case we can further decompose S(C ⊥ u ) = C l ⊕ L, where C l is the rank one distribution spanned by l ♯ and L is any rank one distribution complementary to Π ♯ in C ⊥ u . For any choice of the screen bundle, the restriction of g to S(C ⊥ u ) is non-degenerate and positive-definite and hence admits an orthogonal complement in T M which has the form C u ⊕ C v , where C v ⊂ T M is the rank one distribution spanned by the unique null vector field v ♯ ∈ X(M ) which is orthogonal to S(C ⊥ u ) and satisfies g(u ♯ , v ♯ ) = 1. This gives: , which allows us to write the metric as: and v is the one-form dual to v ♯ . Lemma 5.10. Suppose that (M, g) admits a nontrivial real Killing spinor ǫ ∈ Γ(S) with Killing constant λ 2 = 0 and let (u, l) be a corresponding parabolic pair of one-forms. Around every point in M , there exist local Walker-like coordinates (x v , x u , x 1 , x 2 ) with u ♯ = ∂ x u in which the metric takes the form: where F , K, ω i and q ij are locally-defined functions which do not depend on x u and such that K is nowhere-vanishing. In these coordinates, the one-forms u and l can be written as: for some locally-defined function s.
Proof. Since C ⊥ u is integrable and of corank one and C u ⊂ C ⊥ u has rank one, there exist local coordinates (x v , x u , x 1 , x 2 ) on M such that ∂ x u = u ♯ , the vector fields ∂ x u , ∂ x 1 , ∂ x 2 span C ⊥ u and ∂ xv is null. In such local coordinates, we have: Notice that g uv is nowhere-vanishing since u is. The vector fields ∂ x 1 and ∂ x 2 span an integrable local screen S(C ⊥ u ) for C ⊥ u . Let v be the unique null one form which vanishes along S(C ⊥ u ) and satisfies v(∂ x u ) = 1. Then the vector field v ♯ satisfies the assumptions which allow us to write the metric in the form (73). (73) gives: Relabeling coefficients gives (74) with F = 2g uv v v , K = g uv and ω i = 2g uv v i . The coefficients of g do not depend on x u since u ♯ = ∂ x u is a Killing vector field. In these coordinates we have u = K dx v (hence K is nowhere-vanishing) and the first equation of the Pfaffian system (69) becomes: showing that: for some locally-defined function s.
Lemma 5.10 gives existence of Walker-like coordinates on Lorentzian four-manifolds admitting real Killing spinors. These generalize the classical Walker coordinates [82] of Lorentzian manifolds which admit a parallel null line [76,77,82]. The main difference is that our u is not recurrent. On the other hand, our u is Killing -a condition which may not hold on generic Walker manifolds.
Example 5.11. The simply-connected four-dimensional anti-de Sitter space AdS 4 admits Walker-like coordinates (x v , x u , x, y) in which the Anti-de Sitter metric g reads: where c is a positive constant equal to minus the curvature. It is well-known [83][84][85] that AdS 4 admits a four-dimensional space of real Killing spinors.

5.3.
The locally stationary and locally integrable case. Definition 5.12. Suppose that (M, g) admits a nontrivial real Killing spinor ǫ ∈ Γ(S) with nonzero Killing constant. We say that (M, g, ǫ) is: • locally stationary if, around every point, the Walker-like coordinates induced by ǫ are such that ∂ x v is Killing. • locally integrable if, around every point, the Walker-like coordinates coordinates induced by ǫ are such that ω 1 = ω 2 = 0. • locally static if, around every point, the Walker-like coordinates induced by ǫ are such that ∂ x v is Killing, ω 1 = ω 2 = 0 and F = 0.
Remark 5.13. Notice that the locally defined rank two distribution ∆ spanned by ∂ x v and ∂ x u is nondegenerate (and hence admits a non-vanishing timelike section) since ∂ x u is null and K = g(∂ x u , ∂ x v ) is locally nowhere-vanishing. If (M, g, ǫ) is locally stationary in the sense above then some linear combination of ∂ x v and ∂ x u is a timelike Killing vector field, whence (M, g) is locally stationary in the standard sense. If (M, g, ǫ) is locally integrable, then the orthogonal complement of ∆ is an integrable spacelike distribution of rank two. If (M, g, ǫ) is locally static then ∂ x v is null and (M, g) admits a local hypersurface orthogonal to a time-like Killing vector field X (namely is locally static in the standard sense. Theorem 5.14. The following statements are equivalent: (a) There exists a nontrivial real Killing spinor ǫ ∈ Γ(S) with Killing constant λ 2 = 0 on (M, g) such that (M, g, ǫ) is locally stationary and locally integrable. (b) (M, g) is locally isometric to a Lorentzian four-manifold of the form: are Cartesian coordinates on R 2 , X is a non-compact, oriented and simplyconnected surface endowed with the Riemannian metric q and F , K ∈ C ∞ (X) are functions on X (with K nowhere-vanishing) which satisfy: for some function s ∈ C ∞ (M ) and some one-form κ ∈ Ω 1 (M ), where x 1 , x 2 are local coordinates on X. In this case, the formulas: give a parabolic pair of one-forms (u, l) corresponding to the real Killing spinor ǫ, which satisfy equations (66) with respect to the one-form κ. Moreover, (M, g) is Einstein with Einstein constant Λ iff Λ = −3λ 2 and the following equations are satisfied, where Ric q is the Ricci tensor of q: in which situation (X, q) is a hyperbolic Riemann surface.
Remark 5.15. Here, the Laplacian ∆ q is defined through: The second equation in (75): is a "wrong sign" eigenvalue problem for the Laplacian on X. Since ∆ q is negative semidefinite, no nontrivial solutions to (78) exists unless X is non-compact. The function s can be chosen at will as long as equations (75) hold, since different choices produce the same signed polyform square α = u + u ∧ l of ǫ, namely: We can exploit this freedom to choose: which is independent of x v and x u . For this choice of s, equations (75) reduce to: and hence κ is a one-form defined on X which is completely determined by λ, F , K and q. In particular, the condition that (M, g) admits a nontrivial real Killing spinor with Killing constant λ 2 such that (M, g, ǫ) is locally stationary and locally integrable reduces to the last two equations in (81), which involve only λ, K and q but do not involve F . This generalizes a statement made in [84, page 391]. On the other hand, the Einstein condition (77) involves both F and K. Strictly speaking, the "gauge choice" (80) requires: if l is to be well-defined, since the formula for l involves s. However, the real Killing spinor associated to (u, l) is well-defined and satisfies the Killing spinor equations even when s 2 is formally negative somewhere on M , because ǫ is determined by the polyform (79), which is independent of s.
Proof. By Lemma 5.10 and Theorem 5.3, we must solve equations (66) for u and l of the form (76) , which automatically satisfy the first equation of the Pfaffian system (69). The first equation in (66) is equivalent to the condition that the vector field u ♯ = ∂ x u is Killing, together with the condition the first equation in (69) holds. Thus it suffices to consider the second equation in (66). Evaluating this equation on ∂ x v and ∂ x u gives the system: , which reduce to the following equations for u and l as in (76): On the other hand, restricting the second equation in (66) to X and using (76) gives: Furthermore, taking the trace of (83) and combining it with the third equation in (82) we obtain ∆ q K = 6λ 2 K. Together with relations (82) and (83), this establishes the system (75). Consider now the Einstein equation Ric g = Λ g on (M, g). The only non-trivial components are: Direct computation gives: Combining these relations with (75), we conclude. Theorem 5.14 suggests a strategy to construct Lorentzian four-manifolds admitting real Killing spinors. Fix a simply-connected (generally incomplete) hyperbolic Riemann surface (X, q) and consider the eigenspace of the Laplacian on (X, q) with eigenvalue 6λ 2 . In this space, look for a function K which satisfies the first equation in (75). If such exists, it gives a real Killing spinor for any F ∈ C ∞ (X). Below, we give special classes of solutions when (X, q) is the Poincaré half-plane.

5.4.
Special solutions from the Poincaré half-plane. Take: where x, y are global coordinates on the Poincaré half-plane H = (x, y) ∈ R 2 | y > 0 and F , K are real-valued functions defined on H (with K nowhere-vanishing), while c > 0 is a constant. Let q denote the metric c dx⊗dx+dy⊗dy y 2 on H. Theorem 5.14 shows that such (M, g) admits a real Killing spinor with Killing constant λ 2 = 0 iff: in which case (M, g) is Einstein iff: Direct computation shows that equations (85) are equivalent with: , y 2 ∂ 2 x K + ∂ 2 y K = 6 λ 2 c K . This gives the following: Corollary 5.17. The Lorentzian four-manifold (84) admits a nontrivial real Killing spinor with Killing constant λ 2 = 0 iff: In this case, it is Einstein iff F satisfies (86).
Choosing F to not satisfy (86) produces large families of non-Einstein Lorentzian four-manifolds admitting real Killing spinors.
Example 5.18. Taking K = F = c y 2 gives a solution of (87) iff c λ 2 = 1. Hence the Lorentzian four-manifold: admits a real Killing spinor. This is the AdS 4 space with metric written in horospheric coordinates [86], which is well-known to admit the maximal number (namely four) of real Killing spinors [28].
More examples can be constructed by solving in more generality the eigenvector problem for the Laplace operator of the Poincaré half plane and checking which solutions satisfy the first equation in (85). We illustrate this by constructing solutions obtained through separation of variables. Set: where k x ∈ C ∞ (H) depends only on x and k y ∈ C ∞ (H) depends only on y. The second equation in (87) gives: where the dot denotes derivation with respect to the corresponding variable. Whenk x = 0, equations (87) reduce to: with general solution K = c 0 y −2 (where c 0 = 0 is a constant). Ifk x = 0, then (88) givesk y + 2ky y = 0, so k y = c 0 y −2 for a non-zero constant c 0 . Using this in (87) givesk x = 0, a contradiction. Hence every Lorentzian four-manifold of the form: with F a smooth function admits nontrivial real Killing spinors. This gives large families of non-Einstein Lorentzian four-manifolds carrying real Killing spinors by taking F to be generic. The Lorentzian manifold (89) is Einstein when equation (86) is satisfied, which for K = c 0 y −2 reads: To study (90), we try the separated Ansatz: where f x ∈ C ∞ (H) depends only on x and f y ∈ C ∞ (H) depends only on y. Equation (90) gives: for some c ∈ R. If c = 0, this is solved by: where a def.
= (a 1 , . . . , a 4 ) ∈ R 4 . This gives the following family of Einstein Lorentzian metrics on R 2 × H admitting real Killing spinors, where we eliminated c 0 by rescaling x u : For a 1 = a 2 = a 3 = a 4 = 0 we recover the AdS 4 metric written in horospheric coordinates. Thus This gives a four-parameter family (parameterized by (a 1 , a 2 , a 3 , a 4 ) ∈ R 4 ) of Lorentzian Einstein metrics on R 2 × H admitting real Killing spinors.
Remark 5.19. When a 1 a 2 = 0 and a 3 = 0, the Lorentzian four-manifolds constructed above are not isometric to AdS 4 , since their Weyl tensor is non-zero and their Riemann tensor is not parallel.

Supersymmetric heterotic configurations
In this section we consider generalized constrained Killing spinors in an abstract form of heterotic supergravity (inspired by [88]), which is parameterized by a triplet (M, P, c), where M is a spin open four-manifold, P is a principal bundle over M with compact semi-simple Lie structure group G and c is an Ad G -invariant, symmetric and non-degenerate inner product on the Lie algebra g of G. Let g P def.
= P × Ad G g be the adjoint bundle of P . The Killing spinor equations of heterotic supergravity couple a strongly spinnable Lorentzian metric g on M (taken to be of "mostly plus" signature), a closed one-form ϕ ∈ Ω 1 (M ), a three-form H ∈ Ω 3 (M ) and a connection A on P . This system of partial differential equations characterizes supersymmetric configurations of the theory defined by (M, P, c). For simplicity, we assume H 1 (M, Z 2 ) = 0, although this assumption can be relaxed. We refer the reader to Appendix 6 for certain details.  (g, ϕ, H, A), where g is a strongly-spinnable Lorentzian metric on M , ϕ ∈ Ω 1 (M ) is a closed one-form, H ∈ Ω 3 (M ) is a three-form and A ∈ A P is a connection on P such that the modified Bianchi identity holds: The configuration is called supersymmetric if there exists a nontrivial spinor ǫ ∈ Γ(S g ) such that: (94)∇ H ǫ = 0 , ϕ · ǫ = H · ε , F A · ǫ = 0 .
Remark 6.2. Equations (94) encode vanishing of the gravitino, dilatino and gaugino supersymmetry variations. Since we work in Lorentzian signature, supersymmetric configurations need not solve the equations of motion (which are given in Appendix 6). However, the study of supersymmetric configurations is a first step toward classifying supersymmetric solutions. The study of supersymmetric solutions this theory in the physical case of ten Lorentzian dimensions was pioneered in [50,58,60], where their local structure was characterized. The last equation in (94) is formally identical to the spinorial characterization of instantons in Riemannian signature and dimensions from four to eight.
6.2. Characterizing supersymmetric heterotic configurations through differential forms. The metric connection ∇ H is given by (see Appendix 6 for notation): where ∇ g is the Levi-Civita connection of (M, g) and H ♯ is H viewed as a T M -valued two-form.
Hence the first equation in (94) can be written as: where H(X) def.
= * H. In this case, u ♯ ∈ X(M ) is a Killing vector field.
Proof. By Lemmas 6.3, 6.4 and 6.5, it suffices to prove the last equation in the third line of (99). The modified Bianchi identity can be written as: and gives: d * ρ = c(ι u χ A , ι u χ A ) = c(χ A (u), χ A (u)) = 0 , because χ A is orthogonal to u. The first equation in the the third line of (99) amounts to: showing that u ♯ is Killing and the distribution ker u ⊂ T M is integrable, giving a transverselyorientable foliation F u ⊂ M of codimension one.

Some examples.
Example 6.7. Let: (M, ds 2 g ) = R 2 × X, 2 dx v dx u + q(x v ) , where q(x v ) is a flat Riemannian metric on X for all x v ∈ R and take P to be the unit principal bundle over M . Consider the spinor ǫ corresponding to the pair (u, l), where u = dx v and l = l(x v ) depends on x v . Finally, take: ϕ = Ω u , ρ = 0 , where Ω ∈ C ∞ (R 2 × X). A short computation shows that equations (99) reduce to: Applying this to ∂ x v , ∂ x u and restricting to T X gives: where κ X def.
= κ| T X . This implies: showing that (X, q(x v )) is flat for all x v ∈ R. The only remaining non-trivial condition is: This is a linear first order ordinary differential equation for the function x v → l(x v ). For every choice of parallel vector field on (X, q(x v 0 )) with fixed (x v 0 , x u 0 ) ∈ R 2 , its solution with the corresponding initial condition determines a one-parameter family of one forms {l(x v )} x v ∈R on (X, q(x v )). Assuming for instance that X is simply connected and that q(x v ) satisfies: for some function F (x v ) depending only on x v , then the explicit solution is: where l 0 is parallel vector field on (X, q(x v 0 )) and we canonically identify the tangent vector spaces of {(x v , x u )} × X at different points (x v , x u ).
Remark 6.8. Heterotic solutions with exact null dilaton were considered before (see [87]). As remarked earlier, a supersymmetric heterotic configuration need not be a solution of the equation of motion given in Appendix B. The classification of (geodesically complete) supersymmetric heterotic solutions on a Lorentzian four-manifold and the diffeomorphism type of four-manifolds admitting such solutions for fixed principal bundle topology is an open problem. Appendix B gives a brief formulation of abstract bosonic heterotic supergravity and its Killing spinor equations.
Appendix A. Parabolic 2-planes and degenerate complete flags in R 3,1 Let (V, h) be a four-dimensional Minkowski space of "mostly plus" signature. A non-zero subspace W ⊂ V * is called degenerate or nondegenerate according to whether the restriction h * W of h * to W is a degenerate or non-degenerate quadratic form. A non-degenerate subspace W is called: • positive or negative definite, if the restriction h * W to W is positive or negative negative definite, respectively • hyperbolic, if the restriction of h * to W is not positive or negative definite. Notice that W is partially isotropic (i.e. contains nonzero null vectors) iff it is degenerate or hyperbolic. Let L denote the cone of causal (i.e. non-spacelike) vectors in (V * , h * ). A non-zero subspace W ⊂ V * is: • hyperbolic iff dim(W ∩ L) > 1, i.e. iff W meets L along a sub-cone of the latter which has dimension at least two. • degenerate iff dim(W ∩ L) = 1, i.e. iff W is tangent to L along a null line • non-degenerate iff W ∩ L = {0}, in which case W is spacelike (i.e. positive definite). A degenerate subspace W of V * contains no timelike vectors and the set of its null vectors coincides with the kernel K h (W ) def.
= ker(h * W ); accordingly, W decomposes as: where K h (W ) = ker(h * W ) coincides with the unique null line contained in W and U is a spacelike subspace of V * which is orthogonal to K h (W ). In particular, we have W ⊂ K h (W ) ⊥ . For example, a 2-plane Π ⊂ V * can be spacelike, hyperbolic or degenerate, according to whether h * Π is positivedefinite, non-degenerate of signature (1, 1) or degenerate. In the latter case, Π is called a parabolic 2-plane.
Notice that a complete flag 0 ⊂ W (1) ⊂ W (2) ⊂ W (3) ⊂ V * is determined by the increasing sequence of vector spaces W • def.
Proposition A.2. There exists a natural bijection between the set of parabolic 2-planes and the set of degenerate complete flags in (W * , h * ). This associates to each parabolic 2-plane Π ⊂ V * the unique degenerate complete flag 0 ⊂ W (1) ⊂ W (2) ⊂ W (3) ⊂ V * with W (2) = Π, which is given by: = W (2) is degenerate, i.e. parabolic. Since the subspace W (1) of W (2) = Π is degenerate and one-dimensional, it is a null line and hence coincides with K h (Π). Since W (3) is degenerate, the onedimensional subspace K h (W (3) ) is the unique null line contained in W (3) and hence coincides with W (1) . We thus have W (3) ⊂ W ⊥ (1) . This inclusion is an equality because dim W ⊥ (1) = dim V * − 1 = 3 = dim W (3) . Hence any degenerate complete flag has the form (101) for a unique parabolic 2-plane Π, namely Π = W (2) . It is clear that the correspondence thus defined is a bijection. Definition A.3. A co-oriented degenerate complete flag in (V * , h * ) is a pair (W • , L), where W • = (W (1) , W (2) , W (3) ) is a degenerate complete flag in (V * , h * ) and L is a co-orientation of the parabolic two-plane W (2) . Corollary 3.36 and Proposition A.2 imply the following reformulation of Theorem 3.34: Theorem A.4. There exists a natural bijection between P(Σ) and the set of all co-oriented degenerate complete flags in (V * , h * ).

Appendix B. Heterotic supergravity in four Lorentzian dimensions
Let G be a compact semisimple real Lie group whose Lie algebra we denote by g and whose adjoint representation we denote by Ad G : G → GL(g). Let g = g 1 ⊕ . . . ⊕ g k be the decomposition of g into simple Lie algebras. Then any Ad G -invariant non-degenerate symmetric pairing c on g can be written as: where B j is the Killing form of g j and c j are non-zero constants. Let (M, P, c) be a four-dimensional heterotic datum. Let g P def.
= P × Ad G g be the adjoint bundle of Lie algebras of P and A P be the affine space of connections on P . For any connection A ∈ A P , let F A ∈ Ω 2 (M, g P ) denote the curvature form of A. Let c P be the pairing induced by c on the adjoint bundle g P . Since c is Ad G -invariant, the latter can be viewed as a morphism of vector bundles: = ι X α, ι Y β g,c ∀α, β ∈ Ω(M, g P ) ∀X, Y ∈ X(M ) .
For a three-form H ∈ Ω 3 (M ) and the curvature F A ∈ Ω 2 (g P ) of a connection A ∈ A P , we have: