Killing superalgebras for Lorentzian four-manifolds

We determine the Killing superalgebras underpinning field theories with rigid unextended supersymmetry on Lorentzian four-manifolds by re-interpreting them as filtered deformations of ℤ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathbb{Z} $$\end{document}-graded subalgebras with maximum odd dimension of the N = 1 Poincaré superalgebra in four dimensions. Part of this calculation involves computing a Spencer cohomology group which, by analogy with a similar result in eleven dimensions, prescribes a notion of Killing spinor, which we identify with the defining condition for bosonic supersymmetric backgrounds of minimal off-shell supergravity in four dimensions. We prove that such Killing spinors always generate a Lie superalgebra, and that this Lie superalgebra is a filtered deformation of a subalgebra of the N = 1 Poincaré superalgebra in four dimensions. Demanding the flatness of the connection defining the Killing spinors, we obtain equations satisfied by the maximally supersymmetric backgrounds. We solve these equations, arriving at the classification of maximally supersymmetric backgrounds whose associated Killing superalgebras are precisely the filtered deformations we classify in this paper.

1 Introduction A number of impressive exact results [1][2][3][4][5][6][7][8][9] obtained in recent years via supersymmetric localisation have motivated a more systematic exploration of quantum field theories with rigid supersymmetry in curved space. A critical feature in many of these calculations is the non-trivial rôle played by certain non-minimal curvature couplings which regulate correlation functions, so a clear understanding of the general nature of such couplings would be extremely useful.
Several isolated examples of curved backgrounds which support rigid supersymmetry, like spheres and anti-de Sitter spaces (also various products thereof), have been known for some time [10,11]. Beyond these examples, the most systematic strategy for identifying curved backgrounds which support some amount of rigid supersymmetry has hereto been that pioneered by Festuccia and Seiberg in [12]. In four dimensions, they described how a large class of rigid supersymmetric non-linear sigma-models in curved space can be obtained by taking a decoupling limit (in which the Planck mass goes to infinity) of the corresponding locally supersymmetric theory coupled to minimal off-shell supergravity. In this limit, the gravity supermultiplet is effectively frozen out, leaving only the fixed bosonic supergravity fields as data encoding the geometry of the supersymmetric curved background. Following this paradigm, several other works explored the structure of rigid supersymmetry for field theories in various dimensions on curved manifolds in both Euclidean and Lorentzian signature [13][14][15][16][17][18][19].
A well-established feature of supersymmetric supergravity backgrounds is that they possess an associated rigid Lie superalgebra [20][21][22][23][24][25][26][27][28][29][30][31][32][33][34] that we shall refer to as the Killing superalgebra of the background. Indeed, with respect to an appropriate superspace formalism, the construction described in ( [20], section 6.4) (and reviewed in [34]) explains how this Killing superalgebra may be construed in terms of the infinitesimal rigid superisometries of a given background supergeometry. The even part of the Killing superalgebra contains the Killing vectors which preserve the background, whereas the odd part is generated by the rigid supersymmetries supported by the background. The image of the odd-odd bracket for the Killing superalgebra spans a Lie subalgebra of Killing vectors which preserve the background. This Lie subalgebra, together with the rigid supersymmetries, generate an ideal of the Killing superalgebra, which we call the Killing ideal of the background. The utility of this construction is that it often allows one to infer important geometrical properties of the background directly from the rigid supersymmetry it supports. For example, in dimensions six, ten and eleven, it was proved in [35,36] that any supersymmetric supergravity background possessing more than half the maximal amount of supersymmetry is necessarily (locally) homogeneous.
As a rule, the interactions in a non-linear theory with a local (super)symmetry may be constructed unambiguously by applying the familiar Noether procedure to the linearised version of the theory. Indeed, this is the canonical method for deriving interacting gauge theories in flat space, supergravity theories and their locally supersymmetric couplings to field theory supermultiplets. However, depending on the complexity of the theory in question, it may not be the most wieldy technique and it is sometimes preferable to proceed JHEP06(2016)106 with some inspired guesswork, perhaps based on the assumption of a particular kind of symmetry (e.g., conformal coupling in a conformal field theory). Either way, the guiding principle is to deform (in some sense) the free theory you know in the most general way that is compatible with the symmetries you wish to preserve.
One way to motivate the construction we shall describe in this paper is as an attempt to streamline the procedure for deducing which curved backgrounds support rigid supersymmetry directly in terms of their associated Killing superalgebras. Instead of applying the Noether method to obtain some complicated local supergravity coupling, taking a rigid limit, looking for supersymmetric backgrounds and then computing the Killing superalgebras of those backgrounds, our strategy will be to simply start with the unextended Poincaré superalgebra (without R-symmetry) and obtain all the relevant Killing superalgebras directly as filtered deformations (see below for the definition) of its subalgebras. As expected for the deformation problem of an algebraic structure, there is a cohomology theory which governs the infinitesimal deformations. In this case this is a generalised Spencer cohomology theory, studied in a similar context by Cheng and Kac in [37,38]. In the present work, we shall apply this philosophy to the unextended Poincaré superalgebra on R 1,3 , following a similar analysis on R 1,10 pioneered in [39,40] which yielded what might be considered a Lie-algebraic derivation of eleven-dimensional supergravity.
Let us describe more precisely the problem we set out to solve. Let (V, η) denote the Lorentzian vector space on which four-dimensional Minkowski space is modelled, so(V ) the Lie algebra of the Lorentz group and S its spinor representation. The associated N =1 Poincaré superalgebra p has underlying vector space so(V ) ⊕ S ⊕ V and Lie brackets, for all A, B ∈ so(V ), s ∈ S and v, w ∈ V , given by where σ is the spinor representation of so(V ) and κ : ⊙ 2 S → V is such that κ(s, s) ∈ V is the Dirac current of s. (This and other relevant notions are defined in the appendix.) The Poincaré superalgebra is Z-graded by assigning degrees 0, −1 and −2 to so(V ), S and V , respectively and the Z 2 grading is compatible with the Z grading, in that the parity is the degree mod 2. More precisely, the even subalgebra is the Poincaré algebra p0 = so(V ) ⊕ V and the odd subspace is p1 = S. By a Z-graded subalgebra a of p we mean a Lie subalgebra a = a 0 ⊕ a −1 ⊕ a −2 , with a i ⊂ p i . Now recall that a Lie superalgebra g is said to be filtered, if it is admits a vector space filtration g • : · · · ⊃ g −2 ⊃ g −1 ⊃ g 0 ⊃ · · · , with ∪ i g i = g and ∩ i g i = 0, which is compatible with the Lie bracket in that [g i , g j ] ⊂ g i+j . Associated canonically to every filtered Lie superalgebra g • there is a graded Lie superalgebra g • = i g i , where g i = g i /g i+1 . It follows from the fact that g • is filtered that [g i , g j ] ⊂ g i+j , hence g • is graded. We say that a Lie superalgebra g is a filtered deformation of a < p if it is filtered and its associated graded superalgebra is isomorphic (as a graded Lie superalgebra) to a. If we do not wish to mention the subalgebra a explicitly, we simply say that g is a filtered subdeformation of p.

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The problem we address in this note is the classification of filtered subdeformations g of p for which g −1 = S (and hence g −2 = V ). This paper is organised as follows. In section 2 we define and calculate the Spencer cohomology group H 2,2 (p − , p) of the Poincaré superalgebra. This is the main cohomological calculation upon which the rest of our results are predicated. In particular we use it to extract the equation satisfied by the Killing spinors, recovering in this way the form of the (old minimal off-shell) supergravity Killing spinor equation. We will also use this cohomological calculation as a first step on which to bootstrap the calculation of infinitesimal subdeformations of the Poincaré superalgebra. We give two proofs of the main result in section 2 (Proposition 3): a traditional combinatorial proof using gamma matrices and a representation-theoretic proof exploiting the equivariance under so(V ). In section 3 we prove that the (minimal off-shell) supergravity Killing spinors generate a Lie superalgebra, and that this Lie superalgebra is a filtered subdeformation of p. These results are contained in Theorem 7 in section 3.2 and Proposition 8 in section 3.3, respectively. In section 4 we classify, up to local isometry, the geometries admitting the maximum number of Killing spinors. We do this by solving the zero curvature equations for the connection relative to which the Killing spinors are parallel, and this is done by first solving for the vanishing of the Clifford trace of the curvature: this simplifies the calculation and might be of independent interest. Section 4.4 contains the result of the classification of maximally supersymmetric backgrounds up to local isometry: apart from Minkowski space and AdS 4 , we find the Lie groups admitting a Lorentzian bi-invariant metric. In section 5 we finish the determination of maximally supersymmetric filtered subdeformations of p and recover in this way the Killing superalgebras of the maximally supersymmetric backgrounds found in section 4.4. In the case of a Lie group with bi-invariant metric, we note that the Killing ideal is a filtered deformation of a = S ⊕ V and also explicitly describe all other associated maximally supersymmetric filtered subdeformations of p. The main result there is Theorem 14 in section 5.4. Finally, in section 6, we offer some conclusions.
Given the nature of this problem, it is inevitable that we shall recover some known results and observations which it would be remiss of us not to contextualise. In particular, in addition to R 1,3 , our classification of Killing superalgebras for maximally supersymmetric backgrounds yields, up to local isometry, the following conformally flat Lorentzian geometries: • AdS 4 ; • AdS 3 ×R, with AdS 3 identified with SL(2, R) with its bi-invariant metric; • R × S 3 , with S 3 identified with SU(2) with its bi-invariant metric; and • NW 4 , a symmetric plane wave isometric to the Nappi-Witten group with its biinvariant metric.
We prove that the geometries above are indeed realised as the maximally supersymmetric backgrounds of minimal off-shell supergravity in four dimensions, in Lorentzian signature. That is, we do not assume the form of the supergravity Killing spinor equation from the JHEP06(2016)106 outset -we actually derive it via Spencer cohomology! It therefore follows that the first three geometries above are precisely the maximally supersymmetric backgrounds obtained in [12]. Indeed, the classification of maximally supersymmetric backgrounds of minimal off-shell supergravity in four dimensions has been discussed in various other contexts in the recent literature, e.g., see [18] (section 2.1), [41] (sections 4.2-3), [34], [42] (p. 2), [43] (pp. [12][13]. The NW 4 background is rarely mentioned explicitly -perhaps because, unlike the other maximally supersymmetric Lorentzian backgrounds, it has no counterpart in Euclidean signature -but it is noted in ([42] p. 2) as a plane wave limit, albeit in the context of N = 2 supergravity backgrounds. It is also worth pointing out that ( [18] section 2.1) contains several useful identities (e.g., integrability conditions and covariant derivatives of Killing spinor bilinears) that we also encounter in our construction of the Killing superalgebra for minimal off-shell supergravity backgrounds.

Spencer cohomology
In this section we define and calculate the (even) Spencer cohomology of the Poincaré superalgebra. This calculation has two purposes. The first is to serve as a first step in the classification of filtered subdeformations of the Poincaré superalgebra which is presented in section 5. The second is to derive the equation satisfied by the Killing spinors which, as we show in section 3, generate the filtered subdeformation. The main result, whose proof takes the bulk of the section, is Proposition 3.

Preliminaries
Let p = p −2 ⊕ p −1 ⊕ p 0 , where p −2 = V , p −1 = S and p 0 = so(V ), be the Poincaré superalgebra and p − = p −2 ⊕ p −1 the negatively graded part of p. We will now determine some Spencer cohomology groups associated to p. We recall that the cochains of the Spencer complex of p are linear maps ∧ p p − → p or, equivalently, elements of ∧ p p * − ⊗ p, where ∧ • is meant here in the super sense, and that the degree in p is extended to the space of cochains by declaring that p * p has degree −p. The spaces in the complexes of even cochains of small degree are given in table 1, although for d = 4 there are cochains also for p = 5, 6 which we omit.
Let C d,p (p − , p) be the space of p-cochains of degree d. The Spencer differential is the Chevalley-Eilenberg differential for the Lie superalgebra p − relative to its module p with respect to the adjoint action. For p = 0, 1, 2 and d ≡ 0 (mod 2) it is explicitly given by the following expressions:

2)
JHEP06(2016)106 where x, y, . . . are the parity of elements X, Y, . . . of p − and ζ ∈ C d,p (p − , p) with p = 0, 1, 2 respectively. In this section we shall be interested in the groups H d,2 (p − , p) with d > 0 and even. We first recall some basic definitions. A Z-graded Lie superalgebra a = a p with negatively graded part a − = p<0 a p is called fundamental if a − is generated by a −1 and transitive if for any X ∈ a p with p ≥ 0 the condition [X, a − ] = 0 implies X = 0. Lemma 1. The Poincaré superalgebra p = p −2 ⊕ p −1 ⊕ p 0 is fundamental and transitive. Moreover H d,2 (p − , p) = 0 for all even d > 2.
Note that the space of cochains C d,p (p − , p) is an so(V )-module and the same is true for the spaces of cocycles and coboundaries, as ∂ is so(V )-equivariant. This implies that each cohomology group H d,p (p − , p) is an so(V )-module, in a natural way. It remains to compute

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Since End(S) ∼ = Cℓ(V ) ∼ = ∧ • V (where the first isomorphism is one of algebras and the second one of vector spaces), we may write where we have used the last of the duality equations (A. 19) and the symmetry relations (A.11). Inserting this into equation (2.11), which must be true for all s ∈ S, we get that the terms which depend on sΓ ρ s and sΓ ρσ s must vanish separately and we arrive at two equations: and Tracing this last equation with η µν , we learn that (2.21) Multiplying by 1 2 ǫ αβρσ , and using the identities (A.20), we obtain Tracing the expression above with η νβ , we arrive at µν ∈ R. Tracing equation (2.17) with η µν gives while tracing it with η νρ gives
Next we solve the second cocycle condition (2.14). Using the expression for β µ given in equation (2.29), we can rewrite the first term of equation (2.14) as follows: where, using that the Dirac current of s Clifford annihilates s (see Proposition 15), the first term vanishes. Similarly, using Γ µν = −Γ ν Γ µ − η µν and again the fact that (sΓ µ s)Γ µ s = 0, the first term in equation (2.14) becomes We now rewrite the second term in equation (2.14) by inserting the expression for β ν in equation (2.29) into equation (2.16) to obtain 1 2 (sΓ µ β ν s)Γ µν s = 1 2 (sΓ µν (a + bΓ 5 )s)Γ µν s − (sΓ µ s)ϕ µ Γ 5 s , (2.32) where we have again used Γ µν = −Γ ν Γ µ − η µν and the fact that (sΓ µ s)Γ µ s = 0. The first term on the right-hand side vanishes by virtue of the fact that the Dirac 2-form of s and its dual both Clifford annihilate s (see Proposition 15). In summary, equation (2.14) becomes for all s ∈ S, whose general solution is Inserting this into equation (2.29), we arrive at which can be rewritten as from where the result follows.

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Alternative proof. It may benefit some readers to see an alternative proof of this result, which exploits the equivariance under so(V ).
In table 2 above we list the decomposition of Hom(V, From the first decomposition in (2.36) we immediately infer that Hom(V, End(S)) is the direct sum of five different isotypical components, namely JHEP06(2016)106 Note now that for any Θ, Θ ′ ∈ ∧ 2 S the element β ∈ Hom(V, End(S)) defined by and it is therefore a solution of (2.35).
In summary we get that the solution space of equation (2.35) contains an so(V )-module isomorphic to and formed by elements which do not satisfy (2.35). Note that the direct sum of (2.39) and (2.40) gives all the isotypical components (2.37) in Hom(V, End(S)). We now turn to the remaining isotypical components (2.38). We first recall that Hom(V, End(S)) contains a single irreducible submodule of type (V ⊗ ∧ 2 V ) 0 . We fix an orthonormal basis (e µ ) of V , consider the element In other words Υ(β)

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and we have It follows that Υ(β) 2 = 0 unless r 1 = r 2 = 0 and that the solution space of (2.35) does not contain any submodule isomorphic to (V ⊗ ∧ 1 V ) 0 either. In summary we just showed that β ∈ Hom(V, End(S)) solves (2.35) if and only if there exist reals a, b and vectors ϕ 1 , ϕ 2 such that for all v ∈ V and s ∈ S.
We now turn to equation (2.6), with β as in (2.41) and γ expressed in terms of β using (2.5). We remark that from the above discussion we already know that At this point it is convenient to fix an η-orthonormal basis (e µ ) of V and use the Einstein summation convention on indices as in appendix A.2.1.

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again, using that ω (3) · s = 0. Using equation (A. 19) repeatedly, the Clifford relation and Proposition 15 again, we can rewrite the first two terms of (2.43) as Since this must hold for all ϕ ∈ ∧ 1 V and s ∈ S, it follows that r 2 = 3r 1 .

Killing superalgebras
In analogy with the results [39,40] in eleven dimensions, we define a notion of Killing spinor from the component β of the cocycle in Proposition 3. In this section we prove that these Killing spinors generate a Lie superalgebra.

Preliminaries
Let (M, g, a, b, ϕ) be a four-dimensional Lorentzian spin manifold (M, g) with spin bundle S(M ) which is, in addition, endowed with two functions a, b ∈ C ∞ (M ) and a vector field ϕ ∈ X(M ). The main aim of this section is to construct a Lie superalgebra k = k0 ⊕ k1 naturally associated with (M, g, a, b, ϕ).
Note that any non-zero Killing spinor is nowhere vanishing since it is parallel with respect to a connection on the spinor bundle. We set and consider the operation [−, −] : k ⊗ k → k compatible with the parity of k = k0 ⊕ k1 and determined by the following maps: is given by the usual commutator of vector fields, given by the Dirac current of ε ∈ k1, • [−, −] : k0 ⊗ k1 → k1 is given by the spinorial Lie derivative of Lichnerowicz and Kosmann (see [46] and also, e.g., [47]).

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The fact that [−, −] actually takes values in k is a consequence of Theorem 7 below, where we show that [−, −] is the bracket of a Lie superalgebra structure on k. Assuming that result for the moment we make the following is called the Killing superalgebra associated with (M, g, a, b, ϕ).
We recall that the spinorial Lie derivative of a spinor field ε along a Killing vector field X is defined by L X ε = ∇ X ε + σ(A X )ε, where σ : so(T M ) → End(S(M )) is the spin representation and A X = −∇X ∈ so(T M ). It enjoys the following basic properties, for all Killing vectors X, Y , spinors ε, functions f and vector fields Z: (ii) X → L X is a representation of the Lie algebra of Killing vector fields: (iii) L X is compatible with Clifford multiplication: (iv) L X is compatible with the Levi-Civita connection: We note for later use that, from property (iii) and the fact that ⊙ 2 S = ∧ 1 V ⊕ ∧ 2 V , we have for any Killing vector X, spinor ε and vector field Z, which yields the following additional property of the spinorial Lie derivative: (v) the Dirac current is equivariant under the action of Killing vector fields: We first collect a series of important auxiliary results, which will be needed in the proof of the main Theorem 7.
Proposition 6. Let ε be a non-zero section of the spinor bundle S(M ) of (M, g, a, b, ϕ), with associated differential forms JHEP06(2016)106 In particular the Dirac current K = κ(ε, ε) of ε is a Killing vector field satisfying Proof. For any Killing spinor ε and X, Y, Z ∈ X(M ) we compute where, in both cases, the last equality follows from equation (A.11) or, equivalently, that In other words we have

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and applying ⋆, which is a parallel endomorphism of Ω • (M ), on both sides of these identities we also get Claims (i)-(iv) follows then immediately from the fact that for any ω ∈ Ω p (M ) we have where (e µ ) is a fixed local orthonormal frame field of (M, g). Now, for any Killing spinor ε and X, Y ∈ X(M ) we have where the last equality follows from equation (A.11). Since the last term is manifestly i.e., L K b = 0. One shows L K a = 0 in a similar way. If ω = ω (1) , ω (2) , ω (2) , or ω (3) , then ı K ω = 0 by Proposition 15 and from (i)-(iv) we get This proof of (3.3) is thus completed. In order to show (3.4) we use that K is a Killing vector and L K ω (2) = 0 so that for all X, Y, Z ∈ X(M ): where the last identity follows from a direct computation using (3.6) and (3.3).

The Killing superalgebra
We state and prove the main result of section 3.

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Proof. The fact that [k0, k0] ⊂ k0 follows from basic properties of Lie derivatives of vector fields. On the other hand for any X ∈ k0 and Z ∈ X(M ) we have that since D depends solely on the data (g, a, b, ϕ) which is preserved by X ∈ k0. This shows that L X ε is a Killing spinor or, in other words, that [k0, k1] ⊂ k1.
We already know from Proposition 6 that K = κ(ε, ε) is a Killing vector field which satisfies L K a = L K b = 0. To prove K ∈ k0 we still need to show L K ϕ = 0. From Proposition 6 we have (1) and hence, for any ϑ ∈ Ω 1 (M ), In the special case where ı K ϑ = 0 the first three terms of the right-hand side of the above identity are degenerate 4-forms and hence zero. Then equation (3.9) becomes so that L K ϕ = f K, for some f ∈ C ∞ (M ). From this fact, equation (3.4) and ω (1) ∧ ω (2) = 0 we finally get for all X, Y, Z ∈ X(M ), hence f = 0. This proves L K ϕ = 0 and [k1, k1] ⊂ k0.
We finally show that [−, −] : k ⊗ k → k satisfies the axioms of a Lie superalgebra. This is a direct consequence of the following observations: (i) k0 is a Lie algebra: this is just the Jacobi identity of the Lie bracket of vector fields; (ii) k0 acts on k1, by property (ii) of the spinorial Lie derivative; (iii) the Dirac current is a symmetric k0-equivariant map, by property (v) of the spinorial Lie derivative;

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(iv) for any ε ∈ k1, with associated Dirac current K = κ(ε, ε), we have from the definition of Killing spinor and (3.5) that where the last equality holds by Proposition 15. This is equivalent to the component of the Jacobi identity for k with three odd elements.
The proof is thus completed.

The Killing superalgebra is a filtered deformation
We now show that the Killing superalgebra k = k0 ⊕ k1 is a filtered deformation of a Z-graded subalgebra of the Poincaré superalgebra p. To this aim, it is convenient to denote the triple (a, b, ϕ) collectively by Φ and to abbreviate the Killing spinor equation for all Z ∈ X(M ) and ε ∈ Γ(S(M )). The notation is chosen to make contact with that of Proposition 3. The reason for the superscript Φ is to distinguish β Φ from the more general component β of the filtered Lie brackets in (3.15) below. For a similar reason we also introduce the so(T M )-valued symmetric bilinear tensor γ Φ on S(M ) given by

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Parallel sections ζ of a vector bundle with connection are uniquely determined by their value ζ| o at any given point o ∈ M . (We tacitly assume that M is connected.) Let us introduce the following notation Therefore k determines a subspace of E o = V ⊕ so(V ) ⊕ S, which is the underlying vector space of the Poincaré superalgebra p. We recall that p is a Z-graded Lie superalgebra with Lie brackets given in equation (1.1) and that the Z and Z 2 gradings are compatible. Let (ξ, A ξ ), with A ξ = −∇ξ, and (ζ, A ζ ) belong to k0. Their Lie bracket is given by where the bracket on the right-hand side is the commutator in so(T M ). We see that the Riemann curvature measures the failure of k0 to be a Lie subalgebra of the Poincaré algebra p0. If now ε ∈ k1, then the Lie bracket with (ξ, A ξ ) is given by where σ : so(T M ) → End(S(M )) is the spinor representation. Finally, the Dirac current of a Killing spinor ε ∈ k1 is given by We now show that k defines a graded subspace of p = E o . Define ev0 o : k0 → V to be evaluation at o and projection onto V = T o M . More precisely, Similarly, let ev1 o : k1 → S be the evaluation at o. We set S ′ = im ev1 o and V ′ = im ev0 o .
Let h = ker ev0 o . These are the Killing vectors in k0 which take the form (0, A) ∈ V ⊕ so(V ) at o ∈ M . Therefore h defines a subspace of so(V ), but from equation (3.12), we see that it is also a Lie subalgebra: In addition, the conditions L ξ a = L ξ b = L ξ ϕ = 0 that are satisfied by the Killing vectors ξ ∈ k0, when evaluated at o ∈ M , imply that if (0, A) ∈ h then

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In particular, h acts on S by restricting the action of so(V ), and this action preserves S ′ .
The Lie subalgebra h < k0 defines a short exact sequence which yields a vector space isomorphism k0 ∼ = h ⊕ V ′ , and therefore as graded vector spaces, a (non-canonical) isomorphism We now wish to express the Lie superalgebra structure on k in terms of a Lie bracket on the graded vector space h ⊕ S ′ ⊕ V ′ . This requires a choice of splitting of the short exact sequence (3.14). Geometrically, this amounts to choosing for every v ∈ V ′ a Killing vector field ξ ∈ k0 with ξ| o = v. Such a choice gives an embedding of V ′ into V ⊕ so(V ) as the graph of a linear map Σ : The Lie bracket of (0, A) ∈ h and (v, Σ v ) ∈ k0 is given by This allows us to read off the Lie bracket on h ⊕ S ′ ⊕ V ′ . We will let v, w ∈ V ′ , s ∈ S ′ and A, B ∈ h. Then we have Notice that all the under-braced terms have positive filtration degree: λ, α, β and γ have degree 2, whereas δ has degree 4. If we set those maps to zero, which is equivalent to passing to the associated graded superalgebra, then we are left with the Z-graded subalgebra a < p given by the Lie brackets

Zero curvature equations
In this section we calculate the curvature of the connection D on the spinor bundle and solve the zero curvature equations for the metric g and the fields a, b, ϕ. We do this in two steps. In the first step we arrive at a first set of equations obtained by setting the Clifford trace of the curvature to zero. We perform this first step for two reasons. The first reason is by analogy with eleven-dimensional supergravity, where the vanishing of the Clifford trace of the curvature is equivalent to the bosonic field equations (and the Bianchi identity). The second reason is that this first set of equations is easier to solve and already imposes strong constraints on the geometric data which simplify the solution of the zero curvature equations. The second step is the solution of the zero curvature equations, which will yield the maximally supersymmetric backgrounds. The Killing superalgebras of these maximally supersymmetric backgrounds should (and do) agree with the maximally supersymmetric filtered deformations which we classify in section 5.
With regard to the first reason for performing the first step, we must stress that any relation in four dimensions between the equation obtained by setting to zero the Clifford trace of the curvature and the bosonic field equations of minimal off-shell supergravity remains to be seen. If we were to identify (up to constants of proportionality) the fields a, b and ϕ in the connection D in (3.1) with the bosonic fields in the minimal off-shell gravity supermultiplet in four dimensions (as described, say, in [50] section 16.2.3), and identify (up to an overall constant of proportionality) Dε with the supersymmetry variation of the gravitino Ψ in the gravity supermultiplet, evaluated at Ψ = 0, one finds that the purely bosonic terms in the off-shell supergravity Lagrangian density must be proportional to R + 24(a 2 + b 2 + |ϕ| 2 ), where R is the scalar curvature of g. The Einstein equations for JHEP06(2016)106 this supergravity Lagrangian are R µν = −12(a 2 + b 2 )g µν − 24ϕ µ ϕ ν which, after integrating out the auxiliary fields a, b and ϕ, imply that that g must be Ricci-flat. As we will see, the equations obtained by setting to zero the Clifford trace of the curvature are similar but different.

The curvature of the superconnection
Let us write the Killing spinor condition for ε ∈ Γ(S(M )) as ∇ Z ε = β Φ Z ε for all vector fields Z, and where the End(S(M ))-valued one-form β Φ was defined in equation (3.10). In other words, where R is the curvature 2-form of ∇ on the spinor bundle. An explicit calculation shows that vanishes. Here e µ and e µ are g-dual local frames of T M . Another explicit calculation shows that where Ric stands for the Ricci operator and we have introduced the shorthand g( / ∇ϕ, X) = Γ ρ ∇ ρ ϕ µ X µ .

The vanishing of the Clifford trace of the curvature
We now describe the equations arising by demanding that the Clifford trace of the curvature of the spinor connection D vanishes; in other words, that for all vector fields X, Ric D (X) = 0. This is a system of equations with values in End(S(M )), which is isomorphic as a vector bundle to 4 p=0 ∧ p T M . This means that the components of these equations in each summand have to be satisfied separately. The p = 1 component relates the Ricci tensor to the data (a, b, ϕ), whereas the p = 1 components constrain (a, b, ϕ). We start with these first.
Multiplying by b − a vol and since this has to be true for all X, we arrive at (a 2 + b 2 )ϕ = 0 .

The solutions
Let us analyse the type of solutions to these equations. We have seen that there are three branches of solutions stemming from the p = 2 component equation (4.6).
(I) a = b = ϕ = 0. In this case, the p = 1 component equation simply says that g is Ricciflat. In this background, Killing spinors are parallel and therefore the supersymmetric backgrounds are the Ricci-flat manifolds whose holonomy is contained in the isotropy of a spinor. Since the Dirac current of a parallel spinor is null and parallel, these metrics are Ricci-flat Brinkmann metrics. See, e.g., ([51] section 3.2.3) for a discussion of these geometries.
(II) a 2 + b 2 = 0 and ϕ = 0. Putting ϕ = 0, we see from equations (4.4) and (4.5) that da = db = 0, so they are constant and the Ricci tensor is given by so that g is Einstein with negative cosmological constant. The Killing spinors are (up to an R-symmetry which allows us to set b = 0, say) geometric Killing spinors. Such geometries are reviewed in ([52] sections 6-7) and discussed in [53].

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(III) a = b = 0 and ϕ = 0. Then ϕ is a parallel vector field and the Ricci tensor, given by is also parallel. This is a kind of fluid solution. Ricci-parallel geometries have been studied in [54]. The determining factor is the algebraic type of the Ricci endomorphism. In this case, this depends on the causal type of ϕ, which is constant because ϕ is parallel. If ϕ is timelike or spacelike, so that (in our mostly minus conventions) |ϕ| 2 g is positive or negative, respectively, then the Ricci endomorphism is diagonalisable and the geometry decomposes (up to coverings) into a product M = R × N of a line and a three-dimensional Einstein space N , hence a space form. Moreover, upon identifying the spin bundle of M with (an appropriate number of copies of) the spin bundle of N , it is not difficult to see that Killing spinors in these backgrounds correspond to geometric Killing spinors on N (up to an R-symmetry). If ϕ is null, then the Ricci endomorphism is two-step nilpotent and the geometry is Ricci-null. The subbundle of T M of orthogonal vectors to ϕ is also in this case integrable in the sense of Frobenius but the above simple interpretation of Killing spinors is missing since the associated integrable submanifolds N have a degenerate induced metric.

Maximally supersymmetric backgrounds
Maximally supersymmetric backgrounds are those for which the spinor connection D is flat. The zero curvature condition R D X,Y = 0 for all vector fields X, Y becomes a system of equations with values in End(S(M )) and therefore, just as for the vanishing of the Clifford trace of the curvature, the different components of the curvature must vanish separately. We can reuse our calculations above, since if D is flat, the Clifford trace of the curvature certainly vanishes. This means that we can consider the three branches described above. We will meet the geometries we are about to discuss again in the next section, where we classify the maximally supersymmetric filtered subdeformations of the Poincaré superalgebra.
as an equation in End(S(M )). This is equivalent to which says that g is locally isometric to AdS 4 . If a = b = 0, and using that ϕ is parallel, the D-flatness condition is again as an equation in End(S(M )). The corresponding Riemann tensor is given by Since ϕ and g are parallel, so is the Riemann tensor and hence this corresponds to a locally symmetric space. Furthermore, it is conformally flat. Indeed, in four dimensions, the Weyl tensor is given in terms of the Riemann tensor, the Ricci tensor R µν = g ρσ R µρσν and the Ricci scalar R = g µν R µν by Inserting the above expression for R µνρσ into the Weyl tensor we see that it vanishes, so that the geometry is conformally flat. The corresponding Ricci tensor is given by equation (4.11) and the Ricci scalar is R = 24|ϕ| 2 g . This geometry corresponds to a Lorentzian Lie group with a bi-invariant metric. Indeed, the equation (4.12) satisfied by the Riemann tensor is equivalent to the vanishing of the curvature of a metric connection with parallel totally skewsymmetric torsion proportional to the Hodge dual of ϕ. As shown, for instance, in [55,56], the existence of a flat metric connection with closed skewsymmetric torsion is equivalent to the manifold being locally isometric to a Lie group with a bi-invariant metric.
Since ϕ is parallel, its g-norm is constant and in a Lorentzian manifold this can be of three types: 1. |ϕ| 2 g > 0. This is timelike in our conventions. The background is locally isometric to R × S 3 , where we identify the round S 3 with the Lie group SU(2) with its bi-invariant metric.
2. |ϕ| 2 g < 0. This is spacelike and hence the background is locally isometric to AdS 3 ×R, where we identify AdS 3 with SL(2, R) with its bi-invariant metric.
3. |ϕ| 2 g = 0. This is the null case and hence the background is locally isometric to the Nappi-Witten group [57] with its bi-invariant metric.

Maximally supersymmetric filtered deformations
We now resume the analysis of filtered subdeformations of the Poincaré superalgebra by classifying the filtered deformations with maximal odd dimension. We will show that they correspond precisely to the Killing superalgebras of the maximally supersymmetric backgrounds classified in section 4.4.
More precisely, let a = a −2 ⊕ a −1 ⊕ a 0 be a Z-graded subalgebra of the Poincaré superalgebra p = V ⊕ S ⊕ so(V ) with a −1 = S. By Lemma 16, we also have that a −2 = V , JHEP06(2016)106 so that a differs from p only in zero degree, where a 0 = h is a subalgebra of so(V ). The aim of this section is to classify, for any possible given h, the filtered deformations g of a. We will see that they are essentially governed by the h-invariant elements H 2,2 (a − , a) h of the Spencer group H 2,2 (a − , a) of a, where a − = a −2 ⊕ a −1 is the negatively graded part of a.
In section 5.1 we set up the calculation of H 2,2 (a − , a), which will be described in section 5.2. This result will then be used in section 5.3 to classify the filtered deformations. The results are summarised in Theorem 14 in section 5.4.

Preliminaries
Here we set up the calculation of the Spencer cohomology H 2,2 (a − , a). We introduce the Spencer complex of a in complete analogy to the Spencer complex of p (cf. section 2): one has simply to replace so(V ) with h in the definitions. For instance any element ζ ∈ C 2,2 (a − , a) can be uniquely written as the sum ζ = α + β + γ, where and the Lie brackets of a general filtered deformations of a take the form for some maps λ : h ⊗ V → h and δ : We recall that a transitive and fundamental Z-graded Lie superalgebra a = a p with negatively graded part a − = p<0 a p is called a full prolongation of degree k if H d,1 (a − , a) = 0 for all d ≥ k.
Lemma 9. Let a = a −2 ⊕ a −1 ⊕ a 0 be a Z-graded subalgebra of the Poincaré superalgebra which differs only in zero degree. Then a is fundamental, transitive and H d,2 (a − , a) = 0 for all even d > 2. Furthermore it is a full prolongation of degree k = 2.
Proof. We only show the last claim, the others follow as in the proof of Lemma 1. Any ζ ∈ C 2,1 (a − , a) satisfies ζ(V ) ⊂ h, ζ(S) ⊂ a 1 = 0 and for all s 1 , s 2 ∈ S, v 1 , v 2 ∈ V . The first equation directly implies that ζ = 0 is the only cocycle and hence H 2,1 (a − , a) = 0. If d > 2 then C d,1 (a − , a) = 0 for degree reasons.
Remark. One can actually prove that a is a full prolongation of degree k = 1, based on the non-trivial fact that the so-called "maximal prolongation" g ∞ of a − = V ⊕S is a simple Lie superalgebra of type sl(1|4) with a special Z grading of the form g ∞ = g ∞ −2 ⊕ · · · ⊕ g ∞ 2 , cf. [58]; but the simpler result of Lemma 9 suffices for our purposes.

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To state the main first intermediate result on filtered deformations g of a we recall that the Lie brackets of g have components of nonzero degree: the sum µ : a ⊗ a → a of all components of degree 2 and the unique component δ : Proposition 10. Let a = V ⊕ S ⊕ h be a Z-graded subalgebra of the Poincaré superalgebra p = V ⊕ S ⊕ so(V ) which differs only in zero degree. If g is a filtered deformation of a then: 1. µ| a − ⊗a − is a cocycle in C 2,2 (a − , a) and its cohomology class is h-invariant (that is, the cocycle µ| a − ⊗a − is h-invariant up to coboundaries); and isomorphic to g as a filtered Lie superalgebra.
Proof. The first claim follows directly from Proposition 2.2 of [37]. Let now g and g ′ be filtered deformations of a such that [µ| Spencer coboundary and we may first assume without any loss of generality that µ| a − ⊗a − = µ ′ | a − ⊗a − by Proposition 2.3 of [37]. Moreover, since a is a fundamental and transitive full prolongation of degree k = 2 by Lemma 9, Proposition 2.6 of [37] applies and we may also assume µ = µ ′ without any loss of generality. In other words we just showed that g ′ is isomorphic as a filtered Lie superalgebra to another filtered Lie superalgebra g ′′ which satisfies µ ′′ = µ. Now, given any two filtered deformations g and g ′ of a with µ = µ ′ it is easy to see that δ − δ ′ = (δ − δ ′ )| a − ⊗a − is a Spencer cocycle (use e.g., [37] equation 2.6). However H 4,2 (a − , a) = ker ∂| C 4,2 (a − ,a) = 0 by Lemma 9 and hence δ = δ ′ . This proves that any two filtered deformations g and g ′ of a with [µ ′ | a − ⊗a − ] = [µ| a − ⊗a − ] are isomorphic.
In other words, filtered deformations are determined by the space H 2,2 (a − , a) h of hinvariant elements in H 2,2 (a − , a). In particular the components of non-zero filtration degree λ = µ| h⊗V : h⊗V → h and δ : ∧ 2 V → h are completely determined by the class [µ| a − ⊗a − ] ∈ H 2,2 (a − , a) h , up to isomorphisms of filtered Lie superalgebras.
We will now describe H 2,2 (a − , a). We recall that this group has already been determined in Proposition 3 when a = p is the Poincaré superalgebra. Therein we also described the kernel H 2,2 of the Spencer operator acting on Hom(V ⊗ S, S) ⊕ Hom(⊙ 2 S, so(V )): it consists of the maps β +γ ∈ Hom(V ⊗S, S)⊕Hom(⊙ 2 S, so(V )) which are of the form given by Proposition 3. To avoid confusion with the general components (5.1) we will denote these maps by β Φ + γ Φ from now on, that is we set Φ = (a, b, ϕ) ∈ 2R ⊕ V and for all v ∈ V and s ∈ S, according to Proposition 3. In addition we set for all v ∈ V , s ∈ S.

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We will also determine the h-invariant classes in H 2,2 (a − , a), the Lie subalgebras h ⊂ so(V ) for which H 2,2 (a − , a) h = 0, hence the graded subalgebras a = V ⊕S ⊕h of p admitting nontrivial filtered deformations. The condition H 2,2 (a − , a) h = 0 has strong consequences and, as we will now see, gives rise to a dichotomy: either ϕ = 0 and a 2 + b 2 = 0 or ϕ = 0 and a = b = 0. H 2,2 (a − , a) We start with the following Proposition 11. Let a = V ⊕ S ⊕ h be a Z-graded subalgebra of the Poincaré superalgebra p = V ⊕ S ⊕ so(V ) which differs only in zero degree. Then

The cohomology group
is trivial if and only if Φ = 0; (ii) the condition γ Φ (s, s)− ψ(κ(s, s)) ∈ h is satisfied for all s ∈ S if and only if separately

4)
for all s ∈ S; is an h-invariant cohomology class then h leaves ϕ invariant, that is h ⊂ h ϕ where h ϕ = so(V ) ∩ stab(ϕ) and stab(ϕ) is the Lie algebra of the stabiliser of ϕ in GL(V ). a) is a nontrivial and h-invariant cohomology class then exactly one of the following two cases occurs:
We will now have a closer look at condition (5.8), using that with the first two terms (resp. last term) in the r.h.s. of the above equation acting on the component ∧ 2 V (resp. ∧ 1 V ) of ⊙ 2 S but trivially on the other component ∧ 1 V (resp. ∧ 2 V ). In particular condition (5.8) splits into (5.3) and (5.4), proving claim (ii).
By the results of Proposition 10 and Proposition 11, we need only to consider the filtered deformations associated to h-invariant cohomology classes in H 2,2 (a − , a) with Φ = 0. Indeed if Φ = 0 then [µ| a − ⊗a − ] = 0 and the associated filtered Lie superalgebras are just the Z-graded subalgebras of the Poincaré superalgebra.
We now investigate separately the cohomology classes in family (1) and (2) of Proposition 11.
Proof. First of all, as a 2 + b 2 = 0 by Proposition 11, we have that right multiplication by a + b vol in Cℓ(V ) is a linear isomorphism. In particular it restricts to a linear isomorphism of ∧ 2 V ⊂ Cℓ(V ). On the other hand, from Proposition 3: for all w ∧ v ∈ ∧ 2 V ⊂ Cℓ(V ) and s ∈ S. Since γ Φ (s, s) ∈ h for all s ∈ S from (1) of Proposition 11 and ⊙ 2 S = ∧ 1 V ⊕ ∧ 2 V , the claim follows.

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To proceed further, we need to consider the case where ϕ = 0, a = b = 0. It is however sufficient to consider ϕ up to the action of CSO(V ) = R × × SO(V ). To see it, we note that the group CSpin(V) with Lie algebra co(V ) is a double-cover of CSO(V ) and it naturally acts on the Poincaré superalgebra p = V ⊕S ⊕so(V ) by 0-degree Lie superalgebra automorphisms (t Id ∈ CSpin(V) acts with eigenvalues 0, e −t and e −2t on, respectively, so(V ), S and V ). In particular any element c ∈ CSpin(V) sends a Z-graded subalgebra a = V ⊕ S ⊕ h of p into an (isomorphic) Z-graded subalgebra a ′ = c · a = V ⊕ S ⊕ (c · h) of p and, if g is a filtered deformation of a associated with ϕ then g ′ = c · g is a filtered deformation of a ′ , which is associated with ϕ ′ = c · ϕ.
We will distinguish ϕ according to whether it is spacelike, timelike or lightlike and denote by Π ⊂ V the line defined by the span of ϕ. In the first two cases we can decompose V = Π ⊕ Π ⊥ into an orthogonal direct sum and h ϕ = so(Π ⊥ ) ⊂ so(V ). If ϕ is lightlike, we choose an η-Witt basis for V such that V = R e + , e − ⊕ W and ϕ = e + . Our plane is Π = R e + and h ϕ = so(W ) (e + ∧ W ) ⊂ so(V ), where e + ∧ W is the abelian Lie subalgebra of so(V ) consisting of null rotations fixing e + . In this case we decompose any Proof. We already know from (2) of Proposition 11 that Im(γ Φ ) ⊂ h ϕ . In addition: for all v, w ∈ V . Using (5.14) and ⊙ 2 S = ∧ 1 V ⊕ ∧ 2 V , we first see that γ Φ (∧ 2 V ) = 0. We now break our arguments into two cases, depending on whether or not the line Π corresponding to ϕ is degenerate. If ϕ is spacelike or timelike then from (5.14) we see that γ Φ (Π) = 0 whereas is an so(Π ⊥ )-equivariant monomorphism, hence an isomorphism by dimensional reasons.
If ϕ is lightlike we decompose
We collect here for later use different equivalent characterizations of the map ψ : V → so(V ) associated to the unique cocycle representative of Lemma 13: (i) ψ(κ(s, s)) = γ ϕ (s, s) for all s ∈ S; (ii) ψ(u) = 2ı u ı ϕ vol for all u ∈ V ; (v) ψ(u)s = −(ϕ ∧ u) · vol ·s for all u ∈ V and s ∈ S; We also remark that ψ is an h ϕ -equivariant map with the kernel Π and image h ϕ .

Integrability of the infinitesimal deformations
In this section we construct a filtered deformation g for any of the nontrivial h-invariant elements in H 2,2 (a − , a). Our description of g will be very explicit and rely on a direct check of the Jacobi identities. To describe the Lie superalgebra structure of g, it is convenient to introduce a formal parameter t which keeps track of the order of the deformation. In particular, the original graded Lie superalgebra structure on a subalgebra a = V ⊕ S ⊕ h of the Poincaré superalgebra p = V ⊕ S ⊕ so(V ) has order t 0 whereas the infinitesimal deformation has order t.
From Proposition 10, Proposition 11, and Lemma 12, Lemma 13 we know that there are two different families of non-trivial filtered deformations g. The first family has ϕ = 0, a 2 + b 2 = 0 and h = so(V ), that is a = p. In this case γ Φ : ⊙ 2 S → so(V ) is surjective and by (2) of Proposition 11 the filtered Lie superalgebra g has the brackets of the form where A, B ∈ so(V ), s ∈ S, v, w ∈ V , for some maps λ : so(V ) ⊗ V → so(V ) and δ : ∧ 2 V → so(V ) to be determined. In other words the brackets on V ⊗ S and ⊙ 2 S are respectively given by β Φ and γ Φ and we can always assume α = 0 without any loss of generality.

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The second family has ϕ = 0, a = b = 0 and h is a Lie subalgebra of the stabiliser h ϕ = so(V ) ∩ stab(ϕ), see (iii) of Proposition 11. We recall that ϕ ∈ ∧ 1 V can be either spacelike, timelike or lightlike. In this case the bracket on ⊙ 2 S is simply given by the Dirac current and the filtered Lie superalgebra g has the form To go through all the Jacobi identities systematically, we use the notation [ijk] for i, j, k = 0, 1, 2 to denote the identity involving X ∈ a −i , Y ∈ a −j and Z ∈ a −k . We first consider the second case (5.17), which is slightly more involved, and claim that the Jacobi identities are satisfied if we set both λ and δ to be zero. To show this, it is first convenient to note that [V, V ] ⊂ V , [V, S] ⊂ S and [S, S] ⊂ V and prove that the putative bracket operations restricted on V ⊕ S satisfy the Jacobi identities. We have: • the [112] identity is satisfied by virtue of the characterization (i) of ψ, the h ϕequivariance of the Dirac current and the first cocycle condition (2.5); • the [111] identity is satisfied by virtue of the characterization (i) of ψ and the second cocycle condition (2.6); • the [122] identity is satisfied provided for all v, w ∈ V and s ∈ S; • the [222] identity is satisfied provided S(α(u, α(v, w))) = 0 , (5.19) where S is the cyclic sum on u, v, w ∈ V . Now using characterization (iv) of ψ one can check that
Let g − = (V ⊕S, [−, −]) be the filtered Lie superalgebra structure on V ⊕S we have just described. Note that the Lie bracket of g − is defined in terms of ψ, Clifford multiplication, Dirac current of spinors and the vector ϕ, so that the stabilizer h ϕ = so(V ) ∩ stab(ϕ) of ϕ in so(V ) acts naturally on g − by outer derivations. It is then clear from (5.17) that, for any subalgebra h of h ϕ , the semidirect sum g = h g − is the required filtered deformation of a = V ⊕ S ⊕ h.
We now consider the first case (5.16) and set λ to be zero. We have: • the [000] identity is satisfied since so(V ) is a Lie algebra; • In this paper, we have considered the supersymmetries of rigid supersymmetric field theories on Lorentzian four-manifolds from the viewpoint of their Killing superalgebras. We showed that the relevant Killing spinor equations, which we identify with the defining condition for bosonic supersymmetric backgrounds of minimal off-shell supergravity in four dimensions, admit a cohomological interpretation in terms of the Spencer group H 2,2 (p − , p) of the N =1 Poincaré superalgebra p in four dimensions. This result is in analogy with a similar result in eleven dimensions [39,40].
We then gave a self-contained proof of the fact that supergravity Killing spinors generate a Lie superalgebra, and that this Lie superalgebra is a filtered subdeformation of p. Finally we classified, up to local isometry, the geometries admitting the maximum number of Killing spinors: Minkowski space, AdS 4 and the nonabelian Lie groups with a Lorentzian bi-invariant metric, namely AdS 3 ×R, R × S 3 and the Nappi-Witten group NW 4 . Our approach here is based on two independent arguments. In section 4 we solved the flatness equations for the connection defining the Killing spinor equations and described the corresponding Lorentzian geometries. In section 5 we used again Spencer cohomology techniques to describe the filtered subdeformations of p with maximum odd dimension and recovered in this way the Killing superalgebras of the maximally supersymmetric backgrounds.
None of the geometries in our classification are new. The novelty in this paper lies in our approach, which systematises the search for backgrounds on which one can define rigid supersymmetric field theories by mapping it to an algebraic problem on which we can bring to bear representation-theoretic techniques. In forthcoming work, we shall apply these techniques to a broader class of field theories with rigid supersymmetry in higher dimensions.

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The Clifford algebra Cℓ(V ) associated to (V, η) is the real, associative, unital algebra generated by V (and the identity 1) subject to the Clifford relation (please notice the sign!) As a vector space, Cℓ(V ) ∼ = ΛV = 4 p=0 ∧ p V . If v ∈ V and φ ∈ ∧ p V , their Clifford product, denoted by ·, is given by where v ♭ ∈ V * is the dual covector defined by the inner product: v ♭ (w) = η(v, w), for all w ∈ V . We will often drop the superscript ♭ if it is unambiguous to do so. The Clifford algebra is not commutative: Continuing in this way we may derive the Clifford product of φ ∈ ∧ p V with bivectors: Let us introduce the volume element vol = e 0 ∧ e 1 ∧ e 2 ∧ e 3 ∈ ∧ 4 V . It obeys vol 2 = −1 and vol ·φ = (−1) p φ · vol , for φ ∈ ∧ p V . In particular, it is not central. Clifford multiplication by the volume element agrees (up to a sign) with Hodge duality: ⋆ φ = (−1) p(p+1)/2 φ · vol , (A.5) for φ ∈ ∧ p V . It follows that ⋆ 2 = (−1) p+1 on ∧ p V . In particular, it is a complex structure on bivectors, as expected. The Lie algebra so(V ) of η-skewsymmetric endomorphisms of V is isomorphic, as a vector space, to ∧ 2 V . If v ∧ w ∈ ∧ 2 V , then the corresponding endomorphism is defined by Indeed, one checks that the Clifford commutator
A second remark is that it is possible to prove the above proposition without resorting to the Fierz identity, by exploiting the representation theory of the spin group. The group Spin(V ) sits inside the Clifford algebra Cℓ(V ) and hence S becomes a Spin(V )-module by restriction. The volume element defines a complex structure on S which is invariant under the spin group. The identity component of the spin group is isomorphic to SL(2, C) under which S is the fundamental 2-dimensional complex representation. The orbit structure of S under Spin(V ) is therefore very simple; namely, there are two orbits: a degenerate orbit consisting of the zero spinor and an open orbit consisting of all the nonzero spinors. The stabiliser of a nonzero spinor s is the abelian subgroup H s consisting of the null rotations in the direction of its Dirac current κ, and it is a subgroup of the stabiliser of any object we can construct from s in a Spin(V )-equivariant fashion: e.g., the Dirac current and the Dirac 2-form. Now the H s -invariant 2-forms can be seen to be of the form κ ∧ θ, for some "transverse" 1-form θ, and hence the Dirac 2-form ω (2) has this form. By equivariance under H s < Spin(V ), the Clifford product of ω (2) on s must be again proportional to s, but by squaring we see that the constant of proportionality must be zero. Finally, Clifford multiplication by the spacelike θ is invertible, so it must be that κ Clifford-annihilates s.

A.4 A further property of the Dirac current
For completeness we discuss a further algebraic properties of the Dirac current. Recall that if s ∈ S, its Dirac current κ is defined by equation (A.13). Let us define a symmetric bilinear map κ : S ⊗ S → V by κ(s 1 , s 2 ) = 1 2 (κ s 1 +s 2 − κ s 1 − κ s 2 ) , (A. 31) where κ s denotes the Dirac current of s. It follows from the representation theory of so(V ) that the map κ is surjective onto V . Now consider a linear subspace S ′ ⊂ S and let V ′ ⊂ V denote the image of the map κ restricted to S ′ ⊗ S ′ . For which S ′ do we still have that V ′ = V ? The following lemma, which is a modification of the similar result in [35] for eleven dimensions, shows that this holds provided dim S ′ > 2.