Killing superalgebras for Lorentzian fourmanifolds
Abstract.
We determine the Killing superalgebras underpinning field theories with rigid unextended supersymmetry on Lorentzian fourmanifolds by reinterpreting them as filtered deformations of graded subalgebras with maximum odd dimension of the 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 offshell 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 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.
Contents
 1 Introduction
 2 Spencer cohomology
 3 Killing superalgebras
 4 Zero curvature equations
 5 Maximally supersymmetric filtered deformations
 6 Conclusions
 A Conventions and spinorial algebraic identities
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 nontrivial rôle played by certain nonminimal 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 antide 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 nonlinear sigmamodels 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 offshell 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 wellestablished 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, §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 oddodd 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 nonlinear 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 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 Rsymmetry) 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 , following a similar analysis on pioneered in [39, 40] which yielded what might be considered a Liealgebraic derivation of elevendimensional supergravity.
Let us describe more precisely the problem we set out to solve. Let denote the Lorentzian vector space on which fourdimensional Minkowski space is modelled, the Lie algebra of the Lorentz group and its spinor representation. The associated Poincaré superalgebra has underlying vector space and Lie brackets, for all , and , given by
(1) 
where is the spinor representation of and is such that is the Dirac current of . (This and other relevant notions are defined in the Appendix.) The Poincaré superalgebra is graded by assigning degrees , and to , and , respectively and the grading is compatible with the grading, in that the parity is the degree mod . More precisely, the even subalgebra is the Poincaré algebra and the odd subspace is . By a graded subalgebra of we mean a Lie subalgebra , with .
Now recall that a Lie superalgebra is said to be filtered, if it is admits a vector space filtration
with and , which is compatible with the Lie bracket in that . Associated canonically to every filtered Lie superalgebra there is a graded Lie superalgebra , where . It follows from the fact that is filtered that , hence is graded.
We say that a Lie superalgebra is a filtered deformation of if it is filtered and its associated graded superalgebra is isomorphic (as a graded Lie superalgebra) to . If we do not wish to mention the subalgebra explicitly, we simply say that is a filtered subdeformation of .
The problem we address in this note is the classification of filtered subdeformations of for which (and hence ).
This paper is organised as follows. In Section 2 we define and calculate the Spencer cohomology group 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 offshell) 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 representationtheoretic proof exploiting the equivariance under . In Section 3 we prove that the (minimal offshell) supergravity Killing spinors generate a Lie superalgebra, and that this Lie superalgebra is a filtered subdeformation of . 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 , we find the Lie groups admitting a Lorentzian biinvariant metric. In Section 5 we finish the determination of maximally supersymmetric filtered subdeformations of 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 biinvariant metric, we note that the Killing ideal is a filtered deformation of and also explicitly describe all other associated maximally supersymmetric filtered subdeformations of . 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 , our classification of Killing superalgebras for maximally supersymmetric backgrounds yields, up to local isometry, the following conformally flat Lorentzian geometries:

;

, with identified with with its biinvariant metric;

, with identified with with its biinvariant metric; and

, a symmetric plane wave isometric to the NappiWitten group with its biinvariant metric.
We prove that the geometries above are indeed realised as the maximally supersymmetric backgrounds of minimal offshell supergravity in four dimensions, in Lorentzian signature. That is, we do not assume the form of the supergravity Killing spinor equation from the 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 offshell supergravity in four dimensions has been discussed in various other contexts in the recent literature, e.g., see [18, §2.1], [41, §§4.23], [34], [42, p. 2], [43, pp.1213]. The 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 supergravity backgrounds. It is also worth pointing out that [18, §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 offshell supergravity backgrounds.
2. 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.
2.1. Preliminaries
Let , where , and , be the Poincaré superalgebra and the negatively graded part of . We will now determine some Spencer cohomology groups associated to . We recall that the cochains of the Spencer complex of are linear maps or, equivalently, elements of , where is meant here in the super sense, and that the degree in is extended to the space of cochains by declaring that has degree . The spaces in the complexes of even cochains of small degree are given in Table 1, although for there are cochains also for which we omit.
deg  
0 


2 



4 


Let be the space of cochains of degree . The Spencer differential
is the Chevalley–Eilenberg differential for the Lie superalgebra relative to its module with respect to the adjoint action. For and it is explicitly given by the following expressions:
(2)  
(3)  
(4) 
where are the parity of elements of and with respectively.
In this section we shall be interested in the groups with and even. We first recall some basic definitions. A graded Lie superalgebra with negatively graded part is called fundamental if is generated by and transitive if for any with the condition implies .
Lemma 1.
The Poincaré superalgebra is fundamental and transitive. Moreover for all even .
Proof.
The first claim is a direct consequence of the fact that and that the natural action of on is faithful. For any one has
where and . The first equation implies , since is fundamental, and therefore . Finally and for degree reasons, for all even . ∎
Note that the space of cochains is an module and the same is true for the spaces of cocycles and coboundaries, as is equivariant. This implies that each cohomology group is an module, in a natural way. It remains to compute
and, in particular, to describe its module structure. We consider the decomposition
into the direct sum of submodules and write any accordingly; i.e., with
We denote the associated equivariant projections by
(5) 
Lemma 2.
The component of the Spencer differential is an isomorphism. In particular, , where is the kernel of acting on , and every cohomology class has a unique cocycle representative with .
Proof.
The image of under is given by
where and the first claim of the lemma follows from classical arguments (see [44]; see also e.g., [45, 39]).
Now for any given , there is a unique such that , for some and . Hence, given any cocycle , we may add the coboundary without changing its cohomology class and resulting in the cocycle , which has no component in . This proves the last claim of the lemma. The decomposition is clear. ∎
2.2. The cohomology group
Lemma 2 gives a canonical identification of modules. Furthermore it follows from equation (4) that is an element of if and only if the following pair of equations are satisfied:
(6) 
and
(7) 
Note that (6) fully expresses in terms of , once the integrability condition that takes values in has been taken into account. The solution of the integrability condition and of equation (7) is the content of the following
Proposition 3.
Let . Then if and only if there exist and such that

[label=()]

,

,
for all and . In particular there is a canonical identification
of modules.
Proof.
We find it convenient to work relative to an orthonormal basis for . In particular the formalism of Section A.2.1 is in force, as is the Einstein summation convention.
Let us contract the cocycle condition (6) with . The lefthand side becomes
(8) 
whereas the righthand side becomes
(9) 
where we have introduced . In summary, the first cocycle condition becomes
(10) 
which must hold for all , so that they can be abstracted to arrive at
(11) 
Symmetrising we obtain the “integrability condition”
(12) 
whereas skewsymmetrising and using that , we arrive at
(13) 
Notice that, as advertised, this last equation simply expresses in terms of . Acting on ,
(14) 
and inserting this equation into the second cocycle condition (7), we arrive at
(15) 
Since (where the first isomorphism is one of algebras and the second one of vector spaces), we may write
(16) 
with so that
(17) 
where we have used the last of the duality equations (117) and the symmetry relations (109).
Inserting this into equation (12), which must be true for all , we get that the terms which depend on and must vanish separately and we arrive at two equations:
(18) 
and
(19) 
Tracing this last equation with , we learn that
(20) 
whereas tracing (19) with and using (20), results in
(21) 
for .
Substituting the expressions above back into equation (19), we find
(22) 
Multiplying by , and using the identities (118), we obtain
(23) 
Tracing the expression above with , we arrive at
(24) 
for .
Tracing equation (18) with gives
(25) 
while tracing it with gives
(26) 
These two equations together imply
(27) 
which, when inserted into equation (18), yields
(28) 
This implies (i.e., ), so that it can be parametrised by such that
(29) 
In summary, the general solution of equation (12) is
(30) 
where we have used the the last of the identities (117).
Next we solve the second cocycle condition (15). Using the expression for given in equation (30), we can rewrite the first term of equation (15) as follows:
(31) 
where, using that the Dirac current of Clifford annihilates (see Proposition 15), the first term vanishes. Similarly, using and again the fact that , the first term in equation (15) becomes
(32) 
We now rewrite the second term in equation (15) by inserting the expression for in equation (30) into equation (17) to obtain
(33) 
where we have again used and the fact that . The first term on the righthand side vanishes by virtue of the fact that the Dirac form of and its dual both Clifford annihilate (see Proposition 15). In summary, equation (15) becomes
(34) 
for all , whose general solution is
(35) 
Inserting this into equation (30), we arrive at
which can be rewritten as
from where the result follows. ∎
Alternative proof.
It may benefit some readers to see an alternative proof of this result, which exploits the equivariance under .
Let us consider the first cocycle condition (6). Given and any we let to be defined by and rewrite (6) as . Taking the inner product with and using (111) and (107) we arrive at
(36) 
for all , . In other words, for all , the endomorphism of is in or, equivalently, it is fixed by the antiinvolution defined by the symplectic form on . We claim that the solution space of equation (36) is an submodule of . To see this, it is convenient to consider the equivariant map
which sends to given by
for all . We consider also the natural decompositions into submodules
(37) 
which are induced by the usual identification . This allows us to write any elements and as and , where and . The claim then follows from the fact that equation (36) is equivalent to for .
In Table 2 below we list the decomposition of for into irreducible modules, with denoting the kernel of Clifford multiplication .
p  

From the first decomposition in (37) we immediately infer that is the direct sum of five different isotypical components, namely
(38) 
and
(39) 
Note now that for any the element defined by
satisfies
and it is therefore a solution of (36). If instead a similar computation yields . In summary we get that the solution space of equation (36) contains an module isomorphic to
(40) 
where, say, , and that there exists another submodule which is isomorphic to
(41) 
and formed by elements which do not satisfy (36). Note that the direct sum of (40) and (41) gives all the isotypical components (38) in .
We now turn to the remaining isotypical components (39). We first recall that contains a single irreducible submodule of type . We fix an orthonormal basis of , consider the element
and evaluate
In other words , which implies that is not included in the solution space of equation (36). Finally any irreducible submodule in isomorphic to is given by the image into of an equivariant embedding , , where . For instance the image of is
and we have