Eric A Bergshoeff
email: e.a.bergshoeff@rug.nl

Olaf Hohm
email: ohohm@mit.edu

Jan Rosseel
email: j.rosseel@rug.nl

Paul K Townsend
email: p.k.townsend@damtp.cam.ac.uk

On Maximal Massive 3D Supergravity

We construct, at the linearized level, the three-dimensional (3D) N = 4 supersymmetric "general massive supergravity" and the maximally supersymmetric N = 8 "new massive supergravity". We also construct the maximally supersymmetric linearized N = 7 topologically massive supergravity, although we expect N = 6 to be maximal at the non-linear level.

Introduction

The "on-shell" N -extended one-particle supermultiplets available for massless particles of four-dimensional (4D) field theories are well-known. Generically, these supermultiplets will appear in CPT-dual pairs but there are some special supermultiplets that are CPT self-dual; these are unique for a given choice of maximal spin, and have the property that N is maximal for that spin. The corresponding field theories generally have improved ultra-violet (UV) behaviour. For example, N = 4 is maximal for maximum spin 1, and there is a unique N = 4 self-dual supermultiplet, which is realized by the UV-finite N = 4 super-Yang-Mills (SYM) theory. Similarly, N = 8 is maximal for maximum spin 2, and there is a unique N = 8 self-dual supermultiplet, which is realized by N = 8 supergravity; the UV status of this non-renormalizable theory is still a matter of dispute but it is certainly "improved"; see [START_REF] Dixon | Ultraviolet Behavior of N=8 Supergravity[END_REF] for a recent review.

This paper is motivated by the observation that there is a one-to-one correspondence between the massless on-shell supermultiplets of N -extended 4D supersymmetry and the massive on-shell supermultiplets of N -extended three-dimensional (3D) supersymmetry. The N 3D Majorana spinor supercharges Q i , i = 1, • • • , N , obey the anticommutation relations

{Q i α , Q j β } = 2(γ µ C) αβ P µ δ ij , (1.1)
where P µ is the three-momentum, γ µ are the 3D Dirac matrices and C is the charge conjugation matrix. Choosing a real representation of the Dirac matrices with C = γ 0 and choosing the rest frame three-momentum P µ = (-M, 0, 0) for a particle of mass M, one finds that the following combinations

(a i ) † = 1 2 Q i 1 + iQ i 2 , a i = 1 2 Q i 1 -iQ i 2 , (1.2)
obey the anticommutation relations {a i , (a j ) † } = Mδ ij , {a i , a j } = {(a i ) † , (a j ) † } = 0 .

(1.

3)

The operators a and a † can thus be seen as raising and lowering operators. Moreover, it can be checked that they increase or decrease the space-time helicity by 1/2 (see e.g. [START_REF] Hohm | Spontaneous N = 2 → N = 1 supergravity breaking in three dimensions[END_REF]). The construction of the supermultiplets is thus straightforward. Starting from a 'Clifford vacuum' |Ω j with helicity j one can act N times with the raising operators a † . This leads to a massive multiplet with 2 N states of helicities ranging from j to j + N /2, as shown in Table 1 for j + N /2 = 2. Only the multiplets containing the +2 helicities are shown for N = 1, • • • , 7. Parity flips the helicities and hence takes these into multiplets containing a state of helicity -2 but not +2. The N = 8 multiplet is exceptional because it contains both helicities +2 and -2 and is therefore "parity self-dual". One sees that this is formally the same construction as massless particle supermultiplets of four-dimensional N -extended supersymmetry 1 .

Table 1: Some 3D massive supermultiplets helicity +2 +3/2 +1 +1/2 0 -1/2 -1 -3/2 -2 This 4D/3D correspondence holds for each value of N although it should be appreciated that the 3D theory has half the total number of supersymmetries (because the minimal spinor in 3D has half as many independent components as the minimal spinor in 4D). It should also be appreciated that the one-to-one correspondence is for supermultiplets, and not free field theories2 ; this is because a CPT-dual pair in 4D corresponds to a pair of 3D massive supermultiplets paired by parity, but locality in 3D does not require this pairing. This means that there can be parity-violating massive 3D field theories that have no 4D analog; the simplest (N = 0) examples are "topologically massive electrodynamics" (TME) [START_REF] Siegel | Unextended Superfields In Extended Supersymmetry[END_REF][START_REF] Schonfeld | A Mass Term For Three-Dimensional Gauge Fields[END_REF][START_REF] Deser | Topologically massive gauge theories[END_REF] and "topologically massive gravity" (TMG) [START_REF] Deser | Topologically massive gauge theories[END_REF][START_REF] Deser | Three-Dimensional Massive Gauge Theories[END_REF].

N = 1 1 1 N = 2
Thus we can expect to find (at least at the linearized level) parity-preserving massive 3D SYM and massive 3D supergravity theories for each of the corresponding massless 4D SYM and supergravity theories. This expectation is realized in the SYM case by the N -extended 3D SYM-Higgs theories, expanded about a Higgs vacuum in which all particles are massive. This yields 3D theories of massive spin-1 particles with N = 1, 2, 4 supersymmetry. Parity may then be broken by the addition of a supersymmetric Chern-Simons (CS) term, but such a term exists only for N = 1, 2, 3, so N = 3 is maximal for spin 1 if parity is violated [START_REF] Kao | Selfdual Chern-Simons systems with an N=3 extended supersymmetry[END_REF]. This feature can be understood directly from the supermultiplet structure: the N = 4 spin-1 supermultiplet is equivalent to a parity-dual pair of N = 3 spin-1 supermultiplets, so any parity violation that is visible in the linearized theory will split the degeneracy of this pair, thereby breaking N = 4 to N = 3.

In the supergravity case, the 4D/3D analogy leads us to expect parity-preserving 3D supergravity theories propagating massive graviton supermultiplets with N = 1, 2, 3, 4, 5, 6, 8. Until recently, it was far from obvious how such theories could be constructed, but the N = 0 example of "New Massive Gravity" (NMG) [START_REF] Bergshoeff | On Higher Derivatives in 3D Gravity and Higher Spin Gauge Theories[END_REF] has shown the way, because it propagates precisely massive helicity ±2 modes. Furthermore, one can add a Lorentz Chern-Simons (LCS) term to the NMG action to obtain a parityviolating "General Massive Gravity" (GMG) that has both NMG and TMG as limiting cases [START_REF] Bergshoeff | On Higher Derivatives in 3D Gravity and Higher Spin Gauge Theories[END_REF]. The N = 1 supersymmetric extension of these 3D gravity theories has now been constructed [START_REF] Andringa | Massive 3D Supergravity[END_REF][START_REF] Bergshoeff | More on Massive 3D Supergravity[END_REF] as has the linearized N = 2 extension, which nicely combines NMG with the Proca action for spin 1, and combines both CS and LCS terms in a single N = 2 superinvariant [START_REF] Andringa | Massive 3D Supergravity[END_REF]. The main purpose of this paper is to present results for the N > 2 massive supergravity theories. 3We shall work exclusively at the linearized level, leaving the problem of interactions to future work. Even so, there are a number of issues that we are able to address, and resolve. Firstly, we recall that NMG consists of the Einstein-Hilbert action, with the "wrong" sign, and a curvature-squared invariant constructed from the scalar K = G µν S µν , where G µν is the Einstein tensor and S µν the Schouten tensor. We therefore need an N = 8 extension of the Einstein-Hilbert action (the lower N cases can then be obtained by truncation). This involves coupling the N = 8 offshell Weyl supermultiplet [START_REF] Howe | New supergravities with central charges and Killing spinors in (2+1)-dimensions[END_REF] (which contains the graviton field subject to linearized diffeomorphisms and linearized Weyl rescalings) to 8 compensating N = 8 scalar supermultiplets (containing only scalar and spinor fields). Each off-shell N = 8 scalar supermultiplet has an infinite number of auxiliary fields (arising from the expansion of a field defined on N = 8 harmonic superspace [START_REF] Howe | On harmonic superspace[END_REF]). These should be eliminated only after the addition of an N = 8 extension of the NMG K-invariant, but the compensating supermultiplets decouple from the K-superinvariant at the linearized level as a consequence of an "accidental" linearized superconformal invariance; this allows us to trivially eliminate all auxiliary fields of each compensating supermultiplet and work with the simpler on-shell-supersymmetric4 N = 8 scalar supermultiplet.

Starting with the N = 8 super-NMG model, we might expect to be able to construct a parity-violating N = 8 super-GMG model by the addition of an N = 8 super-LCS term. Let us recall that the LCS term is, by itself, the action of 3D conformal gravity [START_REF] Van Nieuwenhuizen | D = 3 Conformal Supergravity And Chern-Simons Terms[END_REF] and that there exists an N = 8 superconformal gravity [START_REF] Gran | Three-dimensional N=8 superconformal gravity and its coupling to BLG M2-branes[END_REF]. However, the N = 8 spin-2 supermultiplet is equivalent to a degenerate parity-dual pair of N = 7 spin-2 supermultiplets, and this degeneracy is lifted if parity is violated. It follows that, at best, N = 7 is maximal for super-GMG and also for its TMG limit. In fact, for reasons that we will give later, we believe that N = 6 is actually maximal 5 for the non-linear TMG and generic GMG. However, we verify that there is an N = 7 linearized super-GMG theory by constructing the linearized off-shell N = 7 super-LCS invariant; this can be added to the N = 8 super-NMG action because it does not couple to the compensating supermultiplet. We also explain why the N = 8 super-LCS invariant cannot be similarly used to construct an N = 8 super-GMG theory; in brief, it is because there is no off-shell-supersymmetric N = 8 super-LCS term.

The method of construction of the linearized N = 7, 8 massive gravities that we have just sketched, can be applied for any N . As a preliminary, we begin with a sketch of how the construction works for each value of N . This involves a determination of the off-shell Weyl supermultiplet and the compensating multiplets needed for the construction of the linearized N -extension of the Einstein-Hilbert term. For N ≤ 3, not only does a single scalar supermultiplet suffice but there is also an SO(N ) singlet in this multiplet than can be identified with the compensating scalar for the local scale invariance of the Weyl multiplet. The situation for N = 4 is quite different. In this case, the scalar multiplet is the 3D version of the 4D hypermultiplet, but there are two distinct versions of it in 3D, and both are needed. Furthermore, the local scale compensator is necessarily composite because neither hypermultiplet contains a singlet of the R-symmetry group. In several respects, the N = 4 case is similar to the N = 8 case, but much simpler, so we discuss N = 4 in detail. This serves to illustrate features that we take over to N = 8, thus obtaining the linearized N = 8 super-NMG theory. Finally, we construct the off-shell N = 7 LCS invariant and hence the maximally-supersymmetry parity-violating super-GMG and super-TMG theories, again at the linearized level. We conclude with a discussion of open problems.

N -extended Weyl and Poincaré supermultiplets

To construct the N -extended supersymmetric LCS term, one needs only the fields of an N -extended "Weyl supermultiplet". At the linearized level, this multiplet contains the metric perturbation h µν and N Majorana anti-commuting vector-spinors ψ i µ (i = 1, . . . , N ) subject to the linearized transformations

δh µν = ∂ (µ v ν) + η µν ω , δψ i µ = ∂ µ η Q + γ µ S , (2.1)
for Minkowski 3-vector v, scalar ω and the anticommuting Majorana spinor parameters of Qand S-supersymmetry. At the linearized level we must distinguish between the Qsupersymmetry gauge invariances and the rigid supersymmetry that relates the various fields of the multiplet; it is only after the inclusion of interactions that these combine to become N local supersymmetries. The linearized Weyl multiplet also includes N (N -1)/2 abelian gauge fields V ij µ in the adjoint irrep of the Spin(N ) R-symmetry group, and subject to the gauge transformation

δV ij µ = ∂ µ Λ ij . (2.2)
For N ≥ 3 there are additional fields in the Weyl multiplet, and for N ≥ 6 these include additional gauge fields, although N = 8 is exceptional in this respect, as we shall see.

To construct the N -extended supersymmetric EH term, or N -extended "Einstein supergravity", we need more than just the Weyl supermultiplet. We need a Poincaré supermultiplet in which the fields are subject only to the standard supergravity gauge transformations and not the additional ones of conformal supergravity. We therefore need to introduce additional degrees of freedom. One way to do this is as follows [START_REF] Siegel | A Polynomial Action For A Massive, Selfinteracting Chiral Superfield Coupled To Supergravity[END_REF][START_REF] Kaku | Poincare Supergravity As Broken Superconformal Gravity[END_REF]. We start from a standard flat space action invariant under rigid superconformal transformations, for a supermultiplet (or supermultiplets) containing physical scalar and spinor fields (and possibly vector or antisymmetric tensor gauge fields); if the field supermultiplets used to construct this action are off-shell supersymmetric then there will generically be auxiliary fields too but their inclusion is optional. This action is then coupled to an off-shell N -extended Weyl supermultiplet (e.g. by the Noether procedure) to give an action that is invariant under local superconformal transformations. Then one fixes the 'unwanted' superconformal symmetries by imposing conditions on the physical scalar and spinor fields of the multiplet(s). To do so, one needs sufficient multiplets so as to have a sufficient number of physical scalar and spinor fields, and with luck there will be none left over. In this case, the conformal coupling to the scalar fields will produce the Einstein-Hilbert term on gauge fixing and the rest of the action will be just what is required for the N -extended supersymmetrization of this term. One says that the original 'scalar' supermultiplets are 'compensating multiplets'. The auxiliary fields of these compensating multiplets become auxiliary fields of the final N -extended Einstein supergravity action.

If we wish to construct generic higher-derivative actions in the way just described then the auxiliary fields of the compensating multiplets must be included because the starting flat space action for the conformal compensator multiplets must be a higherderivative one in which the 'auxiliary' fields also propagate. The LCS term is an exception to this rule because it is superconformally invariant. In contrast, the 4th order curvature-squared term of NMG is not conformally invariant, so one really needs a full superconformal tensor calculus to construct N -extended NMG. Some aspects of this calculus have been worked out for N = 1, 2 [START_REF] Uematsu | Structure Of N=1 Conformal And Poincare Supergravity In (1+1)-Dimensions And (2+1)-Dimensions[END_REF] but, even so, special 'tricks' were needed for the construction of the N = 1 supersymmetric super-NMG model [START_REF] Andringa | Massive 3D Supergravity[END_REF][START_REF] Bergshoeff | More on Massive 3D Supergravity[END_REF] and only linearized results have been found for N = 2 [START_REF] Andringa | Massive 3D Supergravity[END_REF].

However, although we need an N -extended superconformal tensor calculus to construct the full N -extended NMG action, we do not need it to construct the quadratic approximation to this action. This is because the NMG 4th order invariant has the property (shared with an infinite series of yet higher-order invariants) that its quadratic approximation is invariant under the linearized super-Weyl gauge invariances, so compensating multiplets are not needed. Moreover, this quadratic approximation to the 4th order NMG invariant may be added to the quadratic approximation to the Nextended Einstein supergravity to give the quadratic approximation to N -extended NMG. The upshot is that to construct the N -extension of linearized NMG, and more generally the linearized GMG, we do not need a full superconformal tensor calculus. In particular, we need conformal compensator supermultiplets only for the construction of the N -extended Einstein supergravity, and as we need that only to quadratic level, it is sufficient to consider the cubic interaction of the compensator supermultiplets to the Weyl multiplet and a few other quartic interactions that contribute to the quadratic action after fixing the superconformal gauge.

In what follows we will consider sequentially the cases from N = 2, . . . , 8, presenting some details of the Weyl multiplet and determining the type and number of compensating multiplets that are needed. For the convenience of the reader we have summarized some details about this in Table 2. The N = 2 case has already been dealt with in some detail [START_REF] Andringa | Massive 3D Supergravity[END_REF][START_REF] Howe | New supergravities with central charges and Killing spinors in (2+1)-dimensions[END_REF] but it will serve to illustrate the issues involved.

N = 2

The off-shell linearized Weyl multiplet has the field content

(h µν ; ψ i µ ; V µ ) , i = 1, 2. (2.3)
We use semicolons to separate fields of different mass dimensions, which increase by steps of 1/2. The vector field is the SO(2) gauge field. Taking all gauge invariances into account, we are left with 4 + 4 remaining off-shell field components.

To construct a Poincaré supermultiplet suitable for the construction of massive supergravities, we need to add additional degrees of freedom that will allow us to fix the scale and S-supersymmetry transformations, at least, but we may also fix the Rsymmetry gauge invariance. For example, consider the N = 2 scalar multiplet with 4 + 4 off-shell field content (ϕ, ς; λ i ; S, P ). After coupling to the Weyl multiplet, we may fix the scale and SO(2) gauge invariances by setting ϕ = 0 and ς = 0, while the S-supersymmetry gauge invariances may be fixed by setting λ i = 0. This leaves the auxiliary fields S and P , which survive as auxiliary fields of the off-shell Poincaré supermultiplet, which has the 8+8 off-shell field content (h µν ; ψ i µ ; V µ , S, P ), now subject only to diffeomorphisms and local Q-supersymmetry gauge transformations. This multiplet was called the (1, 1) Poincaré supermultiplet in [START_REF] Howe | New supergravities with central charges and Killing spinors in (2+1)-dimensions[END_REF] in order to distinguish it from the (2, 0) Poincaré supermultiplet found by taking the compensator fields to belong to the 4 + 4 off-shell vector multiplet, which has the field content (ϕ, A µ ; λ i ; D). In this case, we use ϕ and λ i to compensate for the scale and S-supersymmetry transformations, as before, but we leave uncompensated the local SO(2) gauge invariance. The linearized Poincaré supermultiplet now has the 8 + 8 off-shell field content (h µν ; A µ ; ψ i µ ; V µ ; D). In the construction of the supersymmetric Einstein-Hilbert term the (2, 0) Poincaré supermultiplet leads to a V dA Chern-Simons type term [START_REF] Howe | New supergravities with central charges and Killing spinors in (2+1)-dimensions[END_REF], and a D 2 term for the auxiliary field D.

Variant choices of conformal compensator multiplets therefore lead to variant versions of the supersymmetric EH action; in this case the (1, 1) or (2, 0) versions. Whatever the choice, one can add to this action any off-shell supersymmetric N = 2 superconformal action, such as LCS. Also, at the linearized level, we can add the quadratic approximation to the N = 2 supersymmetric 4th order invariant of NMG, thereby constructing the N = 2 extension of linearized GMG. This was done for the (1, 1) case in [START_REF] Andringa | Massive 3D Supergravity[END_REF]. The generic GMG model propagates one multiplet of helicities (2, 3/2, 1) with mass m + and another multiplet of helicities (-1, -3/2, -2) with mass m -. The same computations may be done for the (2, 0) case, but with more difficulty because the zero Weyl weight of the vector field A means that the starting flat space action cannot be quadratic. Nevertheless, it can be done and the final spectrum is the same but the spin 1 modes are propagated differently. In the (1, 1) case the action for the spin 1 modes is the Proca action for V with a CS term6 . In the (2, 0) case one finds that the action for the spin 1 modes has the Lagrangian

L 1 = 1 2 F 2 + A µ Gµ + 1 2µ V µ Gµ + 1 2m 2 G2 (2.4)
where

F µ = ε µνρ ∂ ν A ρ , Gµ = ε µνρ ∂ ν V ρ . (2.5)
One may verify that this propagates two spin 1 modes of helicities ±1 and masses m ± given by

m 2 = m + m - µ = m 2 /(m --m + ) , (2.6)
exactly as for the (1, 1) case.

N = 3

The off-shell N = 3 linearized Weyl multiplet has 8 + 8 components and the field content (h µν ; ψ i µ ; V i µ ; χ) , i = 1, 2, 3.

(2.7)

The minimal N = 3 scalar multiplet has the physical (i.e. non-auxiliary) field content (ϕ, ϕ i ; λ, λ i ); i.e. both scalars and spinors are in the 1 ⊕ 3 representation of the SO(3) R-symmetry group. There is an off-shell version of this multiplet with a finite number of auxiliary fields (a 3D version of the relaxed hypermultiplet [START_REF] Howe | The Relaxed Hypermultiplet: An Unconstrained N=2 Superfield Theory[END_REF]) but it is not needed for present purposes. The 1 ⊕ 3 scalars are precisely those needed to compensate for the one scale and three SO(3) gauge invariances. The fermion triplet λ i compensates for the three S-supersymmetries. This leaves the singlet spinor λ. In the context of the N = 3 supersymmetrization of the Einstein-Hilbert action, the pair of spinors (χ, λ) is auxiliary. As the (non-gauge) vectors V ij µ are also auxiliary in this context, only h µν and ψ i µ remain as the 'physical' fields, which actually do not propagate modes. In the context of N = 3 super-TMG there is a χχ term associated with the Lorentz-Chern-Simons term and this means that χ is no longer a Lagrange multiplier for the constraint λ = 0. The combined effect of the 'χλ' terms is to propagate one spin-1/2 mode. Thus, the physical field content in the context of TMG is (h µν ; ψ i µ , λ; V i µ ; χ) subject only to linearized diffeomorphism and local Q-supersymmetry transformations. Note that the effect of the compensating multiplet is no longer merely to reduce the gauge invariances of the Weyl multiplet fields, as is the case for N = 2, but also to provide an additional spinor field. The physical field content for NMG is the same as for TMG but we expect the higher-order interactions to lead to a second massive particle supermultiplet of opposite helicities in exactly the way spelled out for N = 2 in [START_REF] Andringa | Massive 3D Supergravity[END_REF].

As for N = 2, we could consider how things change if we use vector multiplets in place of scalar multiplets, but this is already much more complicated for N = 3 so we shall henceforth restrict the discussion to scalar supermultiplets.

N = 4

The off-shell N = 4 Weyl multiplet, with 16+16 components, has the field content

(h µν ; ψ i µ ; V ij µ , E; χ i ; D) , i = 1, 2, 3, 4. (2.8)
The R-symmetry group is now reducible: Spin(4) ∼ = SU(2)×SU(2). As a consequence, the vector fields may be written as

V ij = V ij + + V ij -, 1 2 ε ijkl V kl ± = ±V ij ± . (2.9)
This sum of self-dual and anti-self-dual terms corresponds to the direct sum representation (3, 1) ⊕ (1, 3). This sum is of course unchanged under a switch in the duality assignments of V ± , which amounts to an exchange of the two SU (2) factors of Spin( 4) by an outer Z 2 automorphism. However, the scalars E, D and the spinors χ i are odd under this Z 2 exchange7 . The minimal scalar N = 4 supermultiplet is the 3D analog of the 4D, N = 2 hypermultiplet with 4 + 4 physical degrees of freedom. However, in 3D there are two distinct versions of the hypermultiplet. The hypermultiplet scalars transform as a complex doublet of one or the other of the SU(2) factors of the R-symmetry group. Whichever SU(2) factor we choose, the spinor fields will transform as a complex doublet of the other SU(2) factor. In other words, we have a hypermultiplet with physical fields (ϕ α , λ α) or one with physical fields (ϕ α, λ α ), where α = 1, 2 and α = 1, 2 are spinor indices for the two SU(2) factors of Spin(4). The SU(2) currents that may be constructed from these scalars will be either self-dual or anti-self-dual, according to which of the two versions of the hypermultiplet we choose. Without loss of generality, we may assume that the scalar fields ϕ α yield a self-dual current J ij + and that the scalar fields ϕ α yield an anti-self-dual current current J ij -. Both types of hypermultiplet are needed for conformal compensation, because we have to compensate for both SU(2) gauge invariances. Another difference from the N = 3 case is that any R-symmetry singlet constructed from the scalar fields of the compensating multiplets must be at least quadratic in the scalar fields. The scalar bilinears

ϕ 2 + ≡ ϕ α ϕ α + ϕ αϕ α , ϕ 2 -≡ ϕ α ϕ α -ϕ αϕ α (2.10)
are respectively even and odd under the Z 2 exchange. The former acts as a compensator for the dilatations. After fixing the dilatation gauge by imposing ϕ 2 + = 1, one finds that the Spin(4) currents take the form

J ij µ ± = ∂ µ ϕ ij ± + . . . , (2.11)
where ϕ ij ± are, collectively, Spin(4) Stueckelberg scalars that may be used to compensate for the local Spin(4) gauge invariance.

In contrast to N = 3, the conformal compensation mechanism is closer to the nonlinear Higgs mechanism than it is to the linear Stueckelberg mechanism, because ϕ 2 can be identified as the field of a residual Higgs boson that survives the 'spontaneous' breaking of symmetries implied by the conformal gauge condition ϕ 2 + = 1. However, this Higgs boson field is set to zero by the D field equation of the Weyl multiplet. The details of the construction will be discussed in the next section.

N = 5

The off-shell linearized Weyl multiplet for N = 5 has 32 + 32 components and the field content (h µν ;

ψ i µ ; V ij µ , E i ; χ ij , χ; D i ) , i = 1, 2, 3, 4, 5.
(2.12)

Fields with multiple indices are antisymmetric in these indices. Thus, the vectors are in the adjoint 10 irrep of Spin(5) ∼ = Sp 2 , as are the spinors χ ij . The minimal scalar supermultiplet now has 8 + 8 physical fields, in the complex 4 representation of Sp 2 . Let (ϕ α , λ α ) (α = 1, 2, 3, 4) be these physical fields. As for N = 4, we need two scalar supermultiplets, ϕ 1 and ϕ 2 . The 16 scalar fields in these two multiplets may be traded for scalar bilinears in the 1 ⊕ 5 representations of Sp 2 , and currents in the adjoint 10 of Sp 2 . The singlet bilinear is Ω αβ ϕ α 1 ϕ β 2 , which we set to unity to fix the local scale invariance. The currents then become derivatives of a 10 of Stueckelberg scalars, to linear order, which may be set to zero to compensate for the local Sp 2 invariance. The 5-plet of scalar bilinears is set to zero by the D i field equation.

N = 6

The off-shell linearized Weyl multiplet now has 64 + 64 components, and the field content (h µν ;

ψ i µ ; V ij µ , V µ , E ij ; χ ijk , χ i ; D ij ) , i = 1, 2, 3, 4, 5, 6. (2.13)
Fields with multiple indices are again antisymmetric in these indices. The R-symmetry group is Spin(6) ∼ = SU(4), but the gauge symmetry is enhanced to U(4), because of the presence of the additional gauge field V µ .

The minimal scalar multiplet again has 8+8 physical fields (ϕ α , χ α ) in the (complex) 4 of SU [START_REF] Siegel | Unextended Superfields In Extended Supersymmetry[END_REF]. We can choose a gauge given by the sum of squares of all the scalars equal to one to fix the dilatations. There will be 15 additional constraints coming from the D-field equation. The U(4) currents then become to linearized order the derivatives of 16 Stueckelberg scalars, which we set to zero to fix the U(4) invariance. We therefore need a total of 32 scalar fields and hence four scalar multiplets.

N = 7

The off-shell linearized Weyl multiplet has 128 + 128 components, and the field content Fields with multiple indices are again antisymmetric in these indices. The R-symmetry group is Spin(7) but the presence of 7 additional vector fields implies an enhancement of the gauge symmetry. We also have an additional Rarita-Schwinger field implying an enhanced local supersymmetry at the non-linear level, but we avoid the associated difficulties here by restricting to the linear theory.

(h µν ; ψ i µ , ψ µ ; V ij µ , V i µ , E ijk ; χ ijk , χ ij ; D ijk ) , i= 1
The minimal scalar multiplet has 8 + 8 components in the spinor representation of Spin [START_REF] Deser | Three-Dimensional Massive Gauge Theories[END_REF]. The D-fields now impose 35 constraints on scalar bilinears. In addition, we need 1 + 21 + 7 = 29 scalars to compensate for the scale and gauge invariances. In total we need 64 scalars and hence eight compensating scalar multiplets.

N = 8

The off-shell Weyl multiplet, again with 128+128 components, has the field content 8) and vice versa for the other version. These are the only possibilities consistent with the supersymmetry parameter being a vector of SO [START_REF] Kao | Selfdual Chern-Simons systems with an N=3 extended supersymmetry[END_REF]. However, only one of the two versions of the scalar multiplet can be consistently coupled to a given version of the Weyl multiplet, and hence all compensating supermultiplets must be of the same type. The D-field will impose 35 constraints on scalar bilinears and we need 1 + 28 scalars to compensate for the scale and SO(8) gauge invariances. We therefore need a total of 64 scalars and hence eight compensating scalar multiplets.

(h µν ; ψ i µ ; V ij µ , E ijkl ; χ ijk ; D ijkl ) , i = 1, 2,

N = 4 Massive Supergravities

In this section we present details of the construction of linearized 3D N = 4 supergravities with various higher-derivative interactions using the superconformal tensor calculus. In particular, we obtain the N = 4 extension of linearized GMG and its limits. Essential to all these constructions is the 16 + 16 component linearized Weyl multiplet of (2.8). The linearized supersymmetry transformation rules (in the same conventions as in [START_REF] Andringa | Massive 3D Supergravity[END_REF]) are

δh µν = ¯ i γ (µ ψ i ν) , δψ i µ = -1 4 γ ρσ ∂ ρ h µσ i -V ij µ j , δV ij µ = 1 2 ¯ [i φ j] µ + ε ijkl ¯ k γ µ χ l , δE = 1 4 ¯ i χ i , (3.1
)

δχ i = 1 8 ε ijkl γ µ l F jk µ (lin) + γ µ ∂ µ E i + D i , δD = 1 4 ¯ i γ µ ∂ µ χ i .
Here F µ ij (lin) and the (dependent) S-supersymmetry gauge field φ i µ are given by

F µ ij (lin) = µνρ ∂ ν V ij ρ , φ i µ = γ ν γ µ R ν i (lin) , R µ i (lin) = µνρ ∂ ν ψ i ρ . (3.2)
We furthermore insist on gauge invariance with respect to the linearized gauge transformations of (2.1) and (2.2). The supersymmetry algebra now closes in the sense that the commutator of two supersymmetries on any field, with parameters 1 and 2 , gives a translation plus (field-dependent) gauge transformations (represented by the dots):

[δ( 1 ), δ( 2 )] = 1 2 ¯ i 2 γ µ i 1 ∂ µ + • • • . (3.3)

Further properties of the Weyl multiplet fields

The transformation laws given above for the independent fields of the Weyl multiplet imply the following supersymmetry transformation of the dependent S-supersymmetry gauge field:

δφ i µ = γ ν i S (lin) µν -γ ν γ µ j F ν ij (lin) , (3.4)
where S µν is the linearized 3D Schouten tensor

S (lin) µν = R (lin) µν - 1 4 η µν R (lin) , R (lin) ≡ η µν R (lin) µν . (3.5)
Using the expression

R (lin) µν = -1 2 2h µν -2∂ (µ h ν) + ∂ µ ∂ ν h , h µ ≡ η νρ ∂ ρ h µν , h = η µν h µν , (3.6)
one can verify that the linearized Weyl transformation of the linearized Schouten tensor takes the following simple form

δ ω S (lin) µν = - 1 2 ∂ µ ∂ ν ω . (3.7)
It follows that the linearized Cotton tensor, defined as

C (lin) µν = ε µ τ ρ ∂ τ S (lin) ρν , (3.8)
is linearized Weyl invariant; this is a consequence of the Weyl invariance of the nonlinear Cotton tensor, which is the 3D analog of the 4D Weyl tensor. The linearized Cotton tensor, which is parity odd, satisfies the identities

∂ µ C (lin) µν ≡ 0 , C (lin) µν ≡ C (lin) νµ , η µν C (lin) µν ≡ 0 , (3.9)
which are also consequences of similar identites satisfied by the full Cotton tensor. The superpartner of the Cotton tensor is the Cottino tensor. The linearized N = 4 Cottino tensor is

C µ i (lin) = γ ν ∂ ν R µ i (lin) + µνρ ∂ ν R i ρ (lin) . (3.10)
It satisfies the identities

∂ µ C µ i (lin) = 0 , γ µ C µ i (lin) = 0 . (3.11)
The linearized Cotton, Cottino and SO(4) curvature tensors have the following supersymmetry transformations:

δC (lin) µν = -1 4 ¯ i γ (µ ρ ∂ ρ C i lin ν) , δC µ i (lin) = γ ν i C µν (lin) + µνρ γ σ γ ν j ∂ ρ F σ ij (lin) , (3.12
)

δF µ ij (lin) = 1 2 ¯ [i C µ j]
(lin) + ¯ k ijkl γ µρ ∂ ρ χ l . These transformation rules define an N = 4 conformal field-strength multiplet with components

C (lin) µν , C µ i (lin) , F µ ij (lin) , χ i , D , E . (3.13)
With the above definitions in hand we may construct various conformal higherderivative actions for the conformal multiplet. Below we give a few examples of such actions where the leading term, bilinear in the graviton field, has 3,4,5 and 6 derivatives.

N = 4 linearized Weyl multiplet actions

Let us now consider invariants that may be constructed from the Weyl multiplet fields alone, at least in the quadratic approximation.

(1) N = 4 Supersymmetric LCS One may verify that the following action containing a linearized Lorentz Chern-Simons term is supersymmetric,

S N =4 3 = d 3 x h µν C (lin) µν + ψi µ C µ i (lin) -2V ij µ F µ ij (lin) + 16 χi χ i -128ED . (3.14)
As this action is the quadratic approximation to an N = 4 superconformal extension of the LCS term, no compensating fields are needed to construct the non-linear invariant.

(2) N = 4 Supersymmetric NMG invariant Using (3.12), one may verify invariance of the following action:

S N =4 4 = d 3 x -1 2 µτ ρ h µ ν ∂ τ C (lin) ρν - 1 2 ψi µ ∂ /C i µ (lin) + F ij µ (lin) F µ ij (lin) + +32E2E -8 χi ∂ / χ i + 32D 2 . (3.15)
The leading term is just the linearization of the fourth-order K invariant of NMG [START_REF] Bergshoeff | On Higher Derivatives in 3D Gravity and Higher Spin Gauge Theories[END_REF].

To see this we note first that K ≡ R µν R µν -3 8 R 2 = G µν S µν . A convenient form for the linearized 3D Einstein tensor is

G µν (lin) = - 1 2 ε µτ ρ ε νησ ∂ τ ∂ η h ρσ . (3.16)
Using this, one may show that

G µν (lin) S (lin) µν = - 1 2 ε µτ ρ h µ ν ∂ τ C (lin) ρν + total derivative . (3.17)
The expression on the right hand side makes manifest the linearized Weyl invariance.

(3) N = 4 Supersymmetric Ricci times Cotton

One can also construct a fifth-order parity-odd action that starts with the product of a Ricci tensor with a Cotton tensor:

S N =4 5 = d 3 x R µν (lin) C (lin) µν + C(lin) µ i C µ i (lin) + µνρ F (lin) ij µ ∂ ν F (lin) ij ρ +16 χi 2χ i -128E2D . (3.18) (4) N = 4 Supersymmetric Cotton Tensor Squared
It is not difficult to construct linearized higher-derivative supersymmetric actions by using the following observation. Schematically the supersymmetric action (3.14) consists of terms that are all of the form

Weyl × Weyl c , (3.19)
where Weyl indicates a field from the Weyl multiplet and Weyl c denotes a field from the conformal field-strength multiplet (3.13). The fact that the supersymmetry transformations are linearized and thus global has the following immediate consequence: Given a set of fields {Weyl} that transforms as (3.1), the set of fields

{Weyl (n) } = {2 n Weyl} (3.20)
transforms in the same way. This means that the action that is obtained from (3.15) by replacing in each term the first factor Weyl by Weyl (n) , such that we obtain

Weyl (n) × Weyl c , (3.21)
also defines a linearized supersymmetric invariant. The case n = 1 is particularly interesting since one can rewrite, using conformal gauge transformations, the action such that each term is bilinear in the fields of the conformal field-strength Weyl multiplet. This leads to the supersymmetric Cotton tensor squared action:

S N =4 6 = d 3 x C µν (lin) C (lin) µν - 1 4 Ci (lin) µ ∂ /C µ i (lin) + F ij µ (lin) 2F µ ij (lin) + +32E 2 2 E -8 χi 2 ∂ / χ i + 32D 2D . (3.22)
This concludes our discussion of linearized higher-derivative actions that can be constructed from the fields of the Weyl multiplet alone.

N = 4 Einstein supergravity

To obtain the N = 4 supersymmetric Einstein action we need two compensating hypermultiplets, one of each type, for reasons explained in the previous section. Following the superconformal approach we couple these compensating hypermultiplets to the Weyl multiplet and take suitable gauge choices for the superfluous (super-) conformal symmetries. This general procedure simplifies due to the fact that we only consider the linearized version of N = 4 Einstein supergravity.

The hypermultiplet with physical fields (ϕ α, λ α ) has the supersymmetry transformations

δϕ α = ¯ i λ β σi αβ , δλ α = 1 4 γ µ ∂ µ ϕ β i σ i α β , (3.23)
where σ i = (½, -iσ a ) and σi = (½, iσ a ) (with σ a the usual Pauli matrices). Similarly, the hypermultiplet with physical fields (ϕ α , λ α) has the supersymmetry transformations

δϕ α = ¯ i λ ασ i α α , δλ α = 1 4 γ µ ∂ µ ϕ α i σi αα . (3.24)
These transformations leave invariant the following action with 8 + 8 on-shell degrees of freedom:

S = d 3 x -∂ µ ϕ α∂ µ ϕ α -∂ µ ϕ α ∂ µ ϕ α -4 λβ γ µ ∂ µ λ β -4 λ βγ µ ∂ µ λ β , (3.25)
where

ϕ α = (ϕ α) * , ϕ α = (ϕ α ) * , λβ = i(λ β ) † γ 0 , λ α = i(λ α) † γ 0 . (3.26)
After coupling to conformal supergravity, the action (3.25) gets extended by many terms which are schematically of the form (matter

) 2 × 1 + (Weyl) + (Weyl) 2 , (3.27)
where the terms in brackets denote factors that are independent, linear or bilinear in the fields of the conformal supergravity multiplet. In particular, the (matter) 2 ×(Weyl) 2 terms contain a term proportional to Rϕ 2 , and this gives rise to the Einstein-Hilbert term on fixing the conformal gauge. Similarly, there will be a term ψφϕ 2 from which the gravitino kinetic term follows, after fixing the conformal gauge, on substitution for φ i µ .

It turns out that many terms in the action are irrelevant for the determination of the final quadratic expression. To keep things simple we only present those terms in the action that contribute to the final answer. These terms, collectively denoted by L N =4 quadr. , are given by

L N =4 quadr. = -8D µ ϕ αD µ ϕ α -8D µ ϕ α D µ ϕ α -32 λβ γ µ D µ λ β -32 λ βγ µ D µ λ β -Rϕ 2 + -1 2 ψk µ γ µν φ k ν ϕ 2 + -64Dϕ 2 --32E 2 ϕ 2 + + 16χ i (σ i αα λ α ϕ α -σ i α αλ αϕ α ) , (3.28)
where ϕ 2 + and ϕ 2 -are defined in (2.10). Note that the Rϕ 2 term is precisely such that the scalar wave equation is conformally coupled to the background metric. Note also that the action is invariant under the exchange of the two SU(2) factors of the R-symmetry group because the D, E and χ i fields are odd under this exchange. The covariant derivatives D µ are covariant with respect to Lorentz and SO(4) rotations only. Defining

V ij µ = (σ ij ) α β V µ α β + (σ ij ) α β V µ α β , (3.29)
the covariant derivative D µ φ α is given by

D µ ϕ α = ∂ µ ϕ α -V µ α β ϕ β . (3.30)
In principle, all coefficients in the action (3.28) follow from the supersymmetry of the full non-linear action. In practice, it is much easier to take the action (3.28) only as a guideline to obtain the final quadratic expression. The coefficients in this final expression are easily determined by requiring (linearized) supersymmetry.

After coupling to conformal supergravity the fields of the two hypermultiplets transform under local dilatations and S-supersymmetry transformations as follows:

δϕ α = ωϕ α , δλ α = σ i α β ϕ β η i , (3.31)
δϕ α = ωϕ α , δλ α = σi αβ ϕ β η i . (3.32)
To fix the dilatations and S-supersymmetry transformations we impose the gauge choices

ϕ 2 + = 1 , σi αα λ α ϕ α + σ i α αλ αϕ α = 0 , (3.33)
respectively. These gauge choices can be used in (3.28). The D and χ fields are Lagrange multipliers that lead to a single bosonic constraint Φ ≡ ϕ 2 -= 0 (3.34) and 4 fermionic constraints

ψ i ≡ σi αα λ α ϕ α -σ i α αλ αϕ α = 0 , (3.35)
respectively. Since we are going to add further higher-derivative terms to the action in the next subsection we leave these Lagrange multipliers in the action.

Using the gauge choices (3.33), the field equation of the 6 vector fields allows for these vector fields to be solved in terms of the scalars:

V ij µ = (σ ij ) α β ϕ α ∂ µ ϕ β + (σ ij ) α β ϕ α∂ µ ϕ β . (3.36)
We next fix the local SO(4) by imposing the following 6 conditions:

ϕ α ∂ µ ϕ β + h.c. = 0 , ϕ α∂ µ ϕ β + h.c. = 0 , (3.37)

N = 4 massive supergravities

In this subsection we will consider the sum of the supersymmetric Einstein action and the conformal higher-derivative actions constructed in subsection 3.2 with 3,4 and 5 derivatives thereby introducing three mass parameters µ , m and M, respectively. This leads to the following action that is left invariant under the same transformation rules (3.41) derived in the previous subsection:

S N =4 = d 3 x 1 2 h µν G (lin) µν + µνρ ψi µ ∂ ν ψ i ρ -V ij µ V µ ij -32E 2 -8 ψ∂ / ψ + 16 ψχ -64DΦ + 32Φ2Φ + 1 µ h µν C (lin) µν + ψi µ C µ i (lin) -2V ij µ F µ ij (lin) + 16 χi χ i -128ED + 1 m 2 -1 2 µτ ρ h µ ν ∂ τ C (lin) ρν - 1 2 ψi µ ∂ /C i µ (lin) + F ij µ (lin) F µ ij (lin) + (3.42) +32E2E -8 χi ∂ / χ i + 32D 2 + 1 M 3 R µν (lin) C (lin) µν + C(lin) µ i C µ i (lin) + µνρ F (lin) ij µ ∂ ν F (lin) ij ρ +16 χi 2χ i -128E2D .
To analyze which massive supermultiplets are propagating for the different values of the mass parameters, it is enough to consider the scalar equations of motion:

E + 2 µ D - 1 m 2 2E + 2 M 3 2D = 0 , Φ + 2 µ E - 1 m 2 D + 2 M 3 2E = 0 , (3.43)
2Φ -D = 0 , and the fermionic equations of motion:

∂ /ψ i = χ i , ψ i + 2 µ χ i - 1 m 2 ∂ /χ i + 2 M 3 2χ i = 0 . (3.44)
We may now analyze the following models

(1) Supersymmetric Einstein : µ , m , M → ∞

We find

E = D = Φ = ψ i = χ i = 0 , (3.45)
and hence there are no propagating multiplets.

(2) Supersymmetric TMG : m , M → ∞

We find one independent propagating scalar that satisfies

2Φ - µ 2 4 Φ = 0 , (3.46)
and hence there is one propagating massive spin 2 supermultiplet. For the fermions we find

∂ /ψ i = χ i , ψ i = - 2 µ χ i , (3.47)
and hence

∂ /ψ i = - µ 2 ψ i . (3.48)
Applying an extra ∂ / leads to

2ψ i - µ 2 4 ψ = 0 , (3.49)
which is in agreement with the N = 4 spin 2 supermultiplet content displayed in Table 1.

(3) Supersymmetric NMG : µ , M → ∞

We find two independent scalars with

2D -m 2 D = 0 , 2E -m 2 E = 0 , (3.50)
and hence there are two propagating N = 4 massive multiplets, of the same mass but with opposite helicity. For the fermions we find:

∂ /ψ i = χ i , ∂ /χ i = m 2 ψ i . (3.51)
Upon diagonalization of these equations one infers that their mass eigenvalues are ±m, which implies that they describe opposite helicities, as required. By applying ∂ /, one finds

2χ i -m 2 χ i = 0 , 2ψ i -m 2 ψ i = 0 , (3.52)
which is again in agreement with the scalar analysis.

(4) Supersymmetric GMG : M → ∞

We now have

E + 2 µ D - 1 m 2 2E = 0 , Φ + 2 µ E - 1 m 2 D = 0 , (3.53)
2Φ -D = 0 .

Eliminating the scalar Φ and replacing E → 1/µE we obtain

2D -(m 2 + 4m 4 µ 2 )D - 2m 4 µ 2 E = 0 , 2E -2m 2 D -m 2 E = 0 . (3.54)
After a diagonalization we find two propagating multiplets with different masses :

m 2 ± = m 2 2m 2 + µ 2 ± 2 m 2 (m 2 + µ 2 ) µ 2 , (3.55) such that 8 m + m -= m 2 , m --m + = 2 m 2 µ . (3.56)
The corresponding eigenscalars are D + b ± E, where

b ± = -m 2 ± m 4 + m 2 µ 2 µ 2 . (3.57)
For the fermions we find

∂ /ψ i = χ i , ∂ /χ i = m 2 ψ i + 2m 2 µ χ i . (3.58)
By applying ∂ / and similar manipulations as done for the scalars, one arrives at

2ψ i -2m 2 χ i -m 2 ψ i = 0 , 2χ i -m 2 + 4m 4 µ 2 χ i - 2m 4 µ 2 ψ i = 0 . (3.59)
This is the same system of equations as encountered in the scalar case, with D → χ i , E → ψ i . The diagonalisation procedure and mass 2 eigenvalues are thus the same as in the scalar case.

(5) Supersymmetric NTMG : M → ∞ , µ , m → 0 such that m 2 /µ = const. of (2.1). Finally, the supersymmetry algebra closes in the same way as in the N = 4 case.

N = 8 conformal higher-derivative actions

Using the same definitions of the Cotton and Cottino tensor as in the N = 4 case we may construct the following parity-even conformal higher-derivative actions for the N = 8 conformal multiplet.

(

) N = 8 supersymmetric R 2 1
There exists a supersymmetric action of the N = 8 conformal multiplet that starts with the linearized version of the R 2 µν -3 8 R 2 term of NMG [START_REF] Bergshoeff | On Higher Derivatives in 3D Gravity and Higher Spin Gauge Theories[END_REF]. More precisely, the transformation rules (4.2) leave the following action invariant:

S N =8 4 = d 3 x - 1 2 µτ ρ h µ ν ∂ τ C (lin) ρν - 1 2 ψi µ ∂ /C µ (lin) i + F ij µ (lin) F µ ij (lin) + + 2 3 E ijkl 2E ijkl - 4 3 χijk ∂ / χ ijk + 2 3 D ijkl D ijkl . (4.3)
Under the rigid supersymmetry rules (4.2) the Cotton, Cottino and SO(8) curvature tensors transform as follows

δC (lin) µν = -1 4 ¯ i γ (µ ρ ∂ ρ C i lin ν) , δC µ i (lin) = γ ν i C µν (lin) + µνρ γ σ γ ν j ∂ ρ F σ ij (lin) , (4.4
)

δF µ ij (lin) = 1 2 ¯ [i C µ j]
(lin) + ¯ k γ µρ ∂ ρ χ ijk . These transformation rules define a N = 8 conjugated Weyl multiplet with components

C (lin) µν , C µ i (lin) , F µ ij (lin) , χ ijk , D ijkl , E ijkl . (4.5)
(2) N = 8 supersymmetric Cotton tensor squared Following the same procedure as in the N = 4 case, making use of the conjugated Weyl multiplet (4.5), we obtain the following supersymmetric Cotton tensor squared action:

S N =8 6 = d 3 x C µν (lin) C (lin) µν - 1 4 Ci (lin) µ ∂ /C µ i (lin) + F ij µ (lin) 2F µ ij (lin) + + 2 3 E ijkl 2 2 E ijkl - 4 3 χijk 2 ∂ / χ ijk + 2 3 D ijkl 2D ijkl . (4.6)
This concludes our discussion of the (parity-even) conformal higher-derivative actions. In the next subsection we consider the case of (non-conformal) N = 8 Einstein supergravity.

the conformal multiplet is off-shell we retain supersymmetry. The combined action is given by

S N =8 NMG = d 3 x 1 2 h µν G (lin) µν + µνρ ψi µ ∂ ν ψ i ρ -V ij µ V µ ij - 2 3 E ijkl E ijkl - 4 3 ψijk ∂ / ψ ijk + 8 3 ψijk χ ijk - 4 3 D ijkl Φ ijkl + 2 3 Φ ijkl 2Φ ijkl + 1 m 2 -1 2 µτ ρ h µ ν ∂ τ C (lin) ρν - 1 2 ψi µ ∂ /C µ (lin) i + F ij µ (lin) F µ ij (lin) - 4 3 χijk ∂ / χ ijk + 2 3 E ijkl 2E ijkl + 2 3 D ijkl D ijkl . (4.11)
It is invariant under the same supersymmetry transformations (4.10). The only thing that changes with respect to the pure Einstein supergravity action is that the equations of motion of the conformal fields E, D, V and χ, receive 1/m 2 corrections which leads to propagating massive degrees of freedom. To be precise, the corrected equations of motion read:

E ijkl - 1 m 2 2E ijkl = 0 , (4.12
)

V µ ij - 1 m 2 ∂ λ F (lin) λµ ij (V ) = 0 , (4.13)
Φ ijkl - 1 m 2 D ijkl = 0 , (4.14)
ψ ijk - 1 m 2 ∂ / χ ijk = 0 . (4.15)
The first equation shows that the E scalars describe 35 massive helicity 0 d.o.f. The third equation, together with the uncorrected equation for Φ ijkl ,

2Φ ijkl -D ijkl = 0 , (4.16)
can be used to show that the D scalars satisfy

D ijkl - 1 m 2 2D ijkl = 0 , (4.17)
and hence describe another 35 helicity d.o.f. of the same mass m. From the second equation it follows that the vector fields describe 28 helicity ±1 states:

V ij µ - 1 m 2 2V ij µ = 0 . (4.18)
Concerning the helicity ±1/2 degrees of freedom we end up with the following two equations of motion:

∂ /χ ijk -m 2 ψ ijk = 0 , ∂ /ψ ijk -χ ijk = 0 . (4.19)
By taking sums and differences we see that this system describes 56 +1/2 helicity states and 56 -1/2 helicity states of the same mass m, i.e.

χ ijk - 1 m 2 2χ ijk = 0 , ψ ijk - 1 m 2 2ψ ijk = 0 . (4.20)
Adding up all massive degrees of freedom, including the helicity ±2 and ±3/2 states, we precisely obtain the content of the N = 8 massive super multiplet given in Table 1, as it should be.

Linearized N = 7 TMG

In this section we study maximal supersymmetry for parity-odd actions. We first consider the Lorentz Chern-Simons (LCS) term. It turns out that there does not exist a N = 8 supersymmetric LCS action for the full conformal multiplet. The reason for this is that such an action would require a term of the form E ijkl D ijkl but, due to the opposite dualities of the E and D fields, such a term does not exist. The best one can do is write down the following so-called "pseudo-action" assuming that E ijkl and D ijkl are not (anti-)selfdual (the Cotton tensor C and the Cottino tenor C are defined below):

S N =8 3 = d 3 x h µν C (lin) µν + ψi µ C µ i (lin) -2V ij µ F µ ij (lin) + 8 3 χijk χ ijk . (5.1)
This action is invariant under supersymmetry up to terms proportional to E ijkl or D ijkl . Indeed, up to a total derivative, the variation of the above action is given by

δS N =8 3 = d 3 x 16 3 χijk γ µ l ∂ µ E ijkl + 16 3 χijk l D ijkl . (5.
2)

The (anti-)selfduality of the E and D fields is only imposed at the level of the equations of motion. Note that setting E ijkl = D ijkl = 0 in the conformal multiplet leads to equations of motion for the remaining fields that can be integrated to an action which is precisely the above action [START_REF] Gran | Three-dimensional N=8 superconformal gravity and its coupling to BLG M2-branes[END_REF]. We have not been able to write down a similar pseudoaction for the combined Einstein-Chern-Simons system. This is only possible if one can impose (anti-) selfduality in the equations of motion for D ijkl and E ijkl . This seems unlikely since we expect an equation of motion of the form E ijkl = 1 µ D ijkl + • • • where µ is a mass parameter. Such an equation obviously is inconsistent with the duality properties of the E and D fields. The non-existence of such a N = 8 topologically massive supergravity theory can also be anticipated from the fact that such a theory would lead to a N = 8 supersymmetric massive supermultiplet with broken parity. According to Table 1 such a multiplet does not exist.

Although a supersymmetric N = 8 LCS action does not exist, one can construct a supersymmetric N = 7 version. In order to do this, one decomposes the R-symmetry index i as i = {I, 8}, where I = 1, • • • , 7. Performing this decomposition, one arrives (5.7)

S N =7 TMG = d 3 x 1 2 h µν G (lin) µν + µνρ ψi µ ∂ ν ψ i ρ -V ij µ V µ ij - 2 3 E ijkl E ijkl - 4
One may verify that the equations of motion lead precisely to a N = 7 spin 2 massive supermultiplet, see Table 1. The analysis is similar to the N = 4 case treated in the section 3.

Conclusions and Outlook

The correspondence between massless supermultiplets of 4D supersymmetry and massive supermultiplets of 3D supersymmetry, and the existence of N -extended 4D supergravity theories for N = 1, 2, 3, 4, 5, 6 and N = 8 suggests the existence of analogous parity-preserving 3D massive supergravity theories, with N now counting the number of 3D two-component Majorana spinor supercharges. In particular, the analogy suggests the existence of an N = 8 maximally supersymmetric extension of "new massive gravity" [START_REF] Bergshoeff | On Higher Derivatives in 3D Gravity and Higher Spin Gauge Theories[END_REF]. We also expect additional parity-violating supergravity theories, such as "topologically massive supergravity" [START_REF] Deser | Three-Dimensional Massive Gauge Theories[END_REF][START_REF] Deser | Topologically Massive Supergravity[END_REF] but representation theory only allows N ≤ 7 in this case. The general massive N = 1 3D supergravity was constructed in [START_REF] Andringa | Massive 3D Supergravity[END_REF][START_REF] Bergshoeff | More on Massive 3D Supergravity[END_REF], as was one version of the linearized N = 2 3D supergravity. A simplifying feature of these unitary higher-derivative models is that the higher-derivative terms have a quadratic approximation (in an expansion about the Minkowski vacuum) that is invariant under linearized superconformal gauge invariances. As a consequence, the superconformal compensating multiplets needed for the construction of generic higher-derivative invariants are needed, at the linearized level, only for the construction of the supersymmetric extension of the Einstein-Hilbert term. This means that we do not need a full superconformal tensor calculus to construct the linearized theories, which is fortunate because this has not been worked out for N ≥ 3 and must involve an infinite number of auxiliary fields for N ≥ 5.

We have discussed the general picture of compensating fields for all N . In particular N = 3 is similar to the N = 2 case discussed in [START_REF] Andringa | Massive 3D Supergravity[END_REF], in that we only need one compensating multiplet which is a Stueckelberg multiplet in the sense that its coupling to the Weyl multiplet is bilinear. For N ≥ 4, we need multiplet copies of the compensating multiplet, and the Weyl multiplet couples to bilinears of them. The R-symmetry group gets spontaneously broken when we fix the scale transformations and the corresponding Goldstone bosons are the Stueckelberg fields. We have presented the details of how things work out for N = 4, thereby constructing the general linearized massive 3D N = 4 supergravity, as well as a number of (non-unitary) linearized models with yet higher derivative terms.

Maximally supersymmetric supergravity models are of particular interest, and we have constructed the linearized maximally-supersymmetric N = 8 "new massive supergravity". The inclusion of the parity-violating Lorentz-Chern-Simons term necessarily breaks N = 8 supersymmetry, so that N = 7 is maximal for models such as topologically massive gravity and we have constructed a linearized N = 7 topologically massive supergravity action. However, it is not clear that N = 7 is realizable beyond the linear level, because there are eight Rarita-Schwinger fields in the N = 7 multiplet; we think it likely that N = 6 is maximal for TMG.

The extension of our results to the full non-linear level is a challenging task that we leave to the future. Given the striking properties of N = 8 supergravity in four dimensions, one may hope that the 3D "new massive" N = 8 supergravity will have similar nice properties. For example, a cosmological extension of the N = 8 massive gravity might allow an AdS vacuum preserving all 16 supersymmetries, in which case it might have a holographic dual 2D conformal field theory with maximal (4, 4) supersymmetry.

Table 2 :

 2 Some properties of the N -extended Weyl multiplets. The fourth column indicates the number of basic compensating supermultiplets needed to obtain (Einstein, TMG or GMG) supergravity.

	N # off-shell d.o.f.	R-symmetry	# multiplets
	2	4 + 4	S O ( 2 )	1
	3	8 + 8	S O ( 3 )	1
	4	16+16	Spin(4) ∼ = SU(2) × SU(2)	2
	5	32+32	Spin(5) ∼ = Sp 2	2
	6	64+64	Spin(6) ∼ = SU(4)	4
	7	128+128	Spin(7)	8
	8	128+128	SO(8)	8

  Fields with multiple indices are again antisymmetric in these indices, and E ijkl and D ijkl have 'opposite SO(8) dualities': they are, respectively, self-dual and anti-selfdual, or vice-versa. As there are two choices of duality assignments, there are two (equivalent) versions of the N = 8 Weyl multiplet.The minimal scalar multiplet has 8 + 8 physical components and also comes in two versions. One version has the scalars in the spinor representation and the fermions in the conjugate spinor representation of Spin(

	3, 4, 5, 6, 7, 8.	(2.15)

Table 3 :

 3 Some 3D higher-derivative sugra models. The second column indicates the number of derivatives in the different terms for the spin 2 field. The 3rd and 4th column indicate the massive spin 2 multiplets described by the corresponding supergravity model. The last 2 columns indicate the masses of these N = 4 multiplets, respectively. The boldface number in the lowest row indicates a ghost multiplet.

  IJKL D IJKL -8 χIJ χ IJ + 4V I µ F µI (lin) -ψµ C µ (lin) .

		3	ψijk ∂ / ψ ijk +	8 3	ψijk χ ijk -	4 3	D ijkl Φ ijkl +	2 3	Φ ijkl 2Φ ijkl
	+	1 µ		h µν C (lin) µν + ψI µ C µ I (lin) -2V IJ µ F µ IJ (lin) +	8 3	χIJK χ IJK
	-	16 3	E

If the anticommutator (1.1) is modified to allow additional (non-central) charges then one can find additional, parity-preserving, multiplets that do not correspond to 4D multiplets because they are acted upon by twice as many supercharges as those discussed here[START_REF] Bergshoeff | A Topologically Massive Gauge Theory with 32 Supercharges[END_REF].

Interactions must be considered on a case-by-case basis.

A "disjoint" family of massive 3D N = 1 supergravity theories was recently found in[START_REF] Lu | Massive Three-Dimensional Supergravity From R + R 2 Action in Six Dimensions[END_REF] by dimensional reduction of a 6D theory with curvature-squared terms. These may also have N > 1 extensions but we shall have nothing to say about that here.

By "on-shell-supersymmetric" we mean that the equations of motion are needed for closure of the supersymmetry algebra, due to the implicit elimination of auxiliary fields.

This recalls the conjecture that N =

is maximal for an off-shell realization of N = 8 4D supergravity[START_REF] Howe | Supersymmetry counterterms revisited[END_REF].

But here we use a different notation: what was called A in[START_REF] Andringa | Massive 3D Supergravity[END_REF] is V here.

It was suggested in[START_REF] Howe | New supergravities with central charges and Killing spinors in (2+1)-dimensions[END_REF] that there should exist an

8+8 Weyl multiplet that contains only V + or V -, and without the higher dimension fields (E; χ i ; D), but it does not appear to be possible to close the supersymmetry algebra on this smaller set.

We have fixed the sign of m --m + in accordance with[START_REF] Bergshoeff | On Higher Derivatives in 3D Gravity and Higher Spin Gauge Theories[END_REF]. To compare, use that μ = 2m 2 /µ 2 .

Acknowledgements

We thank Paul Howe for helpful correspondence. The work of OH is supported by the DFG -The German Science Foundation and in part by funds provided by the U.S. Department of Energy (DoE) under the cooperative research agreement DE-FG02-05ER41360. PKT thanks the EPSRC for financial support.

which implies V ij µ = 0. We note that for pure Einstein supergravity (no further higher-derivative terms added to the action) the 8+8 matter fields of the compensating hypermultiplets are all fixed by the constraints and gauge choices, as the following counting shows: bosons : 8 -1 (D-gauge) -6 (SO(4)-gauge ) -1 (constraint) = 0 , fermions : 8 -4 (S-gauge) -4 (constraints) = 0 .

(3.38)

Using the gauge choices (3.33) and (3.37), one can show that it is possible to rewrite the kinetic terms of the compensating matter scalars and fermions, appearing in the conformal action, as standard kinetic terms for Φ and ψ i only. One thus obtains the following linerarized N = 4 supersymmetric Einstein action

Using the fact that the gauge-fixing requires a compensating S-transformation and SO(4) transformation with parameters

respectively, we find that the action (3.39) is invariant under the following supersymmetry rules

)

Note that the action (3.39) does not describe any massive degrees of freedom. Several fields could be integrated out to leave us with only the Einstein term and the kinetic term of the gravitino. We do not do this because the equations of motion of these fields change as soon as we add one of the superconformal higher-derivative actions constructed in the previous subsection.

We now have one propagating scalar,

and hence one propagating multiplet. Similarly, we find for the fermions:

from which one derives

again in agreement with the scalars.

(

This is the case of 'Extended New Massive Gravity' (ENMG) discussed in the third reference of [START_REF] Bergshoeff | On Higher Derivatives in 3D Gravity and Higher Spin Gauge Theories[END_REF]. This case leads to the scalar equations

Since in the action we have

we end up with one physical and one ghost multiplet, each with (mass) 2 = M 3 /µ.

For the fermions we find 1

which also implies one physical and one ghost multiplet.

We have summarized the analysis of the different models in Table 2.

N = 8 Maximal Supergravity

In this section we consider the case of maximal supersymmetry. Since the application of the superconformal tensor calculus has been explained in the previous section we will be brief whenever there is overlap with the N = 4 case.

sugra model action N = 4 opposite helicity (mass) 2 (mass)

The N = 8 conformal multiplet

The D = 3, N = 8 linearized conformal supergravity multiplet has 128+128 field degrees of freedom and contains the following components:

with h µν = h νµ the linearized graviton, ψ i µ (i = 1, • • • , 8) the 8 gravitini and V ij µ = -V ji µ the SO(8) R-symmetry gauge field. The matter fields E ijkl and D ijkl (of different mass dimension) are in the antisymmetric selfdual 35 + and anti-selfdual 35 -representation of SO [START_REF] Kao | Selfdual Chern-Simons systems with an N=3 extended supersymmetry[END_REF], respectively, while the fermions χ ijk form an antisymmetric 56 -plet of SO [START_REF] Kao | Selfdual Chern-Simons systems with an N=3 extended supersymmetry[END_REF].

The linearized supersymmetry transformations are given by [START_REF] Howe | New supergravities with central charges and Killing spinors in (2+1)-dimensions[END_REF] (using the same conventions as in [START_REF] Andringa | Massive 3D Supergravity[END_REF])

)

Here F µ ij (lin) and φ i µ are defined in an analogous manner as in (3.2) for N = 4. We insist on gauge invariance with respect to the analogue of the linear gauge transformations

N = 8 Einstein supergravity

To obtain supersymmetric Einstein we introduce 8 scalar multiplets and couple them to conformal supergravity. The supersymmetry transformation rules of 8 scalar multiplets with components (φ αA , χ A α ) (A = 1, • • • , 8) are given by

where (i, α, α) = (1, • • • , 8) denote the 8-dimensional v, s and c representations of SO [START_REF] Kao | Selfdual Chern-Simons systems with an N=3 extended supersymmetry[END_REF], respectively. For the SO( 8) Dirac matrices we use the notation of [START_REF] Green | Superstring Interactions[END_REF]. The transformation rules (4.7) leave the following action invariant

We next couple the above action to N = 8 conformal supergravity and follow precisely the same steps as in the N = 4 case. From now on we identify A = α and write ϕ αB = ϕ αβ . This leads to the following Einstein supergravity action

We find that this action is invariant under the following supersymmetry rules:

)

N = 8 new massive supergravity

To obtain N = 8 new massive supergravity we add the supersymmetric R 2 action (4.3), which we give a coefficient 1/m 2 , to the supersymmetric Einstein action. Since at the following fields:

(5.

3)

The transformation rules can be found from the N = 8 ones, by making the above decomposition and by putting 8 = 0. In this way one finds

where φ µ = φ 8 µ . The (anti-)self duality conditions for E IJKL and D IJKL no longer hold but are instead replaced by

(One could as well remove E IJK and D IJK using these rules.) The combination E IJKL D IJKL is thus no longer zero and can be added to the action. One obtains that the following action is invariant


J B Gutowski
email: jan.gutowski@kcl.ac.uk

W A Sabra

Solutions of Minimal Four Dimensional de Sitter Supergravity

Keywords: numbers: 04, 65, +e, 12, 60, Jv

Pseudo-supersymmetric solutions of minimal N = 2, D = 4 de Sitter supergravity are classified using spinorial geometry techniques. We find three classes of solutions. The first class of solution consists of geometries which are fibrations over a 3-dimensional manifold equipped with a Gauduchon-Tod structure. The second class of solution is the cosmological Majumdar-Papapetrou solution of Kastor and Traschen, and the third corresponds to gravitational waves propagating in the Nariai cosmology.

Introduction

The classification of supersymmetric solutions has attracted considerable attention in recent years due to the important role these solutions play in string and M-theory. Many years ago, Tod was able to find all metrics admitting supercovariantly constant spinors in N = 2, D = 4 ungauged minimal supergravity [1]. In recent years and motivated by the work of [1], progress has been made in the classification of supersymmetric supergravity solutions. For example, the classification of supersymmetric solutions of minimal N = 2, D = 4 gauged supergravity has been performed in [START_REF] Caldarelli | High Energy Phys[END_REF]3]. The bosonic part of N = 2, D = 4 gauged supergravity is basically Einstein-Maxwell theory with a negative cosmological constant. The supersymmetric solutions of this theory are obtained by solving the Killing spinor equation obtained from the vanishing of the gravitini supersymmetry variation. In this paper, we will be interested in finding solutions of Einstein-Maxwell theory with a positive cosmological constant. This theory cannot be embedded in a supergravity theory, as supersymmetry restricts the cosmological constant to be negative. However, one can obtain a fake Killing spinor equation by analytic continuation. Therefore we will be using fake supersymmetry as a solution generating technique. De Sitter supergravities can also be obtained from type IIB* theory [START_REF] Liu | High Energy Phys[END_REF]. Solutions of five dimensional De Sitter supergravity were recently analyzed in [5]. We shall use spinorial geometry techniques to investigate solutions of the minimal four-dimensional de Sitter supergravity. These have been used to analyse supersymmetric solutions in ten and eleven dimensional supergravity theories [START_REF] Gillard | [END_REF]7,8,[START_REF] Gran | High Energy Phys[END_REF] as well as in lower dimensions [START_REF] Cacciatori | High Energy[END_REF].

The plan of the paper is as follows. In Section two, a summary of the basic equations of the theory of N = 2, D = 4 minimal de Sitter supergravity and a brief description of the representatives of the Killing spinors are presented. Sections three, four and five contain a detailed analysis of the Killing spinor equations for the three canonical forms of the Dirac spinors, making use of the linear system presented in the Appendix. We conclude in Section six.

N = 2, D = 4 Minimal de Sitter Supergravity

In this section, we present a summary of N = 2, D = 4 minimal de Sitter supergravity. The bosonic action associated with this theory is [12,13]

S = d 4 x √ -g ( 1 4 R - 1 4 F µν F µν - 3 2 2 ) ( 1 )
where F = dA is the Maxwell field strength and is a non-zero real constant. The signature of the metric is (-, +, +, +). The Einstein and gauge field equations are

R µν -3 -2 g µν -2F µρ F ν ρ + 1 2 F αβ F αβ g µν = 0 d F = 0 . (2)
We shall consider solutions which are pseudo-supersymmetric i.e. those which admit a non-zero Killing spinor satisfying the Killing spinor equation:

D µ = ∂ µ + 1 4 ω µ,ν 1 ν 2 Γ ν 1 ν 2 + i 4 F ν 1 ν 2 Γ µ Γ ν 1 ν 2 -iF µν Γ ν - i 2 -1 Γ µ --1 A µ = 0 . (3)
The Killing spinor is a Dirac spinor. We follow the conventions of [14] in dealing with such spinors; for convenience we summarize a number of useful results here. Dirac spinors can be written as complexified forms on R 2 ; a generic spinor η can therefore be written as [START_REF] Lawson | Spin Geometry[END_REF]

η = λ1 + µ i e i + σe 12 (4)
where e 1 , e 2 are 1-forms on R 2 , and i = 1, 2; e 12 = e 1 ∧ e 2 . λ, µ i and σ are complex functions. It will be particularly useful to work in a null basis, and set

Γ + = √ 2i e 2 Γ -= √ 2e 2 ∧ Γ 1 = √ 2i e 1 Γ1 = √ 2e 1 ∧ (5)
where in the null basis the metric is

ds 2 = 2e + e -+ 2e 1 e 1 . (6)
We recall from [14] that a spinor can be written, using Spin(3, 1) gauge transformations, as one of three possible simple canonical forms:

= e 2 (7)
or

= 1 + µ 1 e 1 (8) or
= 1 + µ 2 e 2 . (9)
Note that by making use of a Spin(3, 1) transformation generated by Γ +-, combined with an appropriately chosen U (1) gauge transformation of A which together leave 1 invariant, one can without loss of generality take |µ 2 | = 1 in (9).

To proceed, we evaluate the Killing spinor equation (3) acting on the spinor

= λ1 + µ i e i . (10)
The resulting equations are summarized in Appendix A. We then consider the three cases (7), ( 8) and (9) separately.

Solutions with = e 2

In order to analyse solutions with = e 2 , evaluate the equations in Appendix A with λ = µ 1 = 0 and µ 2 = 1. One obtains

F +-+ F 1 1 + -1 = 0 (11)
and

F +-+ F 1 1 --1 = 0 . ( 12
)
It is clear that these equations admit no solution; hence there are no supersymmetric solutions with = e 2 .

Solutions with = 1 + µe 1

On evaluating the equations in Appendix A with λ = 1, µ 1 = µ, µ 2 = 0 one one obtains the conditions:

∂ 1 µ = ∂ + µ = 0 ∂1µ = √ 2i -1 (1 + µμ) ( 1 3
)

ω 1,+1 = ω -,+1 = ω 1,+ 1 = ω +,1 1 = ω +,+ 1 = 0 ω -,1 1 = (µ∂ -μ -μ∂ -µ) (1 + µμ) ω 1,1 1 = -µ √ 2i -1 (14)
-1 A -= 1 2 (μ∂ -µ + µ∂ -μ) (1 + µμ) - 1 2 ω -,+- -1 A 1 = - 1 2 ω 1,+-- 1 √ 2 µi -1 -1 A + = - 1 2 ω +,+- (15)
F -1 = - i √ 2
∂ -µ (1 + µμ) F +-= -1 F 1 1 = F +1 = 0 . ( 16
)
To analyse these conditions, observe first that

de -= -ω A,+-e A ∧ e - (17)
and hence e -is hypersurface orthogonal; one can introduce a co-ordinate u and function H such that e -= Hdu .

Next, we examine the constraints on the gauge potential A. Note that

-1 A = 1 2 d log(1 + |µ| 2 ) + 1 2 d log H + P du (19)
for some function P . Hence, by making a gauge transformation in A, combined with an appropriately chosen Spin(3, 1) transformation generated by Γ +-, which together with the A-gauge transformation leave 1 + µe 1 invariant, one can without loss of generality work in a gauge for which

-1 A = 1 2 (μ∂ -µ + μ∂ -µ) (1 + µμ) - 1 2 ω -,+-e - (20)
and moreover,

ω +,+-= 0 (21)
and

- 1 2 ω 1,+-- 1 √ 2 µi -1 = 0 . (22)
In this gauge, we then find

de -= -d log(1 + |µ| 2 ) ∧ e - (23)
and hence it is most convenient to introduce a local co-ordinate u such that

e -= 1 1 + |µ| 2 du . (24)
Next, consider the exterior derivative of e 1 restricted to hypersurfaces u = const.. One finds that de

1 = -d log(1 + |µ| 2 ) ∧ e 1 ( 25
)
where d denotes the restriction of the exterior derivative to u = const.. It follows that one can introduce a complex co-ordinate z such that

e 1 = 1 1 + |µ| 2 (dz + ξdu) ( 2 6 )
for ξ ∈ C. The metric can be simplified further by making use of the Spin(3, 1) gauge transformation generated by βΓ +1 + βΓ + 1, for β ∈ C, which leaves the Killing spinor 1 + µe 1 invariant. This gauge transformation corresponds to a change of basis of the form

(e -) = e - (e + ) = e + + 4|β| 2 e --2 βe 1 -2βe 1 (e 1 ) = e 1 -2βe -. (27)
By choosing β appropriately, one can, without loss of generality, set ξ = 0 in (26). So, on introducing a final local co-ordinate v such that the vector field dual to e - is ∂ ∂v , one finds that one can write the basis in the u, v, z, z co-ordinates as

e -= 1 1 + |µ| 2 du e 1 = 1 1 + |µ| 2 dz e + = dv + Hdu + Gdz + Ḡdz ( 28
)
where H is a real function, G is a complex function, and µ does not depend on v.

Next consider the constraints implied by (13). In particular, µ depends only on z and u, with

∂µ ∂ z = √ 2i -1 . (29)
Hence

µ = √ 2i -1 z + h(u) ( 3 0 )
where h is a function of u. By making a change in co-ordinates of the form z = z + ψ(u) together with an appropriately chosen Spin(3, 1) transformation generated by βΓ +1 + βΓ + 1, one can without loss of generality take the basis given in (28) with

µ = √ 2i -1 z . ( 31
)
Observe that, in this basis, ∂ -µ = 0.

To proceed, consider the conditions ω -,+1 = ω -,1 1 = 0 on the geometry. It is straightforward to show that these imply that

G = - 2 -2 vz 1 + 2 -2 z z + φ (32)
where φ(u, z, z) is a complex function satisfying

∂ z φ -∂ z φ = 0 . (33)
Next, noting that

ω -,+-= -(1 + 2 -2 z z) ∂H ∂v (34)
we impose the Bianchi identity F = dA to obtain the constraints

∂ 2 H ∂v 2 = 2 -2 1 + 2 -2 z z , 1 2
∂ 2 H ∂z∂v = -2 1 + 2 -2 z z ( - 2 -2 vz 1 + 2 -2 z z + φ) . (35) These can be solved to find H = -2 v 2 1 + 2 -2 z z + Θ 1 v + Θ 2 (36)
where Θ 1 , Θ 2 do not depend on v, and

φ = 1 2 2 (1 + 2 -2 z z) ∂Θ 1 ∂z (37) with Θ 1 constrained by z ∂Θ 1 ∂ z -z ∂Θ 1 ∂z = 0 . ( 38
)
One can simplify the solution considerably by making the co-ordinate transformation

v = (1 + 2 -2 z z)(v - 1 2 2 Θ 1 ) . ( 39
)
On dropping the prime on v the solution can then be written as

ds 2 = 2du dv + ( -2 v 2 + Ψ)du + 2 (1 + 2 -2 z z) 2 dzdz (40) with F = -1 dv ∧ du . (41)
The function Ψ = Ψ(u, z, z) appearing in the metric is constrained to be harmonic on R 2 by the Einstein equations:

∂ 2 Ψ ∂z∂ z = 0 . ( 42
)
Observe that the gauge field equations d F = 0 hold with no further constraints.

Solutions with

= 1 + e iα e 2
On evaluating the equations in Appendix A with λ = 1, µ 1 = 0, µ 2 = e iα , one obtains the components of the gauge field strength as:

F +-= √ 2( sin αω +,+--∂ + α cos α) --1 F 1 1 = i √ 2( cos αω +,+-+ sin α∂ + α) F -1 = i √ 2 e iα ω -,-1 F +1 = i √ 2 e -iα ω +,+1 . (43)
The components of the gauge potential are given by:

-1 A -= - 1 2 ω -,+-, -1 A 1 = 1 2 i∂ 1 α -ω 1,1 1 , -1 A + = 1 2 ω +,+-. ( 44
)
The geometric constraints are given by

ω +,+-+ ω -,+-= √ 2 -1 sin α ∂ -α -∂ + α = √ 2 -1 cos α ω1 ,1 1 = 2i∂1α + ω +,+ 1 = ω -,- 1
ω1 ,+1 = -ω +,+--i∂ + α -i √ 2e iα -1 ω1 ,-1 = -ω +,+-+ i∂ + α ω 1,+- = -i∂ 1 α, ω +,1 1 = 2i∂ + α ω -,1 1 = ω +,-1 = ω -,+1 = ω 1,+1 = ω 1,-1 = 0 . ( 45
)
Thus we can write

de 1 = -ω +,+-+ i √ 2e -iα -1 -i∂ + α e 1 ∧ e + -(ω +,+-+ i∂ + α) e 1 ∧ e --2i∂1α + ω +,+ 1 e 1 ∧ e 1 de + = ω -,-+ e + ∧ e -+ ω -,-1 e 1 ∧ e --i∂ 1 αe 1 ∧ e + + ω -,-1e 1 ∧ e -+ i∂1αe 1 ∧ e + + 2i∂ + αe 1 ∧ e 1 de -= -ω ++-e + ∧ e -+ i ∂ 1 αe 1 -∂1αe 1 ∧ e - + ω +,+1 e 1 + ω +,+ 1e 1 ∧ e + -2i ∂ + α + √ 2 cos α -1 e 1 ∧ e 1 . ( 46
)

Solutions with cos α = 0

For these solutions, it is convenient to define the 1-form

V = 1 cos α (e + -e -) ( 4 7 )
and introduce a local co-ordinate t such that V = ∂ ∂t . It is straightforward to see that the supersymmetry constraints imply that

∂α ∂t = √ 2 -1 (48)
and furthermore

L V e 1 = √ 2i -1 e -iα cos α e 1 L V (e + + e -) = √ 2 -1 tan α(e + + e -) . (49)
These constraints imply that one can write

e 1 = (1 + i tan α)ê 1 e + + e -= √ 2 cos α e 2 (50)
where

L V ê1 = 0, L V e 2 = 0 . (51)
Note, furthermore, that

dê 1 = √ 2 sec αω +,+-- 1 √ 2 sin α cos 2 α (∂ + α + ∂ -α) --1 tan α -2i -1 e 2 ∧ ê1 + sec 2 α(-i∂1α -cos αe -iα ω +,+ 1)ê 1 ∧ ê1 . (52)
and

de 2 = 1 2 sec 2 α (e 2iα ω +,+1 + ω -,-1 )ê 1 + (e -2iα ω +,+ 1 + ω -,-1)ê 1 ∧ e 2 -2i -1 ê1 ∧ ê1 ( 53
)
Note then that the metric can be written as

ds 2 = - 1 2 (e + -e -) 2 + sec 2 α ds 2 GT ( 54
)
where

ds 2 GT = (e 2 ) 2 + 2ê 1 ê1 (55)
The metric on the manifold GT does not depend on t, and (52) and (53) imply that GT admits a t-independent basis E i for i = 1, 2, 3 satisfying

dE i = B ∧ E i --1 ijk E j ∧ E k (56)
where

B = 1 2 sec 2 α (ω -,-1 + e 2iα ω +,+1 )ê 1 + (ω -,-1 + e -2iα ω +,+ 1)ê 1 + √ 2 sec αω +,+-- 1 √ 2 sin α cos 2 α (∂ + α + ∂ -α) --1 tan α e 2 (57)
Note in particular that (56) implies that B must be independent of t, and furthermore, must satisfy

dB = 2 -1 3 B ( 58
)
where 3 denotes the Hodge dual on GT (in our conventions, the volume form on GT is iê 1 ∧ ê1 ∧ e 2 ). In turn, (58) implies that

d 3 B = 0 . ( 59
)
The condition (56) imples that GT admits a Gauduchon-Tod structure. Such structures arise in the context of 4-dimensional hyper-Kähler with torson manifolds which admit a tri-holomorphic isometry, and have been analysed in [START_REF] Gauduchon | [END_REF] and [17].

To proceed further, we next consider the constraints which (46) impose on e +e -. It will be convenient to write

e + -e -= -2 sec α(dt + Ω) (60)
and to set

α = √ 2 -1 t + Φ ( 61
)
where Ω is a t-dependent 1-form on GT and Φ is a t-independent function. Then (46) implies that

L V Ω + 2 √ 2 -1 tan α Ω -B -2 tan α dΦ = 0 . ( 62
)
This condition can be integrated up, and on changing co-ordinates from t to α, one obtains

e + -e -= - √ 2 sec αdα - √ 2 sin α B -2 cos α ψ ( 63
)
where ψ is an α-independent 1-form on GT . Note that ψ is defined in terms of the basis e 1 , ê1 , e 2 as

ψ = 2 √ 2 cos 2 α i(-ω -,-1 + e 2iα ω +,+1 )ê 1 -i(-ω -,-1 + e -2iα ω +,+ 1)ê 1 + 1 2 -sec α(∂ + α + ∂ -α) - √ 2 sin α cos 2 α ( √ 2ω +,+---1 sin α) e 2 . ( 64
)
The remaining content of (46) imposes an additional condition on ψ:

dψ + B ∧ ψ -2 -1 3 ψ = 0 . ( 65
)
It remains to consider the constraints on the fluxes. Note first that (44) implies that

A = - 2 tan α dα + 2 cos 2α B - 1 √ 2 sin 2α ψ . ( 66
)
It is straightforward to show that on applying the exterior derivative to (66), one obtains the components of the field strength given in (43), with no further constraint. In order to evaluate the gauge field equations, observe that the above conditions imply that the Hodge dual of F is given by

F = - √ 2 dα ∧ 1 √ 2 cos 2α B --1 sin 2α ψ + √ 2 cos 2 α B ∧ ψ -sin 2α 3 B - √ 2 -1 cos 2α 3 ψ . ( 67
)
The conditions obtained previously imply that the RHS of this expression is closed with no additional constraint, hence the gauge equations are satisfied.

In order to examine the Einstein equations we follow the reasoning presented (in the context of solutions of the anti-de-Sitter minimal gauged supergravity) in Appendix E of [START_REF] Caldarelli | High Energy Phys[END_REF]. In particular, the integrability conditions of the Killing spinor equation associated with a pseudo-supersymmetric solution for which the Maxwell field strength F satisfies the Bianchi identity and gauge field equations imply that

E µν Γ ν = 0 (68)
where

E µν = R µν -3 -2 g µν -2F µρ F ν ρ + 1 2 F αβ F αβ g µν . (69)
Evaluating (68) acting on the Killing spinor = 1 + e iα e 2 , one finds that all components of E µν are constrained to vanish, i.e. the Einstein equations hold automatically.

To summarize, the solutions with Killing spinor 1 + e iα e 2 and cos α = 0 have metric

ds 2 = -( sec α dα + sin α B + √ 2 cos α ψ) 2 + sec 2 αds 2 GT ( 70
)
where ds 2 GT is an α-independent metric on a 3-dimensional manifold which has a Gauduchon-Tod structure. The 3-manifold GT admits a α-independent basis E i and an α-independent 1-form B satisfying (56) (with associated integrability conditions (58) and ( 59)). GT also admits an α-independent 1-form ψ satisfying (65). The flux is then given by

F = d 2 cos 2α B - 1 √ 2 sin 2α ψ . (71)
Finally, we remark that on making the co-ordinate transformation t = tan α, one finds that the metric can be written in the form originally obtained in [11].

Solutions with cos α = 0

Suppose that sin α = ±1, then

F = ( ± √ 2ω +,+---1 )e + ∧ e - ± 1 √ 2 (e + -e -) ∧ (ω +,+1 e 1 + ω +,+ 1e 1) ( 7 2 )
and

-1 A = 1 2 ω +,+-e + - 1 2 ω -,+-e -+ 1 2 ω +,+1 e 1 + 1 2 ω +,+ 1e 1 (73)
and

ω +,+-+ ω -,+-= ± √ 2 -1 ω1 ,1 1 = ω +,+ 1 = ω -,- 1
ω1 ,+1 + ω +,+-= ± √ 2 -1 ω1 ,-1 = -ω +,+- ω 1,+-= ω +,1 1 = ω -,1 1 = ω +,-1 = ω -,+1 = ω 1,+1 = ω 1,-1 = 0 .( 74
)
It follows that de + = -ω -,+-e + ∧ e -+ (ω +,+1 e 1 + ω +,+ 1e 1) ∧ e - de -= -ω +,+-e + ∧ e -+ (ω +,+1 e 1 + ω +,+ 1e 1) ∧ e +

de 1 = (ω +,+-∓ √ 2 -1 )e + + ω +,+-e -+ ω +,+ 1e 1 ∧ e 1 . ( 75
)
To proceed, note that (75) implies that (e + + e -) ∧ d(e + + e -) = 0, (e +e -) ∧ d(e +e -) = 0 .

Hence, there exist real functions H, B, z, t such that

e + = 1 √ 2 (Hdz -Bdt), e -= 1 √ 2 (Hdz + Bdt) . (77)
Next, note that (72) and (75) imply that

F = ± 1 √ 2 d(e + -e -) . (78)
On comparing this expression with (73), one finds that there exists a function C such that 1 √ 2 (e +e -) = 2 (ω +,+-e + -ω -,+-e -+ ω +,+1 e 1 + ω +,+ 1e 1)

- 2 d log C . (79)
Substituting this expression back into (75) one finds

de 1 = d log C ∧ e 1 (80)
and so there exist real functions C, x, y such that

e 1 = 1 √ 2 C(dx + idy) . (81)
It is then straightforward to show that (75) implies that

H = Cf 1 (z), B = C -1 f 2 (t) ( 8 2 )
where f 1 and f 2 are arbitrary functions of z, t. By making appropriate z, t co-ordinate transformations, one can without loss of generality take f 1 = f 2 = 1. Furthermore, (75) implies that ∂C ∂t = ± -1 (83) so that

C = ± -1 t + V (84)
for V = V (x, y, z). The metric and gauge field strength are then given by

ds 2 = - 1 (V ± -1 t) 2 dt 2 + (V ± -1 t) 2 (dx 2 + dy 2 + dz 2 ) ( 8 5 )
and

F = ∓d 1 V ± -1 t dt . (86)
Finally, we impose the gauge field equations d F = 0, which imply that V is harmonic on R 3 :

∂ 2 ∂x 2 + ∂ 2 ∂y 2 + ∂ 2 ∂z 2 V = 0 , (87)
and we remark that, from the reasoning used in the previous sub-section, this condition is sufficient to ensure that the Einstein equations hold automatically. This solution is the cosmological Majumdar-Papapetrou black hole solution found in [18]. Observe that on taking the limit → ∞ one recovers the standard Majumdar-Papapetrou solution. The cosmological solution (85) is obtained by shifting the harmonic function by a term linear in t; this method of obtaining solutions in de Sitter supergravity has also been investigated in [5,19]

Conclusions

Using spinorial geometry techniques, all pseudo-supersymmetric solutions of minimal de Sitter N = 2, D = 4 supergravity have been classified. There are three classes of solutions:

(i) The first class of solution has metric and field strength

ds 2 = 2du dv + ( -2 v 2 + Ψ)du + 2 (1 + 2 -2 z z) 2 dzdz (88) with F = -1 dv ∧ du . ( 89
)
where Ψ = Ψ(u, z, z) satisfies

∂ 2 Ψ ∂z∂ z = 0 . (90)
(ii) The second class of solution has metric

ds 2 = -( sec α dα + sin α B + √ 2 cos α ψ) 2 + sec 2 αds 2 GT (91)
where ds 2 GT is an α-independent metric on a 3-dimensional manifold which has a Gauduchon-Tod structure. The 3-manifold GT admits a α-independent basis E i and an α-independent 1-form B satisfying

dE i = B ∧ E i --1 ijk E j ∧ E k (92)
together with an α-independent 1-form ψ satisfying

dψ + B ∧ ψ -2 -1 3 ψ = 0 . (93)
The gauge field strength is

F = d 2 cos 2α B - 1 √ 2 sin 2α ψ . (94)
(iii) The third class of solution consists of the cosmological Majumdar-Papapetrou black hole solution found in [18] with

ds 2 = - 1 (V ± -1 t) 2 dt 2 + (V ± -1 t) 2 (dx 2 + dy 2 + dz 2 ) ( 9 5 )
and

F = ∓d 1 V ± -1 t dt (96)
where V = V (x, y, z) satisfies

∂ 2 ∂x 2 + ∂ 2 ∂y 2 + ∂ 2 ∂z 2 V = 0 . ( 97
)
We remark that the solution given in (i) for the special case Ψ = 0 was found in [11]; where it is noted that the spacetime is the Nariai solution [20]; the solution (88) corresponds to gravitational waves propagating in this background [21].

Appendix A. The Linear System

In this appendix we present the decomposition of the Killing spinor equation acting on the spinor = λ1 + µ i e i ; we obtain the following constraints:

∂ + λ + λ - 1 2 ω +,+-- 1 2 ω +,1 1 --1 A + - i √ 2 µ 2 F +-+ F 1 1 + -1 = 0 ∂ + µ 1 + µ 1 - 1 2 ω +,+-+ 1 2 ω +,1 1 --1 A + -ω +,-1 µ 2 = 0 ∂ + µ 2 + ω +,+ 1µ 1 + µ 2 1 2 ω +,+-- 1 2 ω +,1 1 --1 A + = 0 ω +,+1 λ + √ 2iF +1 µ 2 = 0 ∂ -λ + λ - 1 2 ω -,+-- 1 2 ω -,1 1 --1 A -- √ 2iF -1µ 1 = 0 ∂ -µ 1 - √ 2iF -1 λ + µ 1 - 1 2 ω -,+-+ 1 2 ω -,1 1 --1 A --ω -,-1 µ 2 = 0 ∂ -µ 2 + i √ 2 λ F +--F 1 1 --1 + ω -,+ 1µ 1 + µ 2 1 2 ω -,+-- 1 2 ω -,1 1 --1 A -= 0 -ω -,+1 λ - i √ 2 µ 1 F +-+ F 1 1 --1 = 0 ∂ 1 λ + λ - 1 2 ω 1,+-- 1 2 ω 1,1 1 --1 A 1 - i √ 2 µ 1 F +-+ F 1 1 + -1 = 0 ∂ 1 µ 1 + µ 1 - 1 2 ω 1,+-+ 1 2 ω 1,1 1 --1 A 1 -ω 1,-1 µ 2 = 0 ∂ 1 µ 2 + ω 1,+ 1µ 1 + µ 2 1 2 ω 1,+-- 1 2 ω 1,1 1 --1 A 1 = 0 ω 1,+1 λ + √ 2iF +1 µ 1 = 0 ∂1λ + λ - 1 2 ω1 ,+-- 1 2 ω1 ,1 1 --1 A1 + √ 2iF -1µ 2 = 0 ∂1µ 1 + λ i √ 2 -F +-+ F 1 1 --1 + µ 1 - 1 2 ω1 ,+-+ 1 2 ω1 ,1 1 --1 A1 -ω1 ,-1 µ 2 = 0 ∂1µ 2 + √ 2iF + 1λ + ω1 ,+ 1µ 1 + µ 2 1 2 ω1 ,+-- 1 2 ω1 ,1 1 --1 A1 = 0 -ω1 ,+1 λ + i √ 2 µ 2 F +-+ F 1 1 --1 = 0

Acknowledgments

J. G. is supported by the EPSRC grant, EP/F069774/1.