Geometric algebra techniques in flux compactifications

We study `constrained generalized Killing (s)pinors', which characterize supersymmetric flux compactifications of supergravity theories. Using geometric algebra techniques, we give conceptually clear and computationally effective methods for translating supersymmetry conditions into differential and algebraic constraints on collections of differential forms. In particular, we give a synthetic description of Fierz identities, which are an important ingredient of such problems. As an application, we show how our approach can be used to efficiently recover results pertaining to N=1 compactifications of M-theory on eight-manifolds.


Introduction
A fundamental problem in the study of flux compactifications of M-and string theory is to give efficient geometric descriptions of supersymmetric backgrounds in the presence of fluxes. This leads, in particular cases, to beautiful connections [1,2] with the theory of G-structures, while in more general situations it translates to difficult mathematical problems involving novel geometric realizations of supersymmetry algebras (see [3][4][5][6] for some examples).
When approaching this subject, one may be struck by the somewhat ad-hoc nature of the methods usually employed, which signals a lack of unity in the current understanding of the subject. This is largely due to the intrinsic difficulty in finding unifying principles while keeping computational complexity under control. In particular, one confronts the lack of general and structurally clear descriptions of Fierz identities, the fact that phenomena and methods which are sometimes assumed to be 'generic' turn out, upon closer inspection, to be relevant only under simplifying assumptions and the insufficient mathematical development of the subject of 'spin geometry [7] in the presence of fluxes'.
The purpose of this paper is to draw attention to the fact that many of the issues mentioned above can be resolved using ideas inspired by a certain incarnation of the theory of Clifford bundles known as 'geometric algebra', which goes back to [8] and [9] (see also [10] and [11][12][13] for an introduction) -an approach which provides a powerful language and efficient techniques, thus affording a more unified and systematic description of flux compactifications and of supergravity and string compactifications in general. In particular, we show that the geometric analysis of supersymmetry conditions for flux backgrounds (including the algebra of those Fierz identities relevant for the analysis) can be formulated efficiently in this language, thereby uncovering structure whose implications have remained largely unexplored.
Though the scope of applications of our approach is much wider, we shall focus here on the study of what we call 'constrained generalized Killing (CGK) (s)pinor equations', which distill the mathematical description of supersymmetry conditions for flux backgrounds. A constrained generalized Killing (s)pinor is simply a (s)pinor satisfying conditions of the type D m ξ = Q 1 ξ = . . . = Q χ ξ = 0, where D m = ∇ S m + A m is some connection on a bundle S of (s)pinors (which generally differs from the connection ∇ S m induced on S by the Levi-Civita connection ∇ m of the underlying pseudo-Riemannian manifold) while Q j are some globally-defined endomorphisms of S. Such equations are abundant in flux compactifications of supergravity (see, for example, [14,15]), where ξ is the internal part of a supersymmetry generator while the equations themselves are the conditions that the compactification preserves the supersymmetry generated by ξ. The quantities A m and Q j are then certain algebraic combinations of gamma matrices with coefficients dependent on the metric and fluxes. An example with a single algebraic constraint Qξ = 0 (arising in a compactification of eleven-dimensional supergravity) is discussed in Section 6, which the reader can consult first as an illustration motivating the formal developments taken up in the rest of the paper.
Using geometric algebra techniques, we show how such supersymmetry conditions can be translated efficiently and briefly into a system of differential and algebraic constraints for a collection of inhomogeneous differential forms expressed as (s)pinor bilinears, thus displaying the underlying structure in a form which is conceptually clear as well as highly amenable to computation. The conditions which we obtain on differential forms provide a generalization of the well-known theory of Killing forms, which could be studied in more depth through methods of Kähler-Cartan theory [16] -even though we will not pursue that avenue in the present work. We also touch on our implementation of this approach using various symbolic computation systems.
As an example, Section 6 applies such techniques to the study of flux compactifications of M-theory on eight-manifolds preserving N = 1 supersymmetry in 3 dimensions -a class of solutions which was analyzed through direct methods in [3] and [4]. In that setting, we have a single algebraic condition Qξ = 0, with Q = 1 2 γ m ∂ m ∆ − 1 288 F mpqr γ mpqr − 1 6 f p γ p γ (9) − κγ (9) and A m = 1 4 f p γ m p γ (9) + 1 24 F mpqr γ pqr + κγ m γ (9) . We show how our methods can be used to recover the results of [3] in a synthetic and computationally efficient manner, while giving a more complete and general analysis. We express all equations in terms of certain combinations of iterated contractions and wedge products which are known as 'generalized products' and whose conceptual role and origin is explained in Section 3. The reader can, at this point, pause to take a look at Subsection 6.2, which should provide an illustration of the techniques developed in this paper.
The paper is organized as follows. In Section 2, we define and discuss constrained generalized Killing (s)pinors. In Section 3, we recall the geometric algebra description of Clifford bundles as Kähler-Atiyah bundles while in Section 4 we explain how pinor bundles are described in this approach. Using our realization of spin geometry, Section 5 presents a synthetic formulation of Fierz rearrangement identities for pinor bilinears, which encodes identities involving four pinors through certain quadratic relations holding in (a certain subalgebra of) the Kähler-Atiyah algebra of the underlying manifold. We also re-formulate the constrained generalized Killing pinor equations in this language and discuss some aspects of the differential and algebraic structure resulted from this analysis, thereby extending the well-known theory of Killing forms. In Section 6, we apply this formalism to the study of N = 1 compactifications of M-theory on eight-manifolds. We conclude in Section 7 with a few remarks on further directions. The Appendices summarize various technical details and make contact with previous work. The physics-oriented reader can start with Section 6, before delving into the technical and theoretical details of the other sections.
Notations. We let K denote one of the fields R or C of real or complex numbers. We work in the smooth differential category, so all manifolds, vector bundles, maps, morphisms of bundles, differential forms etc. are taken to be smooth. We further assume that our connected and smooth manifolds M are paracompact and of finite Lebesgue dimension, so that we have partitions of unity of finite covering dimension subordinate to any open cover. If inhomogeneous form ω ∈ Ω K (M ) expands as: where the symbol = U means that the equality holds only after restriction of ω to U and where we used the expansion: The locally-defined smooth functions ω = γ((e a ) ♯ ) = g ab γ b . We will occasionally assume that the frame (e a ) is pseudo-orthonormal in the sense that e a satisfy: g(e a , e b ) (= g ab ) = η ab , where (η ab ) is a diagonal matrix with p diagonal entries equal to +1 and q diagonal entries equal to −1.
Remark. Physics terminology is often imprecise with the distinction between spinors and pinors which we are making here and throughout this paper. Physically, one typically assumes that (M, g) is both oriented and time-oriented and one is concerned with objects transforming in representations of the orthochronous part Spin ↑ (p, q) of the spin group Spin(p, q) and thus in vector bundles associated with a principal bundle with fiber Spin ↑ (p, q) which is a double cover of the principal SO ↑ (p, q)-bundle consisting of those pseudo-orthonormal frames of (M, g) which are both oriented and time-oriented. Due to the issue of time-orientability, what matters in many physics applications is not a spin structure in the standard mathematical sense (see [18] for a recent discussion with applications to string theory) but rather a 'time-oriented' spin structure.
Constrained generalized Killing (s)pinors. Let us fix a K-pinor or K-spinor bundle S over M , a linear connection D on S and a finite collection of bundle endomorphisms Q 1 , . . . , Q χ ∈ Γ(M, End(S)).
When the algebraic constraints are trivial (χ = 0 or, equivalently, when all Q j vanish), one deals with the generalized Killing (GK) spinor equation Dξ = 0. Since D can be written as the sum ∇ S + A of the spinorial connection ∇ S induced on S by the Levi-Civita connection of (M, g) and an End(S)-valued one-form A on M , the GK (s)pinor equations can be viewed as a deformation of the parallel (s)pinor equation ∇ S ξ = 0, the deformation being parameterized by A. Our terminology is inspired by the fact that the choice A m = −λγ m (with λ a real parameter and γ m ∈ Γ(M, End(S)) the gamma 'matrices' in some local coframe of (M, g)) leads to the ordinary Killing (s)pinor equations ∇ m ξ = λγ m ξ.
Connection to supergravity and string theories. Constrained generalized Killing (s)pinors arise naturally in supergravity and string theory. In particular, they arise in supersymmetric flux compactifications of string theory, M-theory and various supergravity theories. In such setups, ξ is a (s)pinor of spin 1/2 defined on the background pseudo-Riemannian manifold and corresponds to the generator of supersymmetry transformations of the underlying supergravity action (or string theory effective action) while the constrained generalized Killing (s)pinor equations are the conditions that the supersymmetry generated by ξ is preserved by the background. The connection D on S and the endomorphisms Q j are fixed by the precise data of the background, i.e. by the metric and fluxes defining that background. For example, the supersymmetry equations of eleven-dimensional supergravity involve the supercovariant connection D, which acts on sections of the bundle S of Majorana spinors (a.k.a. real pinors) defined in eleven dimensionsthis corresponds to the differential constraint Dξ = 0, without any algebraic constraint. When considering a compactification of eleven-dimensional supergravity down to a lower-dimensional space admitting Killing (s)pinors, the internal part (which now plays the role of ξ) of the generator of the supersymmetry variation is a section of some bundle of (s)pinors (which now plays the role of S) defined over the internal space, while the condition of preserving the supersymmetry generated by the tensor product of this internal generator and some Killing (s)pinor of the non-compact part of the background induces a differential (generalized Killing) constraint as well as an algebraic constraint for the internal part of the supersymmetry generator. A specific example arising from eleven-dimensional supergravity is discussed in Section 6 below. Similarly, the supersymmetry equations for IIA supergravity in ten dimensions (with Minkowski signature) can be written in terms of a supercovariant connection D defined on the real vector bundle S of Majorana spinors (a.k.a. real pinors) in ten dimensions and an endomorphism Q of S; we have S = S + ⊕ S − where S ± are the bundles of Majorana-Weyl spinors of positive and negative chirality. The condition Dξ = 0 for the supersymmetry generator (a section ξ of S, with positive and negative chirality components ξ ± -which are sections of S ± -such that ξ = ξ + + ξ − ) is the requirement that the supersymmetry variation of the gravitino vanishes, while the condition Qξ = 0 encodes vanishing of the supersymmetry variation of the dilatino. When considering compactifications on some internal space down to some space admitting Killing (s)pinors, ξ is replaced by its internal part (a section of some (s)pinor bundle -which now plays the role of S -defined on the compactification space) while D induces a connection defined on this internal (s)pinor bundle as well as a further algebraic constraint -thereby leading once again to a system of equations of constrained generalized Killing type, which is now defined on the internal space. Finally, the supersymmetry equations for type IIB supergravity in ten dimensions (with Minkowski signature) can be formulated 1 (see, for example, [15]) in terms of sections of the real vector bundle S = S + ⊕ S + of Majorana-Weyl spinor doublets, with a supercovariant connection D defined on this bundle as well as two endomorphisms Q 1 , Q 2 of S. The condition Dξ = 0 for sections ξ of S is the requirement that the supersymmetry variation of the gravitino vanishes in the background, while the conditions Q 1 ξ = Q 2 ξ = 0 are, respectively, the requirements that the supersymmetry variations of the axionino and dilatino vanish. When considering a compactification down from ten dimensions, the constraints Q 1 ξ = Q 2 ξ = 0 descend to similar constraints for the internal part of ξ, while the constraint Dξ = 0 induces both a differential and an algebraic constraint for the internal part -hence the compactification procedure produces a differential constraint while increasing the number of algebraic constraints, the resulting equations being again of constrained generalized Killing type, but formulated for sections of some bundle of (s)pinors defined over the internal space of the compactification.
Some mathematical observations. Let us for simplicity consider the case of a single algebraic constraint (Qξ = 0). Let K(Q) denote the C ∞ (M, K)-submodule of smooth solutions to the equation Qξ = 0 and K(D) denote the K-vector subspace of smooth solutions to the equation Dξ = 0. Then the K-vector subspace K(D, Q) of smooth solutions to the CGK spinor equations equals the intersection K(D) ∩ K(Q). In general, the dimension of the subspace ker(Q x ) ⊂ S x of the fiber of S at a point x may jump as x varies inside M , so Q does not admit a subbundle of S as its kernel (in fact, this is one reason why smooth vector bundles do not form an Abelian category) -even though it does admit a kernel in the category of sheaves over the ringed space associated with M . On the other hand, a simple argument 2 using parallel transport shows that any linearly-independent (over K) collection of smooth solutions ξ 1 , . . . , ξ s of the generalized Killing (s)pinor equation must be linearly independent everywhere, i.e. the vectors ξ 1 (x), . . . , ξ s (x) must be linearly-independent in the fiber S x for any point x of M . In particular, there exists a K-vector sub-bundle S D of S such that rk K S D = dim K K(D) and such that Γ(M, S D ) = K(D) ⊗ K C ∞ (M, K); in fact, any basis of the space of solutions K(D) of the generalized Killing (s)pinor equations provides a global frame for S D (which, therefore, must be a trivial vector bundle). Since the restriction of D to S D is flat, the bundle S D is sometimes referred to as 'the D-flat vector sub-bundle of S'. The condition that the generalized Killing (s)pinor equations admit exactly s linearly independent solutions over K (i.e., the condition dim K K(D) = s) amounts to the requirement that S D has rank s -in particular, this imposes well-known topological constraints on S. A similar argument shows that there exists a (topo- As mentioned in the introduction, a basic problem in the analysis of flux compactifications (which is also of mathematical interest in its own right) is to find efficient methods for translating constrained generalized Killing (s)pinor equations for some collection ξ 1 , . . . , ξ s of sections of S into a system of algebraic and differential conditions for differential forms which are constructed as bilinears in ξ 1 , . . . , ξ s . In Section 5, we show how the geometric algebra formalism can be used to provide an efficient and conceptually clear solution to this problem. Before doing so, however, we have to recall the basics of the geometric algebra approach to spin geometry, which we proceed to do next.

The Kähler-Atiyah bundle of a pseudo-Riemannian manifold
This section lays out the basics of the geometric algebra formalism and develops some specialized aspects which will be needed later on. In Subsections 3.1 and 3.2, we start with the Clifford bundle of the cotangent bundle of a pseudo-Riemannian manifold (M, g), viewed as a bundle of unital and associative -but non-commutative -algebras which is naturally associated to (M, g). The basic idea of 'geometric algebra' is to use a certain isomorphic realization of the Clifford bundle in which the underlying vector bundle is identified with the exterior bundle of M . In this realization, the multiplication of the Clifford bundle transports to a fiberwise multiplication of the exterior bundle; when endowed with this associative but non-commutative multiplication, the exterior bundle becomes a bundle of associative algebras known as the Kähler-Atiyah bundle. In turn, the non-commutative multiplication of the Kähler-Atiyah bundle induces an associative but non-commutative multiplication (which we denote by ⋄ and call the geometric product) on inhomogeneous differential forms. The resulting associative algebra is known as the Kähler-Atiyah algebra of (M, g) and can be viewed as a certain deformation of the exterior algebra which is parameterized by the metric g of M . The Kähler-Atiyah algebra is an associative and unital algebra over the commutative and unital ring C ∞ (M, K) of smooth K-valued functions defined on M -so in particular it is a K-algebra upon considering the embedding K ⊂ C ∞ (M, K) which is defined by associating to each element of K the corresponding constant function. The geometric product has an expansion in terms of so-called 'generalized products', which form a collection of binary operations acting on inhomogeneous forms. In turn, the generalized products can be described as certain combinations of contractions and wedge products. The expansion of the geometric product into generalized products can be interpreted (under certain global conditions on (M, g)) as a form of 'partial quantization' of a spin system -the role of the Planck constant being played by the inverse of the overall scale of the metric. In this interpretation, the Kähler-Atiyah algebra is the quantum algebra of observables while the geometric product is the non-commutative composition of quantum observables; the classical limit corresponds to taking the scale of the metric to infinity while the expansion of the geometric product into generalized products can be viewed as a semiclassical expansion. In the classical limit, the geometric product reduces to the wedge product and the Kähler-Atiyah algebra reduces to the exterior algebra of M , which plays the role of the classical algebra of observables. Subsection 3.3 discusses certain (anti-)automorphisms of the Kähler-Atiyah algebra which will be used intensively later on while subsection 3.4 gives some properties of the left and right multiplication operators in this algebra. In Subsection 3.5, we give a brief discussion of the decomposition of an inhomogeneous form into parts parallel and perpendicular to a normalized one-form and of the interplay of this decomposition with the geometric product. Subsection 3.6 explains the role played by the volume form and introduces the 'twisted Hodge operator', a certain variant of the ordinary Hodge operator which is natural from the point of view of the Kähler-Atiyah algebra. Subsection 3.7 discusses the eigenvectors of the twisted Hodge operator, which we call 'twisted (anti-)selfdual forms' -these will play a crucial role in later considerations. In Subsection 3.8, we recall the algebraic classification of the fiber type of the Clifford/Kähler-Atiyah bundle, which is an obvious application of the well-known classification of Clifford algebras. We pay particular attention to the 'non-simple case' -the case when the fibers of the Kähler-Atiyah bundle fail to be simple as associative algebras over the base field. In Subsection 3.9, we discuss the spaces of twisted (anti-)selfdual forms in the non-simple case, showing that -in this case -they form two-sided ideals of the Kähler-Atiyah algebra. We also give a description of such forms in terms of rank truncations, which is convenient in certain computations even though it is not well-behaved with respect to the geometric product. In Subsection 3.10, we show that, in the presence of a globally-defined one-form θ of unit norm, the spaces of twisted selfdual and twisted anti-selfdual forms are isomorphic (as unital associative algebras !) with the space of those inhomogeneous forms which are orthogonal to θ -a space which always forms a subalgebra of the Kähler-Atiyah algebra. We also show that the components of an inhomogeneous form which are orthogonal and parallel to θ determine each other when the form is twisted (anti-)selfdual and give explicit formulas for the relation between these components in terms of what we call the 'reduced twisted Hodge operator'. Some of the material of this section is 'wellknown' at least in certain circles, though the literature tends to be limited in its treatment of general dimensions and signatures and of certain other aspects. The reader who is familiar with geometric algebra may wish to concentrate on Subsections 3.5, 3.6, 3.7, 3.9 and 3.10 and especially on our treatment of parallelism and orthogonality for twisted (anti-)selfdual forms, which is important for applications.

Preparations: wedge and generalized contraction operators
The grading automorphism. Let π be that involutive C ∞ (M, K)-linear automorphism of the exterior algebra (Ω K (M ), ∧) which is uniquely determined by the property that it acts as minus the identity on all one-forms. Thus: Taking wedge products from the left and from the right with some inhomogeneous form ω ∈ Ω K (M ) defines C ∞ (M, K)-linear operators ∧ L ω and ∧ R ω : which satisfy the following identities by virtue of the fact that the wedge product is associative: as well as the following relation, which encodes graded-commutativity of the wedge product: The inner product. Let , denote the symmetric non-degenerate C ∞ (M, K)-bilinear pairing (known as the inner product of inhomogeneous forms) induced by the metric g on the exterior bundle. To be precise, this pairing is defined through: a relation which fixes the convention used later in our computations (cf. Section 6) via the normalization property: Here,ĝ is the metric induced by g on T * K M , which gives the following pairing on one-forms α = α a e a , β = β b e b ∈ Ω 1 K (M ): g(α, β) = g ab α a β b for α = α a e a and β = β b e b .
The fixed rank components of Ω K (M ) are mutually orthogonal with respect to the pairing , : so the rank decomposition Ω K (M ) = ⊕ d k=0 Ω k K (M ) is an orthogonal direct sum decomposition with respect to this pairing. Notice that the restriction of , to Ω 1 K (M ) coincides with (7). Also notice that π is self-adjoint with respect to the pairing , : Interior products. The , -adjoints of the left and right wedge product operators (4) are denoted by ι R ω and ι L ω and are called the right and left generalized contraction (or interior product) operators, respectively: Properties (5) translate into: while relation (6) is equivalent with: We also have ∧ L,R 1 M = id Ω K (M ) and ι L,R 1 M = id Ω K (M ) . Together with (5) and (9), this shows that ∧ L and ∧ R define a structure of (Ω K (M ), ∧)-bimodule on Ω K (M ) while ι L and ι R define another (Ω K (M ), ∧)-bimodule structure on the same space. These two bimodule structures are adjoint to each other with respect to the pairing , .
Identities (6) and (10) show that ∧ L ω and ∧ R ω determine each other while ι L ω and ι R ω also determine each other. Because of this, we from now on choose to work with the left wedgemultiplication and left generalized contraction (= left interior product) operators: = ∧ L ω , ι ω = ι L ω , which satisfy: as well as: and thus define two different structures of left module on the space Ω K (M ) over the ring (Ω K (M ), ∧).
Wedge and interior product with a one-form. For later reference, let us consider the case when θ ∈ Ω 1 K (M ) is a one-form. Recall that the metric g induces mutually-inverse 'musical isomorphisms' ♯ : Γ(M, T K M ) → Ω 1 K (M ) and ♯ : Ω 1 K (M ) → Γ(M, T K M ) defined by raising and lowering of indices, respectively: X = X a e a =⇒ X ♯ = X a e a where X a def.
= g ab X b , θ = θ a e a =⇒ θ ♯ = θ a e a where θ a def.
These isomorphisms satisfy: . We have: where X denotes the ordinary left contraction of a tensor with a vector field. It is not hard to see that the left contraction ι θ with a one-form coincides with the ordinary left contraction θ ♯ with the vector field θ ♯ : Furthermore, the similar property of θ ♯ implies that ι θ is an odd derivation of the exterior algebra: Local expressions. If e a is an arbitrary local frame of M with dual coframe e a (thus e a (e b ) = δ a b ), we let g ab = g(e a , e b ) and g ab =ĝ(e a , e b ), so we have g ab g bc = δ a c . The vector fields (e a ) ♯ satisfy (e a ) ♯ g = e a and are given explicitly by: (e a ) ♯ = g ab e b ; they form the contragradient local frame defined by (e a ). We have e a = g ab (e b ) ♯ and g((e a ) ♯ , (e b ) ♯ ) = g ab . Thus:

Definition and first properties of the Kähler-Atiyah algebra
The geometric product. Following an idea originally due to Chevalley and Riesz [8,9], we identify Cl(T * K M ) with the exterior bundle ∧T * K M , thus realizing the Clifford product as the geometric product, which is the unique fiberwise associative, unital and bilinear binary composition 4 ⋄ : ∧T * K M × M ∧T * K M → ∧T * K M whose induced action on sections (which we again denote by ⋄) satisfies the following relations for all θ ∈ Ω 1 K (M ) and all ω ∈ Ω K (M ): Equations (15) determine the geometric composition of any two inhomogeneous forms via the requirement that the geometric product is associative and C ∞ (M, K)-bilinear.
The unit of the fiber Cl(T * K,x M ) at a point x ∈ M corresponds to the element 1 ∈ K = ∧ 0 T * K,x M , which is the unit of the associative algebra (∧T * Hence the unit section of the Clifford bundle Cl(T * K M ) is identified with the constant function 1 M : M → K given by 1 M (x) = 1 for all x ∈ M . Through this construction, the Clifford bundle is identified with the bundle of algebras (∧T * K M, ⋄), which is known [10] as the Kähler-Atiyah bundle of (M, g). When endowed with the geometric product, the space Ω K (M ) of all inhomogeneous K-valued smooth forms on M becomes a unital and associative (but non-commutative) algebra (Ω K (M ), ⋄) over the ring C ∞ (M, K), known as the Kähler-Atiyah algebra of (M, g). The unit of the Kähler-Atiyah algebra is the constant function 1 M . We have a unital isomorphism of associative algebras over C ∞ (M, K) between (Ω K (M ), ⋄) and the C ∞ (M, K)-algebra Γ(M, Cl(T * K M )) of all smooth sections of the Clifford bundle. The Kähler-Atiyah algebra can be viewed as a Z 2 -graded associative algebra with even and odd parts given by: , since it is easy to check the inclusions: . However, it is not a Z-graded algebra since the geometric product of two forms of definite rank need not be a form of definite rank. We let P ev = 1 2 (1 + π) and P odd = 1 2 (1 − π) be the complementary idempotents associated with the decomposition into even and odd parts: where ω = ω ev + ω odd ∈ Ω K (M ), with ω ev ∈ Ω ev K (M ) and ω odd ∈ Ω odd K (M ).
Generalized products. Connection with quantization of spin systems. The geometric product ⋄ can be viewed as a deformation of the wedge product (parameterized by the metric g) and reduces to the latter in the limit g → ∞; in this limit, the Kähler-Atiyah algebra reduces to the exterior algebra (Ω K (M ), ∧). Under some mild assumptions, the geometric product can be described quite elegantly in the language of supermanifolds, as the star product induced by fiberwise Weyl quantization of a pure spin system [19][20][21][22]. For this, consider the parity-changed tangent bundle ΠT M of M (a supermanifold with body M ) and introduce odd coordinates ζ a on the fibers of ΠT U , corresponding to a coframe e a of M defined on a small enough open subset U ⊂ M . Inhomogeneous differential forms (1) correspond to functions defined on ΠT M having the following local expansion: This allows us to represent the geometric product through the fermionic analogue ⋆ of the Moyal product, using a certain 'vertical' [22] quantization procedure 5 : Expanding the exponential in (18) gives the following expressions for two general inhomogeneous forms ω, η ∈ Ω K (M )(cf. [23][24][25]): as well as the following expansions for the graded ⋄-commutator and graded ⋄-anticommutator of ω with η: where |0 x is the vacuum at x and z k = 1 2 (ζ k + iζ r+k ) are odd complex coordinates along the fibers of ΠT C M . Identifying Ω C (M ) with the algebra of complex functions on ΠT M , we have a k = ∂ ∂z k and a † k =z k , so the bundle of spin Fock spaces can be identified with the sub-bundle ∧W * of ∧T * C M . The star product ⋆ takes the form: which agrees with (18).
For forms of definite ranks, the graded ⋄-commutator and graded ⋄-anticommutator are of course defined through: = ω⋄η+(−1) pq η⋄ω , ∀ω ∈ Ω p K (M ) , ∀η ∈ Ω q K (M ) , being extended by linearity to the entire space Ω K (M ). Using (19) one can easyly deduce the following relation for homogeneous forms: The binary C ∞ (M, K)-bilinear operations ∧ k which appear in the expansions above are the contracted wedge products [23][24][25][26], defined iteratively through: We will mostly use, instead, the so-called generalized products △ k , which are defined by rescaling the contracted wedge products: These have the advantage that the various factorial prefactors in the expansions above disappear when those expansions are re-expressed in terms of generalized products. Expansions (19) and (20) can also be obtained directly from the definition of the geometric product using (15), which shows that the purely mathematical identities given above also hold irrespective of any interpretation through the theory of quantization of spin systems.

(Anti-)automorphisms of the Kähler-Atiyah algebra
Direct computation shows that π is an involutive automorphism (known as the main or grading automorphism) of the Kähler-Atiyah bundle (a property which, in the limit g → ∞, recovers the well-known fact that π is also an automorphism of the exterior bundle). The Kähler-Atiyah bundle also admits an involutive anti-automorphism τ (known as the main anti-automorphism or as reversion), which is given by: It is the unique anti-automorphism of (∧T * K M, ⋄) which acts trivially on all one-forms (i.e. which satisfies τ (θ) = θ for any form θ of rank one). Direct computation (or the fact that the exterior product is recovered from the diamond product in the limit of infinite metric) shows that τ is also an anti-automorphism of the exterior bundle (T * K M, ∧). We also notice that π and τ commute. All in all, we have the relations: Note that Cl ev (T * K M ) identifies with the sub-bundle of unital subalgebras ∧ ev T * K M = ⊕ k=even ∧ k T * K M of the Kähler-Atiyah bundle, whose space of smooth sections Ω ev K (M ) can be described as the eigenspace of π corresponding to the eigenvalue +1:

The left and right geometric multiplication operators
Let L ω , R ω be the C ∞ (M, K)-linear operators of left and right multiplication with ω ∈ Ω K (M ) in the Kähler-Atiyah algebra: These satisfy: as a consequence of associativity of the geometric product. We also have: since π is an involutive algebra automorphism while τ is an involutive anti-automorphism. Identity (15) can be written as: being equivalent with: This shows that the operators ∧ θ and ι θ (and thus -given properties (11) -also the operators ∧ ω and ι ω for any ω ∈ Ω K (M )) are determined by the geometric product.

Orthogonality and parallelism
Let θ ∈ Ω 1 K (M ) be a fixed one-form which satisfies the normalization condition: This condition is equivalent with: a fact which follows from (15) and from the identity θ ∧θ = 0 (which, together, imply θ ⋄θ = ι θ θ).
Proposition. Any inhomogeneous differential form ω ∈ Ω K (M ) decomposes uniquely as: where θ ω and θ ⊥ ω ⊥ . Moreover, the parallel and orthogonal parts of ω are given by: Proof. The statements of the proposition follow immediately from the fact that ∧ θ and ι θ are nilpotent and because ι θ is an odd derivation of the wedge product, which implies: where we used the normalization condition (26).
As an immediate corollary of the proposition, we find the well-known equalities: as well as the characterizations: where we used relations (24). Thus θ ω iff. ω graded anticommutes with θ and θ ⊥ ω iff. ω graded commutes with θ in the Kähler-Atiyah algebra.
Behavior with respect to the geometric product. Consider the following C ∞ (M, K)submodules of Ω K (M ): Using the characterizations in (29), we find: which translate into: We thus have the inclusions: . Together with the identity ι θ (1 M ) = 0 (which shows that θ ⊥ 1 M ), the last property in (30) shows that Ω ⊥ K (M ) is a unital subalgebra of the Kähler-Atiyah algebra. Notice that characterizations (29) imply that the involutions π and τ preserve parallelism and orthogonality to θ: The top component of an inhomogeneous form. The parallel part of ω ∈ Ω K (M ) can be written as: where: This shows that ω determines and is determined by the two inhomogeneous forms ω ⊥ and ω ⊤ , both of which belong to Ω ⊥ K (M ). In fact, any ω ∈ Ω K (M ) can be written uniquely in the form: namely we have α = ω ⊤ and β = ω ⊥ . This gives a C ∞ (M, K)-linear isomorphism: which sends ω ∈ Ω K (M ) into the pair (ω ⊤ , ω ⊥ ) and whose inverse sends a pair (α, β) with α, β ∈ Ω ⊥ K (M ) into the form (35). Since ω ⊤ is orthogonal to θ, we have θ∧ω ⊤ = θ⋄ω ⊤ = π(ω ⊤ )⋄θ and thus ω = θ ⋄ ω ⊤ . It follows that the decomposition of ω can be written entirely in terms of the geometric product: An easy computation using this formula gives:

The volume form and the twisted Hodge duality operator
The ordinary volume form. From now on, we shall assume that M is oriented (in particular, the K-line bundles Λ d T * K M are trivial for K = R, C). Consider the volume form determined on M by the metric and by this orientation, which has the following expression in a local frame defined on U ⊂ M : Here, det g is the determinant of the matrix (g(e a , e b )) a,b=1...d while ǫ a 1 ...a d are the local coefficients of the Ricci density -defined as the signature of the permutation The volume form satisfies: where we used the congruences: The ordinary Hodge operator. Recall that the ordinary C ∞ (M, K)-linear Hodge operator * is defined through: and satisfies the following properties, which we list for convenience of the reader: We also note the identity: which follows by direct computation upon using the congruence: The modified volume form. Consider the following K-valued top form on M : which satisfies: We have the normalization property: and the identity: where π d−1 represents the composition of d − 1 copies of the main automorphism π: In particular, ν is a central element of the Kähler-Atiyah algebra iff. d is odd. = ω ⋄ ν , ∀ω ∈ Ω K (M ) .
Identity (42) shows that (unlike what happens for the ordinary Hodge operator) the square of the twisted Hodge operator is always a scalar multiple of the identity: A simple computation shows that the twisted and ordinary Hodge operators are related through: * = c p,q (K) * • τ .
In particular, the ordinary Hodge operator admits the representation:

Twisted (anti-)selfdual forms
Let us assume that K = C or that K = R and p − q ≡ 4 0, 1, so that the twisted Hodge operator * squares to the identity. In this case, the twisted Hodge operator has real eigenvalues equal to ±1 and we can consider inhomogeneous real forms belonging to the corresponding eigenspaces. A form ω ∈ Ω K (M ) will be called twisted selfdual if * ω = +ω and twisted anti-selfdual if * ω = −ω.
The ideals Ω ± K (M ). The elements p ± def.
= 1 2 (1 ± ν) are complementary idempotents of the Kähler-Atiyah algebra: Notice that these idempotents are central only when ν is central, i.e. only when d is odd. The operators P ± def.
= R p ± defined through right ⋄-multiplication with these elements: are complementary idempotents in the algebra of endomorphisms of the C ∞ (M, K)-module Ω K (M ): Therefore, the images Ω ± K (M ) = P ± (Ω K (M )) = Ω K (M )p ± are complementary left ideals of the Kähler-Atiyah algebra, giving the direct sum decomposition: In particular, (Ω ± K (M ), ⋄) are associative subalgebras of the Kähler-Atiyah algebra. These subalgebras have units (given by p ± ) iff. d is odd, in which case they are two-sided ideals of (Ω K (M ), ⋄).
Local characterization. With respect to a local coframe e a above an open subset U ⊂ M , we have the expansions: where indices are raised with g ab . Using (44), one easily checks that an inhomogeneous form ω ∈ Ω K (M ) with expansion (1) satisfies * ω = ±ω iff. its non-strict coefficients satisfy the conditions: We note here for future reference the expansions for the Hodge dual and the twisted Hodge dual of any k-form ω :

Algebraic classification of fiber types
The fibers of the Kähler-Atiyah bundle are isomorphic with the Clifford algebra Cl K (p, q) = Cl(p, q)⊗ R K, whose classification is well-known. For K = C, we have an isomorphism of algebras = Cl C (d) and the classification depends only on the mod 2 reduction of d; for K = R, it depends on the mod 8 reduction of p − q. The Schur algebra S K (p, q) is the largest division algebra contained in the center of Cl K (p, q); it is determined up to isomorphism of algebras, being isomorphic with R, C or H. The Clifford algebra is either simple (in which case it is isomorphic with a matrix algebra Mat(∆ K (d), S K (p, q))) or a direct sum of two central simple algebras (namely, the direct sum Mat(∆ K (d), S K (p, q)) ⊕ Mat(∆ K (d), S K (p, q))), where the positive integers ∆ K (d) are given by well-known formulas recalled below. We say that the Clifford algebra is normal if its Schur algebra is isomorphic to the base field. It is convenient for our purpose to organize the various cases according to the isomorphism type of the Schur algebra and to whether the Clifford algebra is simple or not: When K = C. In this case, the Schur algebra is always isomorphic with C (so Cl C (p, q) ≈ Cl C (d) is always normal) and we always have ν ⋄ ν = +1 and ∆ and ν is central.
1. The Schur algebras and the numbers ∆ R (d) are as follows (see, for example, [27]): 2. The simple and non-simple cases occur as follows: In this case, we always have ν ⋄ ν = +1 and ν is central.
The situation when K = R is summarized in Table 1. For both K = R and K = C, the Clifford algebra is non-simple iff. ν is central and satisfies ν ⋄ ν = 1. In this case -for both K = R and K = C -the Clifford algebra admits two inequivalent irreducible representations by K-linear operators, which are related by the main automorphism of the Clifford algebra and both of which are non-faithful; their Schur algebra equals C when K = C but may equal either R or H when K = R. Table 1. Properties of ν according to the mod 8 reduction of p−q for the case K = R. At the intersection of each row and column, we indicate the values of p − q (mod 8) for which the modified volume form ν has the corresponding properties. In parentheses, we also indicate the isomorphism type of the Schur algebra for that value of p − q (mod 8). The real Clifford algebra Cl(p, q) is non-simple iff. p − q ≡ 8 1, 5, which corresponds to the upper left corner of the table. Notice that ν is central iff. d is odd.

Twisted (anti-)selfdual forms in the non-simple case
In this Subsection, let us assume that we are in the non-simple case. Then ν ⋄ ν = +1 and (since d is odd in the non-simple case) ν is a central element of the Kähler-Atiyah algebra: Using the fact that ν is central, an easy computation shows that P + and P − are (non-unital) algebra endomorphisms of the Kähler-Atiyah algebra, in fact: and: In this case, Ω ± K (M ) are complementary two-sided ideals of the Kähler-Atiyah algebra (indeed, p ± are central) -in particular, (Ω ± K (M ), ⋄) are unital algebras, their units being given by p ± .
Truncation and prolongation. Since d is odd in the non-simple case, we have the decomposition: where: The corresponding complementary C ∞ (M, K)-linear idempotent operators P < , P > : Ω K (M ) → Ω K (M ) are given by: , we define ω < (the lower truncation of ω) and ω > (the upper truncation of ω) through: We have: When ω is twisted (anti-)selfdual (i.e. ω ∈ Ω ǫ K (M ) with ǫ = ±1), we have: * ω = ǫω, which implies: Hence in this case ω can be re-constructed from its lower truncation as: It follows that the restriction of 2P < to the subspace Ω ǫ K (M ) gives a C ∞ (M, K)-linear bijection between this subspace and the subspace Ω < K (M ), with inverse given by the restriction of P ǫ to Ω < K (M ). We define the twisted (anti-)selfdual prolongation of a form ω ∈ Ω < K (M ) through: Of course, the form ω ∈ Ω < K (M ) can be recovered from its two prolongations as ω = ω + + ω − .
The truncated algebra (Ω < K (M ), ♦ ± ). We stress that P < does not preserve the geometric product on its entire domain of definition Ω K (M ) -in fact, its image Ω < K (M ) is not a subalgebra of the Kähler-Atiyah algebra since it is not stable with respect to ⋄-multiplication. To cure this problem, we use the linear isomorphisms mentioned above to transfer the multiplication ⋄ of the unital subalgebra Ω ǫ K (M ) to an associative and unital multiplication ♦ ǫ defined on Ω < K (M ) through: Since P ǫ is a morphism of algebras on its entire domain of definition Ω K (M ), we have P ǫ (ω) ⋄ P ǫ (η) = P ǫ (ω ⋄ η), so (47) gives: ) and * • P < = P > • * , this implies: a formula which can be used to implement the product ♦ ǫ in a symbolic computation system. Combining everything shows that we have mutually inverse isomorphisms of algebras: Thus (Ω < K (M ), ♦ ǫ ) provides a model for the unital associative algebra (Ω ǫ K (M ), ⋄).
Local expansions. Let us further assume that d ≥ 3. Then the covector fields e a ∈ Ω 1 K (M ) defined by a local pseudo-orthonormal frame above U ⊂ M belong to the subspace Ω < K (M ) and we consider their twisted (anti-)selfdual prolongations: Since e a 1 ...a k = e a 1 ∧ . . . ∧ e a k = e a 1 ⋄ . . . ⋄ e a k and since P ± are endomorphisms of the Kähler-Atiyah algebra, we find that the prolongations of e a 1 ...a k are given by ⋄-monomials in the prolongations of e a : In particular, the twisted (anti-)selfdual forms form a basis of the module Ω < K (U ) and since the operation of taking the prolongation is an isomorphism of C ∞ (U, K)-modules). In fact, any twisted (anti-)selfdual form ω ∈ Ω ± K (M ) (cf. (1),(2)) expands as: (2). The coefficients with k ≥ d 2 +1 are determined by those with k ≤ d 2 through relations (45). The lower truncation of such ω has the local expansion: We also note the explicit expressions: where c p,q (K) was defined in (41).

Orthogonality in the non-simple case
Assuming that we are in the non-simple case, let us consider the situation when we have a distinguished one-form θ ∈ Ω 1 K (M ) which satisfies the normalization condition ι θ θ = 1.
The isomorphism of algebras between Ω ⊥ K (M ) and Ω ǫ K (M ). For any ω ∈ Ω K (M ), consider the decomposition ω = ω + ω ⊥ into parts ω = θ ∧ (ι θ ω) and ω ⊥ = ι θ (θ ∧ ω) = ω − ω parallel and perpendicular to θ. Since θ ν (indeed, we have θ ∧ ν = 0) and * ω = ω ⋄ ν, properties (30) imply: which gives: The subalgebras Ω ǫ K (M ) and Ω ⊥ K (M ) of the Kähler-Atiyah algebra can be identified with each other using the operator 2P ⊥ , which takes a twisted (anti-)selfdual form ω into 2ω ⊥ . Indeed, if * ω = ǫω (with ǫ = ±1), then * (ω ) = ǫω ⊥ and * (ω ⊥ ) = ǫω . Hence the last of equations (31) implies: where we used the fact that ν is central in the Kähler-Atiyah algebra and that it squares to 1 M . Thus: We also have: where we used the fact that θ||ν. These properties show that the restriction ). An easy computation shows that it is an isomorphism whose inverse equals the restriction of P ǫ to Ω ⊥ K (M ). It follows that we have mutually inverse unital isomorphisms of algebras: Combining with the results of Subsection 3.9, we have thus found two isomorphic models for the unital subalgebra (Ω ǫ K (M ), ⋄): The reduced twisted Hodge operator. Since θ ν, we can write: where the reduced volume form ν ⊤ is defined through: The last two equalities in (56) follow from (15) and from the fact that (in the non-simple case) ν is central in the Kähler-Atiyah algebra (since d is odd in this case). Multiplying the last equation with θ in the Kähler-Atiyah algebra and using the fact that θ ⋄ θ =ĝ(θ, θ) = 1 gives: Notice the identity: which follows from (56) using the fact that ν is central, the normalization condition for θ and the property ν ⋄ ν = +1 M , which always holds in the non-simple case. Defining the reduced twisted Hodge operator * 0 through: we have π(ν ⊤ ) = ν ⊤ , so equation (58) implies: For later reference, we note the identities (where we use (57) and the fact that π(ν ⊤ ) = ν ⊤ ): as well as: which follow by easy computation. Using (24), the last identities imply the following anticommutation relations, which will be important below: To find explicit expressions for the parallel and perpendicular parts of * ω, notice that * ω = where we used (15) and the fact that π(ν ⊤ ) = +ν ⊤ . Thus: The decomposition ω = ω + ω ⊥ and the fact that ι θ ν ⊤ = 0 (thus θ ⊥ ν ⊤ ) imply (using (30)): and (using (32) and the fact that π(ν ⊤ ) = +1): These relations show that: as well as: Combining the last relation with (62) gives: Equations (64) and (53) read: while (63) gives: In particular, we have: The relations above imply the following: Lemma. Consider the operators α θ def.
Notice that (65) takes the form: For reader's convenience, we also list a few other properties which follow immediately from the above: Proposition. Let ω ∈ Ω K (M ). Then the following statements are equivalent: (a) ω is twisted (anti-)selfdual, i.e. * ω = ǫω for ǫ = ±1 .
(b) The components ω and ω ⊥ satisfy the following equivalent relations: In this case, ω and ω ⊥ determine each other and thus any of them determines ω.
Proof. The fact that the two relations listed in (69) are equivalent to each other is an immediate consequence of the Lemma. The rest of the proposition follows from (66).

Proof. Idempotency follows by noticing that
. The fact that ϕ ǫ is a morphism of algebras follows since both P ⊥ and P ǫ are such. Finally, unitality of ϕ ǫ follows by computing: where we used P ⊥ (1 M ) = 1 M and P ⊥ (ν) = 0.
It is clear that ϕ ǫ is surjective. Moreover, the last proposition of the previous paragraph implies: It follows that ϕ ǫ restricts to an isomorphism from Ω ǫ K (M ) to Ω ⊥ K (M ) -which, of course, equals the isomorphism 2P ⊥ | Ω ǫ K (M ) of diagram (54). Notice the relations: where the first equality follows from the fact that 2P ǫ • P ⊥ restricts to the identity on Ω ǫ K (M ) (see diagram (54)). Also notice the property: which follows by direct computation upon using P ⊥ (1 M ) = 1 M , P ⊥ (ν) = 0 and the fact that ν ⋄ ν = 1 M .

Describing bundles of pinors
In this section, we discuss the realization of pin bundles within the geometric algebra formalism -focusing especially on the non-simple case, when the irreducible pin representations are nonfaithful. Subsection 4.1 discusses an approach to pinor bundles which is particularly well adapted to the geometric algebra formalism. In this approach (which, in some ways, goes back to Dirac -see [17,28] for a beautiful treatment), one defines pinors as sections of a bundle S of modules over the Kähler-Atiyah algebra, the fiberwise module structure being described by a morphism γ : (Ω K (M ), ⋄) → (End(S), •) of bundles of algebras. For the case when γ is irreducible on the fibers, the well-known representation theory relevant for this construction and its relation with the fiber type classification of the Kähler-Atiyah bundle is recalled in Subsection 4.2, paying attention to characterizing the kernel and image of γ. In Subsection 4.3, we introduce a certain 'partial inverse' γ −1 of γ, which provides a sort of 'vertical dequantization' map. Subsection 4.4 discusses a trace on the subalgebra Ω γ K (M ), which is related by γ to the natural fiberwise trace on the pin bundle.

Basic considerations
We define a bundle of K-pinors to be a bundle S of modules over the Clifford bundle Cl(T * K M ), i.e. a K-vector bundle all of whose fibers S x (x ∈ M ) are modules over the corresponding Clifford algebras Cl(T * K,x M ). In our language, such a bundle is simply a bundle of modules over the Kähler-Atiyah bundle (∧T * K M, ⋄). In the particular case when the morphism γ : (∧T * K M, ⋄) → End(S) induces an irreducible representation of the Clifford algebra Cl(T * K,x M ) on each fiber End(S x ), a bundle of pinors will be called a pin bundle. Bundles of K-spinors and K-spin bundles are defined similarly, but replacing Cl(T * K M ) with Cl ev (T * K M ), i.e. replacing the Kähler-Atiyah bundle with its even sub-bundle. Physically, smooth sections of a (s)pin bundle describe K-valued (s)pinors of spin 1/2 defined over the manifold M . As explained in [17,28], a pin bundle S in our sense exists on M iff. M admits a so-called Clifford c -structure, i.e. a 'reduction' of the structure group of the bundle of pseudo-orthonormal frames of T K M to a certain extension of the Clifford (a.k.a. Lipschitz) group over K. In this case, the choices globally available for S depend -up to isomorphism -on the choices of a Clifford c structure and can be obtained from such a structure by applying the associated bundle construction. The map induced on sections satisfies: as well as: where id S ∈ Γ(M, End(S)) denotes the identity section of the bundle End(S).
In the language of 'vertical quantization' of spin systems, the (L 2 -completion of the) space Γ(M, S) of smooth sections of S plays the role of the Hilbert space (when K = R, one of course has to consider the complexification of S instead). In this interpretation, γ plays the role of quantization map, giving a morphism from the algebra of quantum observables of the system (which is the Kähler-Atiyah algebra) to the (L 2 -completion of the) algebra (Γ(M, End(S)), •), which plays the role of algebra of 'vertical' operators acting in the Hilbert space. These statements can be made quite precise provided that certain global conditions are imposed on (M, g), but the rigorous treatment of this issue falls outside of the scope of this paper.
Notation. If (e a ) a=1...d is a local frame of T M above an open subset U ⊂ M (with dual coframe (e a ) a=1...d ), then any inhomogeneous differential form ω on M expands locally as in (1). We define γ a def.
Injectivity and surjectivity of γ. It is important to note that γ need not be fiberwise injective or surjective, i.e. the morphisms of algebras γ x : (ΛT * K,x M, ⋄ x ) ≈ Cl(T * K,x M ) → End(S x ) need not be injective or surjective. The following characterization is convenient in this regard: • γ is fiberwise injective iff. the fiber of the Kähler-Atiyah bundle is simple as an associative algebra.
• γ is fiberwise surjective iff. the Schur algebra of the fiber of the Kähler-Atiyah bundle is isomorphic with the base field K.
This gives the following classification: The case K = C. Then γ is always fiberwise surjective, being fiberwise injective iff. d is even.
injective non-injective surjective 0(R), 2(R) 1(R) non-surjective 3(C), 7(C), 4(H), 6(H) 5(H) Table 2. Fiberwise character of γ for the case K = R. At the intersection of each row and column, we indicate the values of p − q (mod 8) for which the map induced by γ on each fiber of the Kähler-Atiyah algebra has the corresponding properties. In parentheses, we also indicate the isomorphism type of the Schur algebra for that value of p − q (mod 8). Note that γ is fiberwise surjective exactly for the normal case, i.e. when the Schur algebra is isomorphic with R.
The Schur bundle and the image of γ. The Schur algebra S K (p, q) of Subsection 3.8 is realized naturally in the representation space. Picking a point x on M , let Σ x be the subalgebra of (End(S x ), •) consisting of those endomorphisms T x ∈ End(S x ) which commute with any operator lying in the image of γ x : Then Σ x is isomorphic with S K (p, q) for all x ∈ M . The bundle determined by Σ x when x varies on M will be called the Schur bundle of γ; it is a bundle of sub-algebras of (End(S), •). Of course, the space S x can be viewed as a left Σ x -module via the obvious action of the elements of Σ x -whereby S can be viewed as a bundle of modules over the Schur bundle. The image γ(ΛT * K M ) of γ coincides with the sub-bundle End Σ (S) ⊂ End(S) whose fiber at x ∈ M is given by: We also note that the image of the map induced by γ on sections is given by: Irreducible algebra representations of the fiber of the Kähler-Atiyah bundle. We end by recalling some well-known facts from the representation theory of Clifford algebras: • A simple Clifford algebra admits (up to K-linear equivalence) a single non-trivial irreducible representation by K-linear operators, whose dimension equals ∆ K (d) dim K S K (p, q).
• A non-simple Clifford algebra admits (up to K-linear equivalence) two non-trivial irreducible representations by K-linear operators, whose real dimensions are both equal to ∆ K (d) dim R S K (p, q). The two representations map the Clifford volume element determined by some given orientation into a sign factor times the identity operator of the representation space, and are distinguished one from another by the value of that signed factor.

A partial inverse of γ in the non-simple case
In the non-simple case, the bundle morphism γ has the property: where ǫ γ ∈ {−1, 1} is a sign factor which we shall call the signature of γ. Direct computation using (78) gives: which implies that γ vanishes when restricted to the sub-bundle ∧ −ǫγ T * K M and that its restriction to ∧ ǫγ T * K M gives an isomorphism between this latter sub-bundle of algebras and the sub-bundle of algebras (End Σ (S), •) of End(S). We have ker(γ) = ∧ −ǫγ T * K M . Hence the corresponding map on sections (which we denote again by γ) has kernel K(γ) = Ω −ǫγ K (M ) while its restriction to Ω ǫγ K (M ) gives an isomorphism between this latter subalgebra of the Kähler-Atiyah algebra and the subalgebra Γ(M, End Σ (S)) of (Γ(M, End(S)), •).
The sub-bundle ∧ γ T * K M and the subalgebra Ω γ K (M ). Let us introduce notation which will allow us to treat all cases uniformly.
• In the non-simple case (when the signature ǫ γ is defined), we let = Ω ǫγ K (M ).
• In the simple case, we let ∧ γ T * K M def.
In both cases, we have Ω γ K (M ) = Γ(M, ∧ γ T * K M ). Notice that ∧ γ T * K M is always a sub-bundle of unital algebras of the Kähler-Atiyah bundle while Ω γ K (M ) is always a unital subalgebra of the Kähler-Atiyah algebra.
The partial inverse of γ.
Consider the bundle isomorphism: obtained by restricting the domain of definition of γ to the sub-bundle ∧ γ T * K M of the exterior bundle and the co-domain of definition to the sub-bundle End Σ (S) of End(S). We let: be the inverse of (79). The corresponding maps on sections give the mutually-inverse isomorphisms of algebras displayed in diagram (80).
(Ω γ K (M ), ⋄) Notation. We define: In the context of 'vertical quantization' of spin systems, γ −1 plays the role of a (partial) 'vertical dequantization map', soŤ in equation (81) is the dequantization of a 'vertical' operator T acting in the Hilbert space. The partial inverse of γ allows us to transfer statements about operators acting on pinors to statements about differential forms -an observation which will be used intensively in what follows. Notice the relations: Local expression for γ −1 . Considering a local pseudo-orthonormal coframe e a and recalling that γ(e a ) = γ a , we find: where we have set: where e a 1 ...a k ± are defined in (52) and we used the fact that γ a 1 ...a k = γ a 1 • . . . • γ a k for all mutually-distinct a 1 . . . a k , the fact that γ −1 is an isomorphism of algebras when corestricted to its image and (for the non-simple case) identity (52).

Trace on Ω γ K (M)
The subalgebra Ω γ K (M ) admits a C ∞ (M, K)-linear map S : Ω ǫ K (M ) → C ∞ (M, K) given by: where ω (0) is the rank 0 component of ω (see expansion (1)), S is any of the K-pinor bundles and N p,q equals 1 or 2 according to whether the corresponding fiberwise representation is faithful or not (notice that N p,q rk K S is the dimension of the smallest faithful representation of the fiberwise Clifford algebra). One has: S(ω) = tr(γ(ω)) , ∀ω ∈ Ω γ K (M ) as well as:

The Fierz isomorphism and generalized Killing forms
In this section, we take up the issue of translating constrained generalized Killing pinor equations into conditions on differential forms. To simplify presentation, we shall assume from the outset that the Schur algebra is isomorphic with the base field K, so that either K = C or we are in the normal case with K = R. We start in Subsection 5.1 with a discussion of the bundle of bipinors. Subsection 5.2 considers a certain isomorphism of bundles of algebras (which we shall call the Fierz isomorphism) that provides an identification of the bundle of bipinors with the bundle (∧ γ T * K M, ⋄) and allows for a concise description of those Fierz identities which involve four pinors. This construction makes essential use of a choice of bilinear and non-degenerate 'admissible' form B on the pin bundle (such inner products were classified in [29], see also [30]). Subsection 5.3 extracts some basic properties of this isomorphism which will be useful later on. In Subsection 5.4, we give a brief discussion of completeness relations for the endomorphism algebra of the pin bundle. Subsection 5.5 gives an explicit local expansion of the Fierz isomorphism which depends on the choice of a local pseudo-orthonormal coframe. In Subsection 5.6, we show how algebraic constraints on pinors translate very directly into constraints on differential forms if one uses the basic properties of the Fierz isomorphism. Subsection 5.7 takes up the problem of translating generalized Killing pinor equations into conditions on differential forms. Using the Fierz isomorphism, we show that any connection on the pin bundle which is compatible with B defines a certain algebra connection on the Kähler-Atiyah bundle (i.e. a linear connection which is a fiberwise derivation of the geometric product) such that the Fierz isomorphism is flat with respect to the connections induced on its domain and codomain. Using this property, we show how one can easily translate generalized Killing conditions on pinors into differential constraints on forms defined on M . Subsection 5.8 gives another form of such differential constraints, which is used in Appendix B for comparison with the component approach outlined in [3]. In Subsection 5.9, we discuss some basic aspects of the algebro-differential system of constraints on inhomogeneous forms which results from our analysis. As expected, our formulation allows one to extract basic structural properties of this system, thereby providing a starting point for a natural generalization of the classical theory of Killing forms. Finally, Subsection 5.10 considers the particular cases of one and two independent constrained generalized Killing pinors with a definite and symmetric Spin(d)-invariant metric, the first of which is relevant to the application discussed in Section 6.
Admissible bilinear pairings on the pin bundle. It is well-known that S carries so-called admissible non-degenerate bilinear pairings B whose action on sections of S satisfies: as well as a certain other property which can be found in [30] but will not be relevant for what follows. In the formulas above, we used the following anti-automorphism of the Kähler-Atiyah algebra: where τ is the reversion anti-automorphism defined in (22). The numbers ǫ B (the type of B) and σ B (the symmetry of B) equal +1 or −1, depending on p, q and the precise choice of B; such bilinear pairings were classified in [29,30]. Notice that the first equation in (83) implies: i.e.: where T t denotes the transpose of T ∈ Γ(M, End(S)) with respect to B, which is defined through: This operation satisfies (T t ) t = T and (id S ) t = id S , the first identity being a consequence of the signed symmetry property of B (the second identity listed in (83)). The operation T → T t of taking the B-transpose defines a C ∞ (M, K)-linear anti-automorphism of the algebra (Γ(M, End(S)), •).
Local expressions. Given a local pseudo-orthonormal coframe e a above U ⊂ M , the first of properties (83) amounts to: which means that γ a = γ(e a ) satisfy (γ a ) t = ǫ B γ a , a relation which implies: where ǫ k B = (ǫ B ) k . Since [γ a , γ b ] +,• = 2η ab , we also have: which in turn gives: Combining the above, we find: which implies: These relations will be useful later.
The isomorphism E. The non-degenerate pairing B induces a bundle isomorphism ρ : S ∼ → S * , whose action on sections is given by: On the other hand, we have a natural bundle isomorphism q : S ⊗ S * ∼ → End(S), given on sections as follows: The two maps above combine to give a bundle isomorphism E def.
= E(ξ ⊗ ξ ′ ) ∈ Γ(M, End(S)) for all ξ, ξ ′ ∈ Γ(M, S), we have: an identity which follows from the explicit form: without making use of the signed symmetry property of B. Note that E depends on the choice of B (since ρ does).
The bundle of bipinors and the bipinor algebra of S. The bundle isomorphism E allows us to transfer the fiberwise composition of operators from End(S) to an associative and bilinear fiberwise composition • defined on the bundle of bipinors S ⊗ S, whose action on sections takes the form: This operation satisfies: The composition • makes the bundle of bipinors into a bundle of unital associative algebras which is isomorphic with the bundle of algebras (End(S), •); of course, the unit section id S of End(S) maps to the unit section of S ⊗ S, which we denote by I def.
= E −1 (id S ). The unital associative algebra Γ(M, S ⊗ S) = Γ(M, S) ⊗ C ∞ (M,K) Γ(M, S) consisting of smooth sections of the bipinor bundle will be called the bipinor algebra of S; it is an algebra over the ring C ∞ (M, K).
The bipinor C ∞ (M, K)-algebra of a submodule of sections. If K is any K-linear subspace of Γ(M, S), then the set: The bipinor K-algebra of a B-flat subspace of sections. Another interesting case arises when the K-linear subspace K ⊂ Γ(M, S) is B-flat, by which we mean that K satisfies the condition: In this case, K ⊗ C ∞ (M,K) K is a K-subalgebra of the bipinor algebra (Γ(M, S ⊗ S), •), which we shall call the (flat) bipinor K-algebra determined by K.

The Fierz isomorphism and Fierz identities. Fierz algebras.
The Fierz isomorphismĚ. Let us now assume that we are in one of the cases when the Schur algebra is isomorphic with the base field. Then one can also transport to the bundle of bipinors the isomorphism γ −1 : End(S) ∼ → (∧T * K M ) γ to get an isomorphism of bundles of algebras: , which we shall call the Fierz isomorphism. On sections, this induces a C ∞ (M, K)-linear isomorphism of algebras (denoted, as usual, by the same symbol): , which identifies the bipinor algebra with the subalgebra Ω γ K (M ) of the Kähler-Atiyah algebra. Note thatĚ depends on the choice of admissible form B.
=Ě(ξ ⊗ ξ ′ ) = γ −1 (E ξ,ξ ′ ) ∈ Ω γ K (M ) (for ξ, ξ ′ ∈ Γ(M, S)), equation (87) implies the following identity in the subalgebra Ω γ K (M ) of the Kähler-Atiyah algebra: Equation (91) is the condensed expression of Fierz identities involving four pinors. These identities simply express the fact that γ (and thusĚ) is an isomorphism of bundles of algebras, rather than simply an isomorphism of vector bundles -and are, in fact, equivalent with this property once fiberwise linearity of γ is assumed. The construction ofĚ is summarized in the commutative diagram (92), which applies provided that S K (p, q) ≈ K. In the diagram, we show the action of the various morphisms on sections.
Notation. Let K ⊂ Γ(M, S) be any K-linear subspace of Γ(M, S). The image of the K-linear subspace K ⊗ C ∞ (M,K) K ⊂ Γ(M, S ⊗ S) through the Fierz isomorphism will be denoted by: and is a K-linear subspace of Ω γ K (M ).  K)) of the bipinor algebra of S. Its image (93) through the Fierz isomorphism is a (generally non-unital) subalgebra of the C ∞ (M, K)-algebra (Ω γ K (M ), ⋄), which we shall call the Fierz subalgebra determined by K. This algebra over C ∞ (M, K) encodes the Fierz identities between bilinears constructed from pinors which belong to K. A particular case arises when K = Γ(M, K) where K ⊂ S is some vector sub-bundle of S -in which situation K⊗ C ∞ (M,K) K is the bipinor algebra of K. The corresponding Fierz subalgebraǨ will then be called the Fierz subalgebra determined by the sub-bundle K. With the further assumption that B is a scalar product (as happens in the application of Subsection 6), the morphism E can then be used to identify the Fierz algebra of K with the C ∞ (M, K)-algebra Γ(M, End(K)) of globally-defined endomorphisms of the bundle K; in particular, the Fierz subalgebra is unital in such cases.
The Fierz K-algebra of a B-flat subspace of sections. When K is a B-flat K-linear subspace of Γ(M, K), the vector space K ⊗ C ∞ (M,K) K is a K-subalgebra of the bipinor algebra (Γ(M, S ⊗ S), •). It follows that its image (93) through the Fierz isomorphism is a K-subalgebra of the algebra (Ω γ K (M ), ⋄), which will be called the (flat) Fierz K-algebra determined by K.

Some properties of the Fierz isomorphism
A simple computation using (88) shows that the following identities hold for any T ∈ Γ(M, End(S)): i.e.
where L T and R T are the operators of left and right multiplication with T in the algebra (Γ(M, End(S)), •). Applying γ −1 to identities (94) and settingŤ def.
= γ −1 (T ) ∈ Ω γ K (M ) gives: i.e. (substituting T → T t into the second equation): This also reads: where LŤ and RŤ are the operators of left and right multiplication withŤ in the Kähler-Atiyah algebra. Equation We can thus write the second relation of (96) in the form: We also notice the relation: which follows from the signed symmetry of B together with definition (88). The last identity can also be written as: One easily checks that transp B is an anti-automorphism of the bipinor algebra. For later reference, note the identity: which will be useful below.

Local completeness relations for the endomorphism algebra of the pin bundle
Let tr : End(S) → O K be the natural 6 trace on End(S) (which is a morphism of K-vector bundles). Recall that we assume the Schur algebra to be isomorphic with the base field K. Then a convenient generating set of local sections for the vector bundle End(S) above a sufficiently small open subset U ⊂ M is given (see, for example, [27]) by the operators {γ a 1 ...a k |k = 0, . . . d, 1 ≤ a 1 < . . . < a k ≤ d}, where d = p + q and γ a = γ(e a ) with (e a ) a pseudo-orthonormal local coframe of M above U . The following identity (the 'completeness relation') holds: where Remark. The K-vector bundles End(S) ⊗ End(S) and End(End(S)) can be identified through the bundle isomorphism W : End(S)⊗End(S) → End(End(S)), which acts as follows on sections: Using this isomorphism, we transport the composition • of End(End(S)) to an associative composition • defined on End(S) ⊗ End(S), whose action on sections is given by: End(S)) .
An easy computation shows that • has the explicit form: The unit section id End(S) of the bundle End(End(S)) corresponds via W to the unit section I def.

Explicit expansion of the Fierz isomorphism when the Schur algebra equals the base field
Given a local pseudo-orthonormal coframe of M , explicit expansions for the isomorphismĚ can be derived using the results of [27] (see also [31]) for any choices of the base field K and of the signature type (p, q). A complete discussion is quite involved given the different behavior in various cases and will be taken up in detail in a different publication. Below, we shall consider only the case when the Schur algebra is isomorphic with the base field K, i.e. the case when K = C and the case when K = R with p − q ≡ 8 0, 1, 2.
In this case, the local completeness relation (104) holds. Applying it to the endomorphism T = E ξ,ξ ′ and using relation (103), we find the local expansion: which implies the following local expansion of the Fierz isomorphism upon applying γ −1 to both sides:Ě (an identity holding in Ω γ K (M )), where: (as in (1)), with: where, using (86), we have:

Expressing the algebraic constraints through differential forms
Consider the case when we have a single algebraic constraint Qξ = 0, where Q ∈ Γ(M, End(S)). As in Section 2, we let K(Q) denote the space of solutions of the algebraic constraint, which is a sub-module of the C ∞ (M, K)-module Γ(M, S). Recall that there generally exists no sub-bundle of S whose space of smooth sections equals K(Q).
The dequantized constraint and the C ∞ (M, K)-subalgebra of constrained inhomogeneous forms. The inhomogeneous form: ∈ Ω γ K (M ) will be called the dequantization of Q. Relation (95) gives: where L Q , R Q t are -as above -the operators on Γ(M, End(S)) given by left and right composition with Q and Q t . Using the fact that E is an isomorphism as well as the identities Applying γ −1 to both sides of this relation gives the following description of the Fierz algebra of the sub-module of sections K(Q), which we shall call the C ∞ (M, K)-algebra of Q-constrained inhomogeneous forms: (112) Here, LQ, RQ are the left and right ⋄-multiplication operators of Section 3.4 and the second equality above is obvious. With these definitions, we have the equivalence: i.e.:Q Behavior under τ B . The equivalence (114) can be written in the following form, which follows by applying τ B to the second equation in the left hand side: or (using (100)) in the form: In fact, relation (101) and the obvious fact that K(Q) ⊗ C ∞ (M,K) K(Q) is invariant under the antiautomorphism transp B imply that the algebra of Q-constrained differential forms is invariant under τ B : Remark. Using (99) and (108) and separating ranks shows that (114) can be written as: which amounts to the following description upon using the explicit form (111) ofĚ: In Appendix B.1, we show that equations (117) are equivalent with certain relations which were first discussed in [3] for the special case d = p = 8, q = 0.

Conditions on differential forms implied by the generalized Killing equation
In our formulation, a Clifford connection ∇ S = dx m ⊗ ∇ S m on S is a connection which satisfies: where ∇ m is the connection induced on Notice that P ǫγ • ∇ is the connection induced by ∇ on the sub-bundle (∧T * K M ) ǫγ and that this induced connection is determined by (∇ S ) ad through property (118). In the following, we take ∇ S to be the connection on S induced by the Levi-Civita connection of (M, g); it is wellknown that ∇ S is a Clifford connection in the sense discussed above 7 . A discussion of this and other properties of ∇ S in index language (which also serves to fix our conventions and leads to another derivation of certain identities extracted in this paper) can be found in Appendix A. Equation (118) is compatible with the fact that ∇ m is a derivation of the Kähler-Atiyah algebra (Ω K (M ), ⋄) -a property which can be checked by direct computation using the fact that ∇ m is an even derivation of the exterior algebra which is compatible with the metric. Similarly, we consider the connection D ad induced by D on End(S): Notation. We let K(D) = ∩ d m=1 K(D m ) ⊂ Γ(M, S) be the finite-dimensional K-linear subspace of all generalized Killing pinors with respect to D.
The dequantized connection. In the following, we consider only the case when the Schur algebra is isomorphic with the base field. Then any connection D on S can be written as: where:Ǎ are inhomogeneous differential forms on M . The Clifford connection property (118) of ∇ S implies: where the derivationĎ ad m (which we shall call the adjoint dequantized connection) of the Kähler-Atiyah algebra (Ω K (M ), ⋄) is defined through: Since Composing equation (120) with γ −1 from both sides gives the following relation which will be used below: To arrive at (122), we noticed thatĎ ad m , which follows upon composing the last equality in (120) with γ −1 from the right and using the property γ • γ −1 = id End K (S) . Note that A m can be combined into the object: whileĎ ad m can be combined into the map: where: and we used e m ⊗ ∇ m = ∇.
Flatness of the Fierz isomorphism. For the remainder of this paper, we shall assume that D is compatible with B in the usual sense that B is D-flat: . It follows that the isomorphism E = q • (id S ⊗ ρ) satisfies 8 : In particular, we have: where we have introduced the following K-subalgebra of (Γ(M, End(S)), •): On the other hand, γ −1 satisfies (122). Together with (124), this implies that the Fierz isomor-phismĚ satisfies: Therefore, we find: The K-algebra of generalized Killing forms. We define the K-algebra of generalized Killing forms to be the following K-subalgebra of (Ω γ K (M ), ⋄): The elements ofǨ D ad will be called generalized Killing forms. Relation (125) implies that the flat Fierz K-algebra: defined by the B-flat subspace K(D) ⊂ Γ(M, S) is a subalgebra of the K-algebra of generalized Killing forms: In particular, we have: Behavior under τ B . Spin(d)-invariance of B implies that B is flat with respect to the connection ∇ S : which is easily seen to imply the property: Together with the assumption (123), identity (130) implies that A m are B-antisymmetric endomorphisms of S: In turn, these properties imply the relation Setting T = γ(ω) and applying γ −1 to both sides gives: In particular, the K-algebra of generalized Killing forms is invariant under τ B : a property which (by virtue of (101)) it shares with the flat Fierz K-algebraǨ(D): Together with (100), identity (132) implies thatĎ ad -flatness ofĚ ξ,ξ ′ andĎ ad -flatness ofĚ ξ ′ ,ξ are equivalent statements, so that it suffices to require only one of the two.

Alternate form of the differential constraints
Consider the following local expansion, which results by applying (104) to [T, E ξ,ξ ′ ] −,• , where T ∈ Γ(M, End(S)): An easy computation using cyclicity of tr and identity (103) gives: so that (133) becomes: Setting T = A m = γ(Ǎ m ) in (134) and applying the morphism γ −1 to both sides gives: Using this identity in the definition (121) ofĎ ad m gives: Consider now relation (127), written in terms ofĚ ξ,ξ ′ : . Substituting (136) in the left hand side, this becomes: Separating ranks, we conclude that equation (127) is equivalent with the following system of identities: where ξ, ξ ′ ∈ Γ(M, S) are arbitrary. In particular, we have: which agrees with equation (213) (see appendix B.2).

5.9
The K-algebra of constrained generalized Killing forms As before, we consider the case χ = 1 of the CGK equations: Let K(D, Q) = K(D) ∩ K(Q) denote the (finite-dimensional) K-linear subspace of Γ(M, S) consisting of all solutions to (140). We define the K-algebra of constrained generalized Killing (CGK) forms determined by D and Q to be the following K-subalgebra of (Ω γ K (M ), ⋄): In general,Ǩ D,Q is a non-unital K-algebra. The discussion of the previous subsections shows that the flat Fierz K-algebra determined by the B-flat subspace K(D, Q) ⊂ Γ(M, S): is a subalgebra of the K-algebra of CGK forms: This property of the Fierz isomorphism depends essentially on the assumption that B is D-flat (an assumption which is satisfied in the application discussed in Section 6).
Expression for a basis of solutions of the CGK pinor equations. Let s = dim K K(D, Q) denote the K-dimension of the space of solutions to the CGK equations. Choosing a basis (ξ i ) i=1...s of such solutions, we set: where (cf. equation (111)): ..a k ξ j ) giving the following expression for the homogeneous form-valued bilinears: Since ξ i ⊗ξ j form a basis of the K-vector space K(D, Q)⊗ C ∞ (M,K) K(D, Q), the inhomogeneous differential formsĚ ij form a basis of the K-vector spaceǨ(D, Q). Inclusion (142) amounts to the following system of equations for the inhomogeneous differential formsĚ ij : These can also be written as: upon applying τ B to the last equation in (146) and using the relation τ B (Ě ij ) = ǫ BĚji (cf. (100)). The inhomogeneous differential formsĚ ij also satisfy the Fierz identities: where we defined the following constants: The algebro-differential system consisting of (147) and (148) can be taken as the basis for extending the classical theory of Killing forms, a subject which is of mathematical interest in its own right. It provides a synthetic geometric description of the essential conditions imposed by having a fixed number of unbroken supersymmetries in a flux compactification, formulated in the language of geometric algebra. When expanding the geometric product into generalized products using (19), the innocently-looking equations (147) and (148) take on a form which may seem rather formidable when s is sufficiently large (see Section 6 for an example). Of course, the language of geometric algebra allows one to study such systems starting directly from the synthetic expressions (147), (148), which shows that the problem of characterizing the solutions of such equations belongs most properly to the intersection between Kähler-Cartan theory [16] and the theory of noncommutative associative algebras -a point of view on flux compactifications, which, in our opinion, could lead to a deeper understanding of various problems pertaining to that subject.
Consider the particular case when K = R and B is symmetric (thus σ B = +1) and positivedefinite with ǫ B = +1 (this happens, for example, in the application considered in Section 6).
In this case, we can choose ξ 1 , . . . , ξ s such that B ij = δ ij . Then relations (148) show that the K-algebraǨ(D, Q) is isomorphic with the algebra Mat(s, R) of square real matrices of dimension s, the unit being given byČ def.
ii . An isomorphism to Mat(s, R) is given byĚ ij → e ij , where e ij ∈ Mat(s, R) is the matrix whose only non-vanishing entry equals 1 and is found on the i-th row and j-th column: ii =Ě ii ,Ě ⋄2 ij = 0 for i = j. SinceĚ ij ⋄Ě ji =Ě ii , the K-algebraǨ(D, Q) is generated by the elements (Ě ij ) i =j , with the relations: This system of generators and relations can be reduced further (see the example below) upon using the identityĚ ji = τ (Ě ij ), where we used the fact that σ B = +1 and we noticed that The case of one CGK pinor. When s = 1 (a single unit norm solution ξ of the CGK pinor equations, which is unique up to sign), the flat Fierz K-algebraǨ(D, Q) has dimension one, being isomorphic with R as a unital R-algebra. It is generated by the single basis elemenť E =Ě ξ,ξ (which is the unit ofǨ(D, Q)), with the relation: The condition τ (Ě) =Ě implies that certain rank components ofĚ vanish identically. This situation occurs in the example considered in Section 6, though in that case we prefer to take the squared norm B(ξ, ξ) of ξ to equal 2, for ease of comparison with the results of [3]. This produces an extra factor of two in the right hand side of (152).
The case of two CGK pinors. When s = 2, the flat Fierz K-algebraǨ(D, Q) is generated byĚ 12 andĚ 21 with the relations: We haveĚ 11 =Ě 12 ⋄Ě 21 ,Ě 22 =Ě 21 ⋄Ě 12 andČ =Ě 11 +Ě 22 =Ě 12 ⋄Ě 21 +Ě 21 ⋄Ě 12 . Sincě E 21 = τ (Ě 12 ), we can in fact generate the entire algebra fromĚ 12 up to applying τ . Using the fact that the first and second equation in (153) as well as the first and second equation in (154) are related through reversion, we find that the entire system is equivalent with the following Fierz relations: together with the following Q-and D-constraints: Remark. The algebra Mat(2, R) ≈ Cl(2, 0) is also generated by two real Pauli matrices, for example by: subject to the relations σ 2 1 = σ 2 3 = 1 and σ 1 σ 3 + σ 3 σ 1 = 0. Therefore, the flat Fierz K-algebrǎ K(D, Q) is also generated by the inhomogeneous differential forms: subject to the relations: Noticing 6 Example: flux compactifications of eleven-dimensional supergravity on eight-manifolds In this section, we illustrate the methods developed in the present paper by applying them to the case of N = 1 warped compactifications of eleven-dimensional supergravity on eight-manifolds down to an AdS 3 space, a situation which was studied through direct methods in [3] and [4]. After some basic preparations in Subsection 6.1, Subsection 6.2 gives our translation of the generalized Killing pinor equations into a system of algebraic and differential constraints on inhomogeneous forms defined on the compactification space and shows how our approach allows one to recover the results of [3].

Preparations
Consider supergravity on an 11-manifoldM with Lorentzian metricg (of 'mostly plus' signature). Besides the metric, the action of the theory contains the three-form potential with four-form field strengthG ∈ Ω 4 (M ) and the gravitinoΨ M , which is a real pinor of spin 3/2. We assume that (M ,g) is spinnable. For supersymmetric bosonic backgrounds, both the gravitino VEV and its supersymmetry variation must vanish, which requires that there exists at least one solutioñ η to the equation: where uppercase indices run from 0 to 10 andD M is the supercovariant connection: Here,γ M are the gamma matrices of Cl (10,1) in that 32-dimensional real (Majorana) irreducible representation for whichγ (12) def. =γ 1 . . .γ 11 = +1 and is the connection on the pin bundleS induced by the Levi-Civita connection of (M ,g). The eleven-dimensional supersymmetry generatorη (which is a Majorana spinor field of spin 1/2) is a smooth section of the pin bundleS, which is a rank 32 real vector bundle defined onM . As in [3], we consider compactification down to an AdS 3 space of cosmological constant Λ = −8κ 2 , where κ is a positive real parameter -this includes the Minkowski case as the limit κ → 0. ThusM = N × M , where N is an oriented 3-manifold diffeomorphic with R 3 and carrying the AdS 3 metric while M is an oriented Riemannian eight-manifold whose metric we denote by g. The metric onM is a warped product: 2 11 where ds 2 11 = ds 2 3 + g mn dx m dx n .
Here, the warp factor ∆ is a smooth function defined on M while ds 2 3 is the squared length element on N . For the field strengthG, we use the ansatz: where f = f m e m ∈ Ω 1 (M ), F = 1 4! F mnpq e mnpq ∈ Ω 4 (M ) and vol 3 is the volume form of N . Small Latin indices from the middle of the alphabet run from 1 to 8 and correspond to a choice of frame on M . Forη, we use the ansatz: where ξ is a Majorana spinor of spin 1/2 (a.k.a. a real pinor) on the internal space M and ψ is a Majorana spinor on the AdS 3 space N . Mathematically, ξ is a section of the pinor bundle of M , which is a real vector bundle of rank 16 defined on M , carrying a fiberwise representation of the Clifford algebra Cl(8, 0). Since p − q ≡ 8 0 for p = 8 and q = 0, this corresponds to the simple normal case of Section 4. In particular, the corresponding morphism γ : (∧T * M, ⋄) → (End(S), •) of bundles of algebras is an isomorphism, i.e. it is bijective on the fibers. We set γ m = γ(e m ) for some local frame of M . In dimension eight with Euclidean signature, there exists an admissible [29] (and thus Spin(8)-invariant) bilinear pairing B on the pin bundle S with σ B = +1 and ǫ B = +1 which is a scalar product (i.e., it is fiberwise symmetric and positive-definite).
Assuming that ψ is a Killing pinor on the AdS 3 space, the supersymmetry condition (155) decomposes into a system consisting of the following constraints for ξ: where D m is a linear connection on S and Q ∈ Γ(M, End(S)) is a globally-defined endomorphism of the vector bundle S. As in [3,4], we do not require that ξ has definite chirality (that is, ξ need not satisfy the Weyl condition). The space of solutions of (157) is a finite-dimensional R-linear subspace K(D, Q) of the space Γ(M, S) of smooth sections of S. Of course, this subspace is trivial for generic metrics g and fluxes F and f on M , since the generic compactification of the type we consider breaks all supersymmetry. The interesting problem is to find those metrics and fluxes on M for which some fixed amount of supersymmetry is preserved in three dimensions, i.e. for which the space K(D, Q) has some given non-vanishing dimension, which we denote by s. The case s = 1 (which corresponds to N = 1 supersymmetry in three dimensions) was studied in [3,4] and will be re-considered below. Direct computation using the compactification ansatz gives the following expressions for D and Q (which are equivalent with those derived in reference [3] -see the remark below for ease of comparison): where γ (9) = γ 1 •. . .•γ 8 . Notice that the last terms in (158) and (159) depend on the cosmological constant of the AdS 3 space -and that they vanish in the Minkowski limit κ → 0. Given some desired amount of supersymmetry which we want to be preserved in three dimensions (i.e. given some desired dimensionality s of the space of solutions to (157)), the general aim is to reformulate (157) as equations on differential formsĚ ..am constructed as bilinear combinations of pinors ξ, ξ ′ which satisfy (157). The pinor bilinears will be constrained by Fierz identities. The translation to equations on the differential formsĚ (k) ξ,ξ ′ can be achieved directly by starting from the following equivalent reformulation of the algebraic constraints Qξ = Qξ ′ = 0: and treating the constraints D m ξ = D m ξ ′ = 0 through the method outlined in [3]. The theoretical basis of that approach is explained in detail in Appendix B, where we also show how that method is equivalent with the more general approach which we have developed in the present paper. In the next Subsection, we shall illustrate our approach in the simplest case s = 1 (N = 1 supersymmetry in three dimensions, see [3]), so we will require that (157) admits one non-trivial solution ξ.
Remark. For easy comparison with [3], we note that loc. cit. uses a redundant parameterization of the degrees of freedom described by ξ, which is given by the following sections of S: which satisfy γ (9) ε ± = ε ∓ . Here, ξ ± = P ± ξ are the positive and negative chirality components of ξ, with P ± def.

The case of N = 1 supersymmetry in 3 dimensions
Let us consider the CGK pinor equations (157) on the Riemannian 8-manifold (M, g), assuming that the space of solutions has dimension s = 1 over R. Since p − q ≡ 8 0, we are in the normal simple case. This case is characterized by two admissible pairings B ± on S, which have the properties given in [30], i.e. ǫ B ± = ±1 and σ B ± = +1. Since any choice of admissible pairing leads to the same result, we choose to work with B = B + for convenience; this satisfies ǫ B = +1 and σ B = +1. We can assume that B is a scalar product on S and we denote the corresponding norm by || ||.
As in Subsection 5.9 (see equation (143)), consider the inhomogeneous differential form: defined by a non-trivial solution ξ of (157). Equation (145) gives: Using the properties of B and relations (85) and (86), one easily checks thatĚ where ν is the volume form. Thus the inhomogeneous form which generates the Fierz algebra expands as: .
Note that this case also admits non-vanishing 3-and 7-form bilinears, which are proportional to the Hodge duals of the 5-and 1-forms respectively (but they do not appear in the generator). These other form-valued bilinears can also be expressed as in (145), but upon using the other admissible pairing B − , which is in fact not necessary for this analysis.
Following the notations and conventions of [3], we define: where a is the norm of ξ and b a smooth function on M . We also have a non-trivial 3-form * Z, a 7-form * K and another 4-form * Y . With these notations, (162) becomes: Note that we use the five-formZ instead of its sign-reversed Hodge dualφ def.
which were used in loc. cit. All in all, we have the following correspondence with the notations and conventions of [3]: The dequantizations of A m and Q are given by: As shown in Section 5, the CGK pinor equations imply the following conditions for the generatoř E:Q In turn, relations (167) imply: In this case, (148) amounts to only one quadratic relation for the inhomogeneous formĚ: which encodes the relevant Fierz identities between the form bilinears constructed from ξ. We also remind the reader that one can uniquely decompose any k-form ω ∈ Ω k (M ) into parallel and orthogonal parts with respect to any fixed 1-form θ ∈ Ω 1 (M ) such that ω = ω +ω ⊥ , where ω = θ ∧ ω ⊤ for ω ⊤ ∈ Ω k−1 (M ), ω ⊥ ∈ Ω k (M ) and θ △ 1 ω ⊤ = θ △ 1 ω ⊥ = 0 (see Subsection 3.5). Choosing θ =K, one has Y = Y + Y ⊥ andZ =Z +Z ⊥ . Note that ν = ν andK =K .
Given the decomposition of Y , one has * Y = * Y + * Y ⊥ = α 1 Y ⊥ +α 2 Y , since the volume form is parallel to θ and the Hodge dual of any parallel component is an orthogonal component while the Hodge dual of any orthogonal component is parallel to θ. In the current example (see Table  1), the volume form ν is twisted central (i.e., we have ν ⋄ ω = π(ω) ⋄ ν for any inhomogeneous differential form ω) and satisfies ν ⋄ ν = +1. The latter property amounts to * * Y = Y , which leads to α 2 = 1/α 1 and implies: Expanding (166) into rank components gives the following conditions, which are equivalent with the 'useful relations' discussed in Appendix C of [3]: Similarly, the rank expansion of (167) gives: which in turn imply the following constraints representing the rank components of (168): Finally, expanding (169) and using the properties of ν, we find: We write below the expansions for all those geometric products (see (19)) that appear in (174). Let us first list the geometric products involvingK: Notice thatK ∧Z ⊥ = 0, since there is no nontrivial six-form pinor bilinear that can be constructed in this case. Thus, we must haveZ ⊥ = 0, which meansZ =Z , a relation which we use when performing the following expansions: We mention here that expansions (171)-(176) were performed by using a package of procedures which we created using Ricci [33]. We have also verified these relations (in tensorial form) using Cadabra [34]. We deduce from the Fierz identities in normal cases that any generalized product of formvalued pinor bilinears can be expressed as a form-valued pinor bilinear. In (176), we omitted to write down those terms which are tautologically zero due to the graded antisymmetry of the wedge product. We also omitted writing some other vanishing terms such as Y ⊥ ∧ Y ⊥ , which has rank 8 and thus must be proportional to the volume form, but vanishes since the volume form does not have an orthogonal component. Similarly, Y △ 1Z = 0 since it should be a parallel 7-form and thus proportional to ( * K) = * (K ⊥ ), butK ⊥ = 0. Furthermore, Y ⊥ △ 1Z = 0 since it is an orthogonal 5-form, thus proportional toZ ⊥ = 0. For similar reasons, Y ⊥ △ 3Z = 0 and Y △ 4Z = 0.
Given expansions (175) and (176), identity (174) gives the following relations when separated into rank components: In order to solve the system (177), we introduce the notations: where e, f, g, h, y 1 , y 2 , y 3 , n, r, s, t, u, x are smooth functions on M which we want to determine in terms of a and b. Using (178), the system (177) leads to four independent relations for ranks 0, 1, 5, 8 and two independent identities for the parallel and perpendicular components of rank 4: Using associativity of the geometric product: as well as (175), (176) and (178), we find the relations: , er + f s = e(y 1 + y 3 ) , es + rf = 2y 2 f , n||K|| 2 = gh , g|| Y ⊥ || 2 = t||K|| 2 , 2y 2 n = nr + ts , y 3 = r , y 2 = sα 1 , which need not be independent. We find that the Fierz identities are equivalent to the algebraic system of equations (179)-(180), which can be solved using Mathematica R and give: Substituting this solution into (178), we find the identities: For the particular choice analyzed in [3], namely a = 1 and b = sinζ, we find agreement with the results in loc. cit., as for example: d(e 3∆K ) = 0 , It is easy to see that the first four of these conditions follow directly from the Fierz identities (181), while the last four can be obtained using the algebraic constraints (171) and the differential constraints (173). (182) and (183). From (181), we have:
Proof of relation (186). The third algebraic constraint listed in (171): and the second differential constraint listed in (173): imply the relation: which is easily seen to be equivalent with (186).

Conclusions and further directions
We showed that geometric algebra techniques can be used to give a highly synthetic, conceptually transparent and computationally efficient re-formulation of the constrained Killing pinor equations, which constitute the condition that a flux background preserves a given amount of supersymmetry. This formulation clearly displays the algebraic and differential structure governing the supersymmetry conditions, leading to a description which opens the way for unified studies of flux backgrounds aimed at uncovering their deeper structure. We showed that our general formalism recovers results and methods which were used in [3] and therefore that it provides a powerful way to extend them. Our formulation is highly amenable to implementation in various symbolic computational packages specialized in tensor algebra, and we touched on two particular implementations which we have carried out using Ricci [33] in Mathematica R , as well as Cadabra [34]. We illustrated our approach with the case of the most general compactifications of M-theory which preserve N = 1 supersymmetry in three dimensions [3,4], showing how the results derived through different methods in loc. cit. can be recovered through our techniques. Further applications of our approach are considered in forthcoming publications. We stress that the methods introduced in this paper have much wider applicability than the example considered in Section 6, leading to promising new directions in the study of supergravity backgrounds and supergravity actions. In particular, we believe that many computationally difficult issues in the subject could be understood much better by using such techniques. The connection with a certain form of geometric quantization (which we have only touched upon) also leads to interesting ideas, problems and directions for further research, which are currently under investigation.

A Identities satisfied by the covariant derivative of pinors
Let (M, g) be a pseudo-Riemannian manifold endowed with a local coordinate system (x m ) and a local pseudo-orthonormal frame (e a ) (vielbein) of (T M, g), both defined above an open set U ⊂ M . In this Appendix, both m and a run from 1 to dim M . As usual, pseudo-orthonormality of e a means g(e a , e b ) = η ab , where η ab is a diagonal matrix all of whose diagonal entries equal +1 or −1. We let e a denote the dual coframe of M , defined through e a (e b ) = δ a b ; it is a local frame of T * M which is pseudo-orthonormal with respect to the metricĝ induced on T * M , i.e.ĝ(e a , e b ) = η ab where η ab η bc = δ a c . We have ∂ m def.
= ∂ ∂x m = e a m (x)e a and e a = e m a (x)∂ m for some locally defined functions e a m , e m a which satisfy e a m e m b = δ a b and e a m e n a = δ n m . This implies g(e a , e b ) = e m a e n b g mn = η ab and g(∂ m , ∂ n ) = e a m e b n η ab = g mn . Any tensor field t ∈ Γ(T M ⊗p ⊗ (T * M ) ⊗q ) of type (p, q) expands as: Here and below, indices denoted by letters chosen from the middle of the Latin alphabet refer to the coordinate frame defined by (∂ m ) while indices denoted by letters chosen from the beginning of the Latin alphabet refer to the local pseudo-orthonormal frame (vielbein) defined by (e a ). A differential k-form ω ∈ Ω k (M ) expands locally as in (1). Similarly, a polyvector field α ∈ Γ(M, ∧ k T M ) expands locally as: (with coefficients functions which are totally antisymmetric in the indices). Let ∇ be the Levi-Civita connection of (M, g). Its Christoffel symbols Γ ρ mn in the given local coordinates are defined through ∇ m (∂ n ) = Γ ρ mn ∂ ρ , while its coefficients Ω b ma with respect to the given coordinate system and vielbein are determined by the expansion ∇ m (e a ) = Ω b ma e b . Here and below, we set ∇ m def.
= ∇ ∂m . The two sets of connection coefficients are related through: Ω a mb = e a n e λ b Γ n mλ − e λ b ∂ m e a λ ⇐⇒ Γ n mλ = e n a e b λ Ω a mb + e n a ∂ m e a λ .
The fact that ∇ is torsion-free amounts to the conditions: where Ω mab is defined through: = η ac Ω c mb = g(e a , ∇ m e b ) = −g(∇ m e a , e b ) .

B.2 Alternate derivation of the differential constraints
For simplicity, let us consider only the case K = R. Recall that the pin bundle S of a pseudo-Riemannian manifold (M, g) is endowed with admissible bilinear pairings B, which, in particular, satisfy (83). SinceΩ t m = −Ω m , we find thatΩ m is anti-self-adjoint with respect to the pairing B, which means that the pin covariant derivative ∇ S induced by the Levi-Civita connection of (M, g) is compatible with B in the sense that this pairing is ∇ S -flat: The deformed pin connection takes the form D m = ∇ S m +A m = ∂ m +Ω m +A m . Since A t m = −A m , it follows that A m (and thus alsoΩ m + A m ) is again anti-self-adjoint with respect to the scalar product on S; as a consequence, the deformed pin connection is also compatible with this pairing:  where (e a ) is a local pseudo-orthonormal frame of (M, g) and γ a = γ((e a ) ♯ )) = η ab γ b .