Nilpotent chiral superfield in N=2 supergravity and partial rigid supersymmetry breaking

In the framework of N=2 conformal supergravity in four dimensions, we introduce a nilpotent chiral superfield suitable for the description of partial supersymmetry breaking in maximally supersymmetric spacetimes. As an application, we construct Maxwell-Goldstone multiplet actions for partial N=2 -->N=1 supersymmetry breaking on R x S^3, AdS_3 x S^1 (or its covering AdS_3 x R), and a pp-wave spacetime. In each of these cases, the action coincides with a unique curved-superspace extension of the N=1 supersymmetric Born-Infeld action, which is singled out by the requirement of U(1) duality invariance.


Introduction
Inspired by the work of Antoniadis, Partouche and Taylor [1], Bagger and Galperin [2] constructed the Goldstone-Maxwell multiplet model for partially broken N = 2 Poincaré supersymmetry in four spacetime dimensions (4D). Their model proved to coincide with the N = 1 supersymmetric Born-Infeld action [3,4]. Two years later, Roček and Tseytlin [5] re-derived the model of [2] using N = 2 superfields, building on the earlier formulation due to Roček [6] for the Volkov-Akulov Goldstino model [7] in terms of a nilpotent N = 1 chiral superfield. 1 The N = 2 Minkowski superspace is one of many maximally supersymmetric backgrounds in 4D N = 2 off-shell supergravity. Such superspaces were classified in [10] building on the earlier analysis [11] of maximally supersymmetric backgrounds in 5D N = 1 off-shell supergravity. The construction in [5] is down-to-earth in the sense that it is specifically designed to describe the partial breaking of N = 2 Poincaré supersymmetry. Here we present a theoretical scheme which is suitable for the description of partial supersymmetry breaking in curved maximally supersymmetric backgrounds in 4D N = 2 off-shell supergravity. As an application of this scheme, we construct Maxwell-Goldstone multiplet actions for partial N = 2 → N = 1 supersymmetry breaking on R × S 3 , AdS 3 × S 1 (or its covering AdS 3 × R), and a pp-wave. This paper is organised as follows. In section 2 we introduce a nilpotent chiral superfield coupled to N = 2 conformal supergravity. In section 3 we explain how such a superfield can be used to construct a model for partially broken supersymmetry for certain maximally supersymmetric backgrounds of N = 2 supergravity. The formalism developed is applied in section 4 to re-derive the Roček-Tseytlin construction. In section 5 we construct Maxwell-Goldstone multiplet actions for partial N = 2 → N = 1 supersymmetry breaking on R × S 3 , AdS 3 × S 1 (or its covering AdS 3 × R), and a pp-wave. Concluding comments are given in section 6. The main body of the paper is accompanied by three technical appendices. In Appendices A and B, we present group-theoretic formulations for four-dimensional N = 1 and N = 2 superspaces over U(2) = (S 1 × S 3 )/Z 2 . The maximally N = 2 supersymmetric background over R × S 3 , which is used in section 5, is the universal covering space of the N = 2 superspace over (S 1 ×S 3 )/Z 2 . Appendix A also contains the group-theoretic description of N = 1 superspace over U(1, 1) = (AdS 3 × S 1 )/Z 2 . Appendix C is devoted to the discussion of a unique feature of the anti-de Sitter supersymmetry that distinguishes AdS 4 from the other maximally supersymmetric four-dimensional backgrounds.
2 Nilpotent chiral superfield in N = 2 supergravity In the framework of four-dimensional N = 2 conformal supergravity 2 we introduce a nilpotent chiral superfield constrained byD iα Z = 0 , (2.1a) where G ij is a linear multiplet constrained by G ij G ij = 0. One may interpret G ij as the field strength of a tensor multiplet. The constraints (2.1a)-(2.1c) are invariant under the N = 2 super-Weyl transformations [12,13] if Z is considered to be a primary superfield of dimension 1.
A chiral superfield constrained by (2.1b) was considered in [14] in the context of the dilaton effective action in N = 2 supergravity. In the super-Poincaré case, chiral superfields obeying the constraint (2.1b) with a constant G ij naturally originate in the framework of partial N = 2 → N = 1 supersymmetry breaking [1,15,16].
We recall that the N = 2 tensor multiplet is described in curved superspace by its gauge invariant field strength G ij which is a linear multiplet. The latter is defined to be a real SU(2) triplet (that is, G ij = G ji andḠ ij := G ij = G ij ) subject to the covariant constraints [17,18] These constraints are solved in terms of a chiral prepotential Ψ [19,20,21,22] via which is invariant under Abelian gauge transformations with Λ a reduced chiral superfield,D iα Λ = 0 , (2.5a) 2 In this paper, we use Howe's superspace formulation [12] for N = 2 conformal supergravity and follow the supergravity notation and conventions of [13]. In particular, the superspace covariant derivatives are denoted D A = (D a , D i α ,Dα i ). We make use of the second-order differential operators . The SU(2) triplet S ij = S ji and its conjugateS ij = S ij stand for certain components of the superspace torsion tensor.
We recall that the field strength of an Abelian vector multiplet is a reduced chiral superfield [23].
The constraints on Λ can be solved in terms of the Mezincescu prepotential [24] (see also [19]), U ij = U ji , which is an unconstrained real SU(2) triplet. The curved-superspace solution is [25] Here∆ denotes the chiral projection operator [26,27] where the real unconstrained parameter σ corresponds to the super-Weyl transformations [13]. 3 Here E and E denote the full superspace and chiral densities, respectively.
The constraints (2.1a) and (2.1b) define a deformed reduced chiral superfield. These constraints may be re-cast in the language of superforms as dF = H, where F is a twoform and H is the three-form field strength, dH = 0, describing the tensor multiplet [27], see also [28]. 4 Switching H off, H = 0, turns F into the two-form field strength of the vector multiplet.
The constraint (2.1b) naturally originates as follows. Consider the model for a massive improved tensor multiplet coupled to N = 2 conformal supergravity [29,30]. The action of this model in the form given in [25] is The parameter σ was denoted 2U in [13]. 4 We are grateful to Joseph Novak for this observation.
where µ and e are real parameters, with µ = 0 (the tensor multiplet mass can be shown to be m = µ 2 + e 2 ). The kinetic term involves the composite [31] W which proves to be a reduced chiral superfield. 5 For m = 0 the above action describes the improved tensor multiplet [31]. We introduce a Stückelberg-type extension of the model where W is the field strength of a vector multiplet. The action is invariant under the gauge transformation (2.4) accompanied by The original action (2.9) is obtained from (2.11) by choosing a gauge W = 0. Now one can see that the superfield Z := W + iΨ obeys the constraint (2.1b).
It is well known that the functional is a total derivative. Since the mass term in (2.11) is invariant under the gauge transformation (2.4) and (2.12), it follows that, given a chiral superfield Z constrained by (2.1b), the functional is invariant under the gauge transformation (2.4), δ Λ I = 0.
The constraints (2.1a)-(2.1c) imply that, for certain supergravity backgrounds, the degrees of freedom described by the N = 2 chiral superfield Z are in a one-to-one correspondence with those of an Abelian N = 1 vector multiplet. The specific feature of such N = 2 supergrvaity backgrounds is that they possess an N = 1 subspace of the full N = 2 superspace. This property is not universal. In particular, there exist maximally N = 2 supersymmetric backgrounds with no admissible truncation to N = 1 [10]. 5 The superfield (2.10) is one of the simplest applications of the powerful approach to generate composite reduced chiral multiplets which was presented in [25].
3 Maximally N = 2 supersymmetric backgrounds and partial supersymmetry breaking So far we have discussed an arbitrary supergravity background. Now we restrict our consideration to a maximally supersymmetric background M 4|8 with the property that the chiral prepotential Ψ for G ij may be chosen such that the following two conditions hold. Firstly, the complex linear multiplet is covariantly constant and null, Secondly, the prepotential Ψ may be chosen to be nilpotent, The null condition for G ij + means that G ij + = q i q j , for some isospinor q i . It follows that We are going to show that the following functional is supersymmetric. Here Z is the nilpotent chiral superfield (2.1), which is assumed to be a composite of the dynamical fields. The complex linear multiplet (3.1) and its chiral prepotential Ψ are background fields associated with the background superspace M 4|8 . Since the covariant derivatives D A are invariant under the isometry transformations of M 4|8 , the fields G ij + and Ψ do not change under such transformations. Let ξ be a Killing supervector field for M 4|8 (see section 6.4 of [32] and [33] for general discussions). Then We introduce a reduced chiral superfield W by where U ij is the Mezincescu prepotential for the reduced chiral superfield W . Since Ψδ ξ Ψ = 0, we have In the next two sections, it will be shown that the action describes the Maxwell-Goldstone multiplet for partial N = 2 → N = 1 supersymmetry breaking on the maximally supersymmetric backgrounds specified.
The above derivation does not use the null condition (3.3). The latter is introduced for the N = 2 superspace M 4|8 to possess an N = 1 subspace.

Example: The super-Poincaré case
The simplest maximally supersymmetric background is N = 2 Minkowski superspace. In this superspace, every constant real SU(2) triplet G ij is covariantly constant, where D A = (∂ a , D i α ,Dα i ) are the flat superspace covariant derivatives. Let Ψ be a chiral prepotential for G ij ,D iα Ψ = 0. We represent It is always possible to choose the prepotential Ψ such that the following properties hold: In N = 2 Minkowski superspace, the constraints (2.1a)-(2.1c) turn intō The action (3.10) becomes Since G ij + is constant, it is invariant under the N = 2 supersymmetry transformations. In accordance with the analysis given in the previous section, the action is N = 2 supersymmetric.
For the Grassmann coordinates θ α i andθ jα of N = 2 Minkowski superspace, as well as for the spinor covariant derivatives D i α andDα i , it is useful to label the values of their R-symmetry indices as i, j = 1, 2. Without loss of generality we can choose We can now reproduce the results of [2] from the N = 2 setup described. In order to solve the constraints (4.4), it is useful to carry out a reduction to N = 1 Minkowski superspace.
Given a superfield U(x, θ i ,θ i ) on N = 2 Minkowski superspace, we introduce its barprojection which is a superfield on N = 1 Minkowski superspace with the Grassmann coordinates θ α = θ α 1 andθα =θ 1 α and the spinor covariant derivatives D α = D 1 α andDα =Dα 1 . The background superfield Ψ is characterised by the properties Taking the bar-projection of this constraint gives where we have introduced the N = 1 components of Z: These superfields satisfy the constraints The constraints on W α tell us that it can be interpreted as the field strength of an Abelian N = 1 vector multiplet. The constraint (4.10) is equivalent to the Bagger-Galperin constraint [2]. Its general solution is Upon reduction to N = 1 superspace, the action (4.5) becomes This is the N = 1 supersymmetric Born-Infeld action. Being manifestly N = 1 supersymmetric, the action is also invariant under the second nonlinearly realised supersymmetry transformation [2] (4.14) For completeness, we re-derive this result.
Let U be a scalar superfield on N = 2 Minkowski superspace. Its isometry transformation is where is a Killing supervector field of Minkowski superspace, 6 The Killing supervector field generating the supersymmetry transformation is characterised by the components Applying this transformation to Z gives δ ξ Z = −(ξ a ∂ a + ξ α i D i α )Z. We now consider only the second supersymmetry transformation by choosing ǫ α 1 = 0 and ǫ α 2 = ǫ α . It acts on the N = 1 superfields (4.11a) as follows where we have made use of the constraints obeyed by Z and X. The supersymmetry transformation (4.14) follows from (4.19) upon a rescaling of ǫ α .

Maxwell-Goldstone multiplet for partially broken rigid supersymmetry in curved space
We turn to applying the theoretical framework of section 3 to maximally supersymmetric curved backgrounds in N = 2 supergravity.

Curved N = 2 superspace backgrounds
We consider a maximally supersymmetric background M 4|8 described by the following algebra of covariant derivatives 7 where the torsion tensor G ij a is annihilated by the spinor covariant derivatives, This algebra is obtained from that corresponding to N = 2 conformal supergravity, and given by eq. (2.8) in [13], by (i) switching off the components S ij , Y αβ , W αβ and G αα of the torsion tensor; and (ii) imposing (5.1d). The constraints (5.1d) are required by the theorem [11] that all fermionic components of the superspace torsion tensor must vanish in maximally supersymmetric backgrounds.
In complete analogy with the 5D case [11], the constraints (5.1d) imply the following integrability condition As shown in [11], the general solution of the conditions (5.1d) and (5.2) is for some real vector g b and real SU(2) triplet s kl . The latter may be normalised as Since g 2 = g a g a is constant, D A g 2 = 0, there are in fact three different superspaces described by the above algebra: (iii) in the null case, g 2 = 0, the spacetime geometry is a pp-wave. We will denote these superspaces as M
In order to get some more insight into the structure of the superspace geometry (5.1), a specific value of g 2 has to be fixed. It suffices to consider the superspace M 4|8 T , since the other two cases may be treated similarly. As a supermanifold, M 4|8 T is the universal covering of the 4D N = 2 superspace introduced in Appendix B.
In the case g 2 < 0, it is possible to choose a Lorentz and SU(2) R gauge such that As shown in [10], the algebra of covariant derivatives is equivalent to where we have introduced the "improved" vector covariant derivatives These (anti-)commutation relations correspond to the superalgebra su(2|1) × su(2|1).
The superspace geometry of M 4|8 T can be described, e.g., in terms of the covariant derivativesD A = (D

Goldstone multiplet for partially broken supersymmetry
We consider a maximally supersymmetric background M 4|4 described by the following algebra of N = 1 covariant derivatives: where the torsion tensor G a is covariantly constant, This is a special case of the superspace geometry for N = 1 old minimal supergravity [34] reviewed in [32]. The above algebra is obtained from the supergravity (anti-)commutation relations (5.5.6) and (5.5.7) in [32] by (i) switching off the chiral torsion superfields R and W αβγ and their conjugates; and (ii) imposing the condition (5.8d).
Since G 2 = G a G a is constant, the geometry (5.8) describes three different superspaces, N , respectively, considered in the previous subsection. 8 We recall that the Lorentzian manifolds supported by these superspaces are R×S 3 , AdS 3 × S 1 or its covering AdS 3 × R, and a pp-wave spacetime, respectively. 9 As a supermanifold, The superspace M 4|4 allows the existence of covariantly constant spinors, Such a spinor is constant in a gauge in which the Lorentz connection vanishes.
By analogy with the flat-superspace case, we consider the following N = 1 supersymmetric theory with action where the covariantly chiral superfield X is a unique solution of the constraint The superfield W α is the chiral field strength of an Abelian vector multiplet and, together with its complex conjugateWα, it obeys the Bianchi identity The explicit solution of the constraint (5.11) is a covariantisation of that described in the previous section. It is given, e.g., in [38].
The action (5.10) is invariant under a second supersymmetry given by 8 In the time-like case, G 2 < 0, the graded commutation relations (5.8) are obtained from (5.6) by choosing i, j = 1 and setting G a = g a . 9 N = 1 supersymmetric theories on R × S 3 were studied in the mid-1980s by Sen [35]. At the component level, the maximally N = 1 supersymmetric backgrounds in four dimensions were classified by Festuccia and Seiberg [36]. Their results were re-derived in [37] using the superspace formalism developed in the mid-1990s [32].
with the parameter ǫ α being constrained as in (5.9). Of course, this transformation should be induced by that of W α . The correct supersymmetry transformation of W α proves to be 14) It has the correct flat superspace limit [2], compare with (4.14), and respects the Bianchi identity (5.12), The dynamical system defined by eqs. (5.10) and (5.11) describes the Maxwell-Goldstone multiplet action for partial N = 2 → N = 1 supersymmetry breaking in those curved spacetimes which are supported by the superspace geometry (5.8), including R × S 3 , AdS 3 × S 1 and its covering AdS 3 × R.

Concluding comments
There are five types of maximally supersymmetric backgrounds in four-dimensional N = 1 off-shell supergravity, two of which are well known: Minkowski superspace R 4|4 [39,40] and anti-de Sitter superspace AdS 4|4 [41,42,43]. The remaining three superspaces, In Appendix C we demonstrate that no Maxwell-Goldstone multiplet action for partial N = 2 → N = 1 supersymmetry breaking exists in the case of the anti-de Sitter (AdS) supersymmetry. The reason for this obstruction is the fact that every covariantly constant SU(2) triplet G ij + must be proportional to the torsion tensor S ij , which is real and covariantly constant in AdS 4|8 [44]. As a consequence, the conditions (3.2) and (3.3) are not compatible in AdS 4|8 . Since the N = 1 AdS superspace AdS 4|4 is naturally embedded in AdS 4|8 as a subspace [73], applying the formalism of section 2 to the case of AdS 4|8 allows us to derive a Maxwell-Goldstone multiplet for partially broken N = 2 AdS supersymmetry. The corresponding technical details are spelled out in Appendix C. However, since the conditions (3.2) and (3.3) are not compatible in AdS 4|8 , we cannot use this Maxwell-Goldstone multiplet to construct a supersymmetric invariant action.
There exists a one-parameter family of N = 1 supersymmetric extensions of the Born-Infeld actions [4]. A unique extension is fixed by the requirement that the action should describe the Maxwell-Goldstone multiplet on R 4|4 for partially broken N = 2 Poincaré supersymmetry [2,5]. The same extension is uniquely fixed by the requirement of U(1) duality invariance [45,46], which implies the self-duality under superfield Legendre transform discovered by Bagger and Galperin [2]. A curved-superspace extension of the N = 1 supersymmetric Born-Infeld action is not unique. However, a unique extension is fixed by the requirement of U(1) duality invariance [38]. It is given by the action (5.10) in which X is a unique solution to the constraint with R the chiral scalar torsion superfield. This action was first proposed in [47]. In the case of anti-de Sitter superspace AdS 4|4 , the only non-zero components of the superspace torsion are R andR, which are constant. The corresponding N = 1 supersymmetric Born-Infeld action possesses U(1) duality invariance, however it is not invariant under a second nonlinearly realised supersymmetry, as demonstrated in Appendix C. Therefore, this action is not suitable to describe a partial breaking of the N = 2 AdS supersymmetry.
In addition to the Maxwell-Goldstone multiplet of [2,5], there exist other multiplets for partially broken N = 2 Poincaré supersymmetry [48,5,49]. We believe these models can be generalised to the superspaces M Recently, there has been much interest in models for spontaneously broken local N = 1 supersymmetry [50,51,52,53,54,55,56,57,58], which are based on the use of the nilpotent chiral Goldstino superfield proposed in [59,60]. Other nilpotent Goldstino superfields can be used to describe spontaneously broken N = 1 supergravity [61,62,63] (for an alternative approach to de Sitter supergravity, see [64]). At the moment it is not clear whether the nilpotent N = 2 chiral superfield advocated in the present paper is suitable for the description of partial supersymmetry breaking in N = 2 supergravity. It is certainly of interest to develop a superspace description for the models for spontaneous N = 2 → N = 1 local supersymmetry breaking pioneered in [65,66] and further developed, e.g., in [67,68].

Acknowledgements:
We thank Joseph Novak for comments on the manuscript and suggestions. GT-M is grateful to Daniel Butter for discussions. This work is supported in part by the Australian Research Council (ARC) Discovery Project DP140103925. The work of GT-M is also supported by the Interuniversity Attraction Poles Programme initiated by the Belgian Science Policy (P7/37) and in part by COST Action MP1210 "The String Theory Universe." In this appendix we give supermatrix realisations for two maximally supersymmetric backgrounds in 4D N = 1 supergravity.
It is useful to switch from the variables h and Θ to new ones, ϕ ∈ R, u and θ, defined as follows: We can represent The supermatrix (A.5) is invariant under the Z 2 transformation ϕ → ϕ + π,û → −û and θ → −θ. This is the origin of It turns out that the superspace M 4|4 introduced above can be identified with the group manifold SU(2|1). Indeed, it may be checked that every element g ∈ SU(2|1) has the form (compare with a similar result in [70]) where u is constrained as in (A.5).
The isometry group of M 4|4 is SU(2|1) × U(2). It acts on M 4|4 as follows: These transformations are holomorphic in terms of the variables h and Θ (hence the isometry transformations act on a chiral subspace of the full superspace). The isometry group has two U(1) subgroups that describe R-symmetry transformations and time translations.
One subgroup corresponds to all diagonal supermatrices (A.7) with u = ½ 2 and θ = 0. The other subgroup is spanned by all diagonal matrices e iψ ½ 2 in U(2).
On the group manifold SU(2|1), we can define an action of SU(2|1) × SU(2|1) by the standard rule These transformations leave invariant the supermetric However, such transformations map the chiral subspace (A.3) to itself only if g R ∈ U(2).
The isometry group of M 4|4 is SU(1, 1|1) × U(2). It acts on M 4|4 as follows: These transformations are holomorphic in terms of the variables h and Θ (hence the isometry transformations acts on a chiral subspace of the full superspace), and leave invariant the supermetric Unlike the superspace considered in the previous subsection, the dimension parametrised by ϕ is now space-like.
Let us consider the coset space where the subgroup U(1) of SU(1, 1|1) consists of all diagonal supermatrices (A.13) with u = ½ 2 and θ = 0. This coset space may be seen to coincide with the 3D (2,0) anti-de Sitter superspace [71]. We recall that in three dimensions, N -extended anti-de Sitter (AdS) superspace exists in several incarnations known as (p, q) AdS superspaces, where the non-negative integers p ≥ q are such that N = p + q. The conformally flat (p, q) AdS superspace is In the case p = N ≥ 4 and q = 0, non-conformally flat AdS superspaces also exist [72].
for some group element h ∈ U(2).
The isometry group of M 4|8 is Given a group element g = g L ×g R ×e iψ ∈ G, with ψ ∈ R, it acts on the pair P = (P L , P R ) by the rule: The equivalence relation allows us to choose P R in the form: The above construction can readily be modified in order to describe the N = 2 superspace over U(1, 1) = (AdS 3 × S 1 )/Z 2 .

C Example: The anti-de Sitter supersymmetry
In this appendix we show that the formalism of sections 2 and 3 can be used to define a Goldstone-Maxwell multiplet for partially broken 4D N = 2 anti-de Sitter (AdS) supersymmetry with the following properties: (i) it is the standard Maxwell multiplet with respect to the N = 1 AdS supersymmetry; (ii) it transforms nonlinearly under the second AdS supersymmetry. However, making use of this multiplet does not allow one to construct an invariant action describing the partial N = 2 → N = 1 AdS supersymmetry breaking.
To start with, we recall a few definitions concerning the 4D N = 2 AdS superspace which is a maximally symmetric geometry that originates within the off-shell formulation for N = 2 conformal supergravity developed in [44]. For comprehensive studies of N = 2 supersymmetric field theories in AdS 4 , the reader is referred to [73,74].
The isometry transformations of AdS 4|8 form the supergroup OSp(2|4). In the infinitesimal case, an isometry transformation is described by a Killing supervector field for some real antisymmetric tensor l bc (z) and scalar ρ(z) parameters. It turns out that the Killing equation (C.4) uniquely determines the parameters ξ α i , l cd and ρ in terms of ξ a . A similar property exists for superspace isometry transformations in any number of dimensions [33]. The specific feature of the 4D N = 2 AdS superspace is that the parameters ξ A and l ab are uniquely expressed in terms of ρ [73].
Due to (C.2), the SU(2) gauge freedom can be used to choose the SU(2) connection Φ A ij in (C.1) to look like Φ A ij = Φ A S ij , for some one-form Φ A describing the residual U(1) connection associated with the generator S ij J ij . Then S ij becomes a constant iso-triplet, S ij = const. The remaining global SU(2) rotations can take S ij to any position on the two-sphere of radius S. We make the choice with |µ| = S. This choice must be used in order to embed an N = 1 AdS superspace, AdS 4|4 , into the full N = 2 AdS superspace [73].
As already mentioned, the choice S 12 = 0 is required for embedding AdS 4|4 into AdS 4|8 . By applying certain general coordinate and local U(1) transformations in AdS 4|8 , it is possible to identify AdS 4|4 with the surface θ µ 2 = 0 andθ 2 µ = 0. The covariant derivatives for AdS 4|4 , are related to (C.1) as follows and similarly for the vector covariant derivative. Here the bar-projection is defined by for any N = 2 tensor superfield U(x, θ ı ,θ ı ). It follows from (C.2) that the N = 1 covariant derivatives obey the algebra which indeed corresponds to the N = 1 AdS superspace (see [32] for more details). As a result, every N = 2 supersymmetric field theory in AdS 4|8 can be reformulated as some theory in AdS 4|4 .
The parameters λ and ω bc obey the equation which defines the Killing supervector field of AdS 4|4 [32]. The second and third terms on the right of (C.14) prove to describe the second supersymmetry and U(1) transformations. The corresponding parameters ε α ,εα and ε have the properties The parameter ε was originally introduced in [75].
We are now prepared to analyse the nilpotent N = 2 chiral superfield Z constrained by (2.1) in the case that the background superspace is AdS 4|8 . We recall that a necessary ingredient of the construction described in section 3 is that G ij is covariantly constant, D A G ij = 0. We require this condition to hold in AdS 4|8 , which implies that G ij is proportional to S ij where κ is a real constant. In accordance with (C.5), we have G 12 = 0. The parameter κ can be chosen to have any given non-zero value by means of rescaling the chiral superfield Z. We choose κ = |µ|, and hence G 11 = −|µ|μ and G 22 = −|µ|µ.
The degrees of freedom described by Z are those of an Abelian N = 1 vector multiplet in AdS 4|4 . Indeed, upon reduction to the N = 1 AdS superspace, the N = 2 chiral scalar Z leads to two chiral superfields, X and W α , defined as Eq. (C.19a) tells us that W α is the chiral field strength of a Maxwell multiplet in AdS 4|4 .
Eq. (C.19b) is of the same type as the constraint (6.1), which generates the N = 1 locally supersymmetric Born-Infeld action with U(1) duality invariance. The constraint (C.19b) is uniquely solved by expressing X in terms of W 2 andW 2 and their covariant derivatives, in complete analogy with the general supergravity analysis of [38].
In accordance with (C.10), the infinitesimal OSp(2|4) transformation of Z is δZ = −ξZ. Using this result, it is straightforward to derive the transformation laws of X and W α under the second supersymmetry and U(1) transformations described by the superfield parameter ε. Making use of the constraints obeyed by Z and X, we obtain One can check that δ ε X and δ ε W α preserve the constraints (C. 19). Due to (C.16), the variation δ ε W α can be rewritten in the form where the real scalar V denotes the unconstrained prepotential of the vector multiplet, Eq. (C.23) is a unique feature that distinguishes AdS 4 from the other maximally supersymmetric backgrounds we have studied in this paper. 10 In accordance with (C.19b), the overall coefficient in (C.21) is chosen such that the kinetic term for the vector multiplet is canonically normalised, S = 1 4 d 4 x d 2 θ E W 2 + c.c. + interaction terms. It should be remarked that the functional Re µ d 4 x d 2 θ E X is a total derivative, in accordance with (C.19b).