AdS superprojectors

Within the framework of ${\mathcal N}=1$ anti-de Sitter (AdS) supersymmetry in four dimensions, we derive superspin projection operators (or superprojectors). For a tensor superfield $\mathfrak{V}_{\alpha(m)\dot{\alpha} (n)} := \mathfrak{V}_{(\alpha_1...\alpha_m) (\dot \alpha_1...\dot \alpha_n)}$ on AdS superspace, with $m$ and $n$ non-negative integers, the corresponding superprojector turns $\mathfrak{V}_{\alpha(m)\dot \alpha(n)} $ into a multiplet with the properties of a conserved conformal supercurrent. It is demonstrated that the poles of such superprojectors correspond to (partially) massless multiplets, and the associated gauge transformations are derived. We give a systematic discussion of how to realise the unitary and the partially massless representations of the ${\mathcal N}=1$ AdS${}_4$ superalgebra $\mathfrak{osp} (1|4)$ in terms of on-shell superfields. As an example, we present an off-shell model for the massive gravitino multiplet in AdS$_4$. We also prove that the gauge-invariant actions for superconformal higher-spin multiplets factorise into products of minimal second-order differential operators.

Ref. [15] described the projectors to the spaces of superfields which are constrained by (1.2), (1.3) and (1.4), see eq. (3.10) below. However, these are not supersymmetric extensions of the Behrends-Fronsdal spin projection operators [13,14], since all independent component fields of the resulting superfield are unconstrained. Supersymmetric spin projection operators are required to turn an unconstrained superfield on AdS 4|4 into one with the properties of a conserved current supermultiplet. It is pertinent to describe the structure of conformal supercurrents in AdS 4|4 following the more general analysis of conserved current supermultiplets in a supergravity background [20].
Let m and n be positive integers. A tensor superfield J α(m)α(n) on AdS 4|4 is called a conformal supercurrent of Lorentz type (m/2, n/2) if it obeys the two constraints If m = n, it is consistent to restrict J α(n)α(n) to be real,J α(n)α(n) = J α(n)α(n) . The m = n = 1 case corresponds to the ordinary conformal supercurrent [21]. The case m = n > 1 was first described in Minkowski superspace in [22] (see also [23]) and extended to AdS 4|4 in [24]. The case m = n + 1 > 1 was first described in Minkowski superspace in [23] and extended to AdS 4|4 in [24].
If m > n = 0, the constraints (1.5) should be replaced with The m = 1 case was first considered in [25], where it was shown that the spinor supercurrent J α naturally originates from the reduction of the conformal N = 2 supercurrent [26] to N = 1 superspace.
Finally, for m = n = 0 the constraints (1.6) should be replaced with These constraints describe a flavour current supermultiplet [27] in AdS 4 . Irreducible supermultiplets of the types (1.5), (1.6) and (1.7) have been used in [28] for the covariant quantisation of the massless supersymmetric higher-spin gauge theories in AdS 4 [16].
The Behrends-Fronsdal projectors have been generalised to the case of AdS 4 only recently in [29]. One of the important outcomes of [29] was a new understanding of the so-called partially massless fields in AdS 4 . Such fields in diverse dimensions were studied earlier in [30][31][32][33][34][35][36][37][38][39][40][41][42][43]. Specifically, it was shown in [29] that the partially massless fields are associated with the poles of the spin projection operators in AdS 4 . In the present paper we provide an N = 1 supersymmetric extension of the AdS 4 spin projection operators and describe various applications of the resulting superprojectors. In particular, we will demonstrate that the partially massless N = 1 supermultiplets in AdS 4 , which have recently been analysed in the component settings in [44,45], are naturally associated with the poles of the proposed superprojectors. This paper is organised as follows. Section 2 provides a brief review of the unitary representations of the AdS algebra so (3,2) and its N = 1 supersymmetric extension osp(1|4). We also review the structure of on-shell fields in AdS 4 . Section 3 is devoted to the construction of spin projection operators that map every unconstrained superfield on AdS 4|4 into one with the properties of a conserved current supermultiplet. On-shell supermultiplets in AdS 4 are studied in section 4. Section 5 is devoted to the component structure of the on-shell supermultiplets. In section 6 we present an off-shell model for the massive gravitino (superspin-1) multiplet. Factorisation of the superconformal higherspin actions is described in section 7 and concluding comments are given in section 8. The main body of the paper is accompanied by two appendices. Appendix A contains a list of identities that are indispensable for the derivation of many results in this paper. Appendix B is devoted to the derivation of partially massless gauge transformations, both in the non-supersymmetric and supersymmetric cases.
Throughout this work we will make use of various abbreviations to denote different types of irreducible superfields. In particular, a superfield J α(m)α(n) satisfying (1.5) is simultaneously transverse linear and transverse anti-linear, and will be called TLAL. A superfield J α(m) satisfying (1.6) is simultaneously linear and transverse anti-linear, and will be called LTAL. Finally, a scalar superfield J satisfying (1.7) is simultaneously linear and anti-linear, and will be called LAL.

Representations of the AdS (super)algebra
In this section we collate the well-known facts concerning (i) the unitary representations of the AdS 4 algebra so (3,2) and its N = 1 supersymmetric extension osp(1|4); and (ii) the on-shell fields in AdS 4 .
The irreducible unitary representations of the AdS 4 algebra so (3,2) are specified by the lowest value E 0 of the energy E and spin s, and are traditionally denoted by D(E 0 , s). 3 The allowed spin values are s = 0, 1/2, 1, · · · , the same as in Minkowski space. However, unlike in Minkowski space, unitarity imposes a bound on the allowed values of energy. According to the theorems proved in [52,53], D(E 0 , s) is unitary iff one of the following conditions holds: representations D 1 2 , 0 = Rac and D 1, 1 2 = Di are known as the Dirac singletons [47]. 4 The representations D(s + 1, s) for s > 0 and D(1, 0) ⊕ D(2, 0) are called massless since they contract to the massless discrete helicity representations of the Poincaré group [54]. These representations prove to be restrictions of certain unitary representations of the conformal algebra so(4, 2) to so (3,2) [54,60]. The remaining representations D(E 0 , s) are usually referred to as the massive AdS 4 representations.
Irreducible unitary representations of the N = 1 AdS 4 superalgebra osp(1|4), which we will call supermultiplets, are conveniently described by decomposing them into irreducible representations of so (3,2) in analogy with the case of the super Poincaré algebra in flat space. Each term in the decomposition is identified with a particle (or field) carrying definite energy and spin. Unlike in flat space, particles in AdS 4 belonging to the same This implies that a massless supermultiplet in AdS 4 consists of two physical component fields. Finally, the fourth type of representation occurs for s = 0, E 0 = 1 2 and is called the Dirac supermultiplet [59,61] (or super-singleton) It unifies the bosonic and fermionic singletons.

On-shell fields in AdS
The geometry of AdS 4 can be described in terms of torsion-free Lorentz-covariant derivatives of the form where e a m is the inverse vielbein, and is the Lorentz connection. The Lorentz generators M bc ⇔ (M βγ ,Mβγ) are defined in appendix A. The algebra of AdS 4 covariant derivatives is where the parameter µμ > 0 is related to the AdS 4 scalar curvature via R = −12µμ. The complex parameter µ appears explicitly only in the algebra of AdS 4|4 covariant derivatives given by eq. (3.3).
Below we make use of the quadratic Casimir of the AdS 4 group, whose realisation on tensor fields is One should keep in mind that ∇ a ∇ a = − 1 2 ∇ αα ∇ αα . Given two positive integers m and n, a tensor field h α(m)α(n) of Lorentz type (m/2, n/2) in AdS 4 is said to be on-shell if it satisfies the equations (2.9b) We say that such a field describes a spin s = 1 2 (m + n) particle with pseudo-mass ρ. The following general result holds: see [56,57] for pedagogical derivations.
As the name suggests, the pseudo-mass does not coincide with what is usually considered to be the physical mass ρ phys . Rather, the two are related through where τ (1,m,n) is one of the partial massless parameters (2.17). The on-shell field h α(m)α(n) corresponds to the irreducible representation D E 0 , 1 2 (m + n) where the minimal energy E 0 is related to the physical mass through The unitarity bound for m + n > 1 is E 0 ≥ 1 2 (m + n + 2), which in terms of the masses is With these relations in mind, we usually prefer to use the pseudo-mass as a representation label in place of E 0 . As a caveat we note that there are two distinct values of E 0 leading to the same value of ρ 2 phys , (2.14) However, the solution (E 0 ) − always violates the unitarity bound for m+n > 1. Thus when referring to a unitary representation with s > 1 2 and pseudo-mass ρ 2 , we are implicitly referring to the representation corresponding to (E 0 ) + , For s = 0, 1 2 (or, equivalently, m + n = 0, 1), the unitary bound is E 0 ≥ s + 1 2 , and the solution (E 0 ) − in (2.14) does not violate the unitarity bound for certain values of ρ 2 phys . For s = 0 the values of ρ 2 phys leading to (E 0 ) − ≥ 1 2 are restricted by the condition − 1 4 |µ| 2 ≤ ρ 2 phys ≤ 3 4 |µ| 2 , which is known as the Breitenlohner-Freedman bound [63].

Partially massless and massive fields
Given two positive integers m and n, the tensor field h (t) α(m)α(n) is said to be partially massless with depth-t if it satisfies the on-shell conditions (2.9) such that its pseudo-mass takes the special value [29,38,39,42] where the dimensionless constants τ (t,m,n) are defined by The specific feature of partially massless fields is that, for a fixed t, the system of equations (2.9) and (2.16) admits a depth-t gauge symmetry This is true as long as the gauge parameter ζ is also on-shell with the same pseudo-mass, It was demonstrated in [29] that the parameters τ (t,m,n) defined by (2.17) determine the poles of the off-shell transverse projection operator for fields of Lorentz type (m/2, n/2) in AdS 4 . This observation leads to a new understanding of the partially massless fields. Specifically, the gauge symmetry (2.18) of the field h (t) α(m)α(n) is associated with the pole ρ 2 = τ (t,m,n) µμ of the corresponding spin projection operator in AdS 4 . In appendix B we provide a systematic derivation of this claim using the spin projection operators of [29].
Strictly massless fields carry depth t = 1 and therefore have mass given by This saturates the bound (2.13) and hence defines a unitary representation of so (3,2). However, on account of the inequality the true partially massless representations are non-unitary. In particular, this means that there are two minimal energy values, which are equally valid since they both violate the unitarity bound. In this work, whenever this ambiguity arises, we always implicitly choose the positive branch, E 0 ≡ (E 0 ) + . To distinguish the true partially-massless representations with depth t and Lorentz type (m/2, n/2), we will employ the notation P t, m, n , 2 ≤ t ≤ min(m, n) . (2.23) Such a representation carries minimal energy E 0 = 1 2 (m + n) − t + 2. The massive representation of so(3, 2) with spin s = 1 2 (m + n) is realised on the space of fields h α(m)α(n) satisfying the equations (2.9) in which ρ is constrained by but is otherwise arbitrary. This restriction ensures the unitarity of the representation.

AdS superprojectors
In this section we construct spin projection operators that map every unconstrained superfield on AdS 4|4 into one with the properties of a conserved current supermultiplet.

AdS superspace
Historically, the N = 1 AdS superspace, AdS 4|4 , was originally introduced [15,64,65] as the coset space AdS 4|4 := OSp(1|4)/SO (3,1). The same superspace is equivalently realised as a unique maximally symmetric solution of the two off-shell formulations for N = 1 AdS supergravity: (i) the well-known minimal theory (see, e.g., [6,66] for reviews); and (ii) the more recently discovered non-minimal theory [67]. Here we prefer to use the second definition, since it allows us to obtain all information about the geometry of AdS 4|4 from the well-known supergravity results.
As usual, we denote by z M = (x m , θ µ ,θμ) the local coordinates of AdS 4|4 . The geometry of AdS 4|4 is described in terms of covariant derivatives of the form where E A M is the inverse superspace vielbein, and is the Lorentz connection. The covariant derivatives obey the following graded commutation relations: Here µ = 0 is a complex parameter. We recall that µ is related to the scalar curvature R of AdS 4 by the rule R = −12µμ.
In what follows, we make extensive use of the quadratic Casimir of the AdS superalgebra, whose realisation on superfields takes the form [28] It is an instructive exercise to check, using the relations (3.3), that (3.4b) holds.
As discussed in section 1, a complex tensor superfield Γ α(m)α(n) is said to be transverse linear if it satisfies the constraint (1.2), whilst its conjugate obeys the constraint D βΓ βα(n−1)α(m) = 0 ⇐⇒ (D 2 − 2(n + 2)μ)Γ α(n)α(m) = 0 (3.5) and is called transverse anti-linear. Similarly, a complex superfield G α(m)α(n) is said to be longitudinal linear if it satisfies the constraint (1.3), whilst its conjugate is constrained by and is called longitudinal anti-linear. We recall that for n = 0 the constraint (1.3) defines a chiral superfield,Dα We also recall that for n = 0 the first constraint in (1.2) is not defined, however one can consistently define complex linear superfields constrained by (1.4).
Ref. [15] introduced projectors P ⊥ n and P n which map the space of unconstrained superfields V α(m)α(n) to the space of transverse linear (1.2) and longitudinal linear superfields (1.3), respectively. These operators have the form and satisfy the projector properties P ⊥ n P n = 0 , P n P ⊥ n = 0 , P ⊥ n + P n = 1 . These properties imply that any superfield V α(m)α(n) can be uniquely represented as a sum of transverse linear and longitudinal linear superfields, which agrees with (3.8). It should be pointed out that the projectors (3.10) are nonanalytic in µ.
Finally, let us introduce the following operators: By making use of the identities in appendix A, one may show that when restricted to the space of scalar superfields they satisfy the projector properties for i = 0, +, −. Here P (+) and P (−) are the chiral and antichiral projectors, respectively, while P (0) projects onto the space of LAL superfields. Using these projectors, it is always possible to decompose an unconstrained complex scalar superfield V as where the complex superfield L is simultaneously linear and anti-linear, σ is chiral andρ is anti-chiral If V is real, then L is also real andρ =σ. In the flat superspace limit, the projectors reduce to those constructed by Salam and Strathdee [1].

Construction of superspin projection operators
Our construction of AdS 4 superprojectors will be based on certain properties of the following theories: (i) the massless supersymmetric higher-spin gauge models in AdS 4 [16]; and (ii) the superconformal higher-spin (SCHS) gauge theories in AdS 4 [20,23,68].
Let s be a positive integer. We recall that there are two dually equivalent gauge formulations for the massless superspin-(s + 1 2 ) multiplet [16]. They both involve the same superconformal prepotential, H α(s)α(s) , but different compensating multiplets. The prepotential H α(s)α(s) is real and possesses the gauge freedom with the gauge parameter ζ α(s)α(s−1) being unconstrained. Associated with H α(s)α(s) is the gauge-invariant chiral field strength The field strength W α(2s+1) (H) and its conjugate are the only independent gauge-invariant field strengths which survive on the mass shell.
The massless superspin-s multiplet can be described using a superconformal prepotential Ψ α(s)α(s−1) , its conjugateΨ α(s−1)α(s) and certain compensating multiplets [16,24]. The gauge transformation law of Ψ α(s)α(s−1) is with the gauge parameters ζ α(s)α(s−2) and ξ α(s−1)α(s−1) being unconstrained. Associated with Ψ α(s)α(s−1) is the gauge-invariant chiral field strength The field strength W α(2s) (Ψ) and its conjugate prove to be the only independent gaugeinvariant field strengths which survive on the mass shell. 6 The gauge prepotentials H α(s)α(s) and Ψ α(s)α(s−1) and the corresponding gauge-invariant field strengths (3.18) and (3.20) were used to construct superconformal higher-spin theories [23,68]. More general superconformal higher-spin models in AdS 4 can be introduced by making use of gauge prepotentials Φ α(m)α(n) , with m and n positive integers. 7 The superfield Φ α(m)α(n) is defined modulo the gauge transformations [23,68] with the gauge parameters being unconstrained. Associated with Φ α(m)α(n) and its complex conjugateΦ α(n)α(m) are the chiral field strengths (also known as the linearised higherspin super-Weyl tensors) which are invariant under the gauge transformations (3.21) The gauge-invariant action S (m,n) SCHS [Φ,Φ], which describes the dynamics of Φ α(m)α(n) and its conjugateΦ α(n)α(m) , is typically written as a functional over the chiral subspace of the full superspace [23,68]. The specific feature of AdS 4|4 is the identity [69] which relates the integration over the chiral subspace to that over the full superspace; here E denotes the chiral integration measure and where we have denoted d 4|4 z = d 4 xd 2 θd 2θ . Upon integrating by parts, the action (3.25) may be written in the suggestive forms They are suggestive because, in addition to being gauge invariant, the linearised higherspin super-Bach tensors 8 are simultaneously transverse linear and transverse anti-linear (TLAL) Our ansatz for the TLAL projectors is based on the higher-derivative descendants (3.27a) and (3.27b), since the latter are each TLAL. However, it is clear that any projector must preserve the rank of the tensor superfield on which it acts. This can be achieved by appropriately removing or inserting vector derivatives in (3.27) to convert the indices. With these remarks in mind, for positive integers m and n, we define the following two differential operators by their action on an unconstrained superfield Φ α(m)α(n) From here one can define the operators where the λ (t,m,n) are dimensionless constants defined by The operators (3.30) project onto the space of TLAL superfields The operators (3.29) also satisfy (3.31), but it is only the operators (3.30) which square to themselves and are hence TLAL projectors. Furthermore, one can consider (say) Π (m,n) to be the unique TLAL projector, since it can be shown that the two types of projectors (3.30a) and (3.30b) actually coincide In the case when m > n = 0, the operators analogous to (3.29) take the form These operators square to themselves but, in contrast to (3.31), they project onto the subspace of simultaneously linear and transverse anti-linear (LTAL) superfields, Once again one may show that the two types of projectors coincide, Π α(m) (Φ) = Π α(m) (Φ).
In the case when n > m = 0, the corresponding projectors Πα (n) (Φ) and Πα (n) (Φ) can be obtained by complex conjugation and similar comments apply. This time however they project onto the subspace of simultaneously anti-linear and transverse linear superfields.
When both m = 0 and n = 0, it may be shown that the orthogonal complement acting on Φ α(m)α(n) can be expressed as the following sum for some unconstrained superfields Ψ α(m)α(n−1) and Ω α(m−1)α(n) . We see that Π If instead m > 0 and n = 0, the orthogonal complement may be written as for some unconstrained Ω α(m−1) and chiral Λ α(m) superfields. On the otherhand, if n > 0 and m = 0, then the orthogonal complement splits into the sum of a longitudinal linear superfield and an anti-chiral superfield, As the first application of the the superprojectors, we provide new expressions for the superconformal actions (3.25). For the higher-spin gauge supermultiplets H α(s)α(s) and Ψ α(s)α(s−1) they take the simple and elegant forms S (s,s) These expressions are analogous to those provided for the conformal higher-spin models in Minkowski space [71] and in AdS 4 [29].

Decomposing unconstrained superfields into irreducible parts
In section 3.2 it was shown that the operator Π (m,n) projects every unconstrained superfield Φ α(m)α(n) to the space of TLAL superfields, whilst its complement Π (m,n) || projects onto the union of the spaces of longitudinal linear and anti-linear superfields. This allows us to decompose Φ α(m)α(n) as follows where φ α(m)α(n) is TLAL and is hence irreducible, whilst ζ α(m)α(n−1) and ξ α(m−1)α(n) are unconstrained and thus reducible. One can then repeat this decomposition on the lowerrank unconstrained superfields in (3.41). After reiterating this procedure a finite number of times, one eventually arrives at a decomposition of Φ α(m)α(n) solely in terms of irreducible superfields. In presenting the resulting decompositions, we will use the notational convention introduced in [72] and adopted in [16]: In the m > n case, the result of this process is for some irreducible complex superfields φ, ψ, χ and ϕ whose properties are summarised in table 1. The superfields σ α(m−n) and σ α(m−n−1) in (3.43) are chiral, If instead m = n = s, then we may further impose the reality condition In contrast to (3.43), H α(s)α(s) now decomposes as for some irreducible complex superfield χ α(s−t)α(s−t−1) and irreducible real superfields  The above decompositions were used, albeit without derivation, in [28] for the covariant quantisation of the massless supersymmetric higher-spin models in AdS 4 [16].

On-shell supermultiplets in AdS
In this section we confine our attention to on-shell superfields φ α(m)α(n) . When m = 0 and n = 0 we define these to satisfy the mass-shell equation and the irreducibility conditions Hence the superfield φ α(m)α(n) is simultaneously transverse linear and transverse antilinear (TLAL), and is said to have super-spin s = 1 2 (m + n + 1) and pseudo-mass M. In the case when m > n = 0 and n > m = 0, the condition (4.1b) should be modified to respectively, whilst (4.1a) remains the same. In the case when m = n = 0, the on-shell superfields are discussed in sections 5.4 and 5.5.
In analogy with (2.11), we define the physical mass M phys through As discussed in section 2, the pseudo-mass of an on-shell non-supersymmetric field is subject to a unitarity bound. Since the on-shell supermultiplet φ α(m)α(n) contains a multitude of such fields, this in turn induces a unitarity bound on the pseudo-mass of φ α(m)α(n) : By going to components, we will see in section 5 that the on-shell supermultiplet

Massless supermultiplets
Massless supermultiplets correspond to those on-shell superfields carrying pseudo-mass There are at least two different angles from which this can be understood.
The first is the fact that the system of equations (4.1) with m ≥ n > 0 and with M 2 satisfying (4.6) is compatible with gauge transformations of the form (3.21), where the gauge parameters ζ α(m)α(n−1) and ξ α(m−1)α(n) are TLAL and obey the constraints Thus, from (4.8) and the gauge variation of (4.1a), we see that the gauge parameters are only non-zero if M satisfies (4.6).
When m = n = s one can consistently impose the reality condition H α(s)α(s) := φ α(s)α(s) =H α(s)α(s) , whereupon the gauge transformations (4.7a) take the form whilst (4.7b) and (4.7c) become The presence of the gauge freedom (4.7) means that the OSp(1|4) representation (4.1) with M 2 given by (4.6) is not irreducible. To obtain an irreducible representation, the space of TLAL superfields (4.1) and (4.6) has to be factorised with respect to the gauge modes. More specifically, two superfields φ α(m)α(n) and φ α(m)α(n) are said to be equivalent if they differ by a gauge transformation (4.7). The genuine massless representation is realised on the quotient space of the space of on-shell superfields (4.1) and (4.6) with respect to this equivalence relation. The gauge degrees of freedom are automatically eliminated if one works with the gauge-invariant chiral field strength (3.22a), instead of the prepotential φ α(m)α(n) . Making use of the constraint (4.1b) satisfied by φ α(m)α(n) , one may show that W α(m+n+1) (φ) is constrained by where the second relation follows from the chirality of the field strength. Once again, the latter is consistent with (4.1a) only if M satisfies (4.6). The description in terms of W α(m+n+1) (φ) provides the second way to formulate massless dynamics.
Let us return to the gauge-invariant chiral field strengths (3.18) and (3.20), which originate in the massless models for the superspin-(s + 1 2 ) and superspin-s multiplets in AdS 4 [16], respectively. As was demonstrated in [16], on the mass shell they satisfy the equations which are exactly the on-shell constraints (4.11).

Partially massless supermultiplets
The transverse projection operators for non-supersymmetric tensor fields in AdS 4 were constructed recently in [29]. There it was observed that the poles of the projectors were intimately related to (partially) massless fields. The structure of on-shell N = 1 supermultiplets containing partially massless fields was recently discussed at the component level in [44,45]. However, their realisations in terms of on-shell superfields are not yet known. In this section we address this gap.
In the spirit of [29], it is natural to expect that the partially-massless supermultiplets are associated with the poles of the TLAL projectors (3.30). This motivates the definition of a partially massless superfield to be one satisfying the on-shell constraints (4.1) such that the pseudo-mass takes one of the values (3.30c), We will say that φ α(m)α(n) carries 'super-depth' t. In section 5 we show that, with the above definitions, the surviving component fields of an on-shell partially massless superfield with super-depth t > 1 are themselves partially massless with depths t or t − 1. We would like to point out that λ (t,m,n) with 1 ≤ t ≤ max(m + 1, n + 1) appear in the poles of the projectors (3.30). The reason that we have chosen t = min(m + 1, n + 1) as an upper bound on t in the definition (4.13) is because, as we will see, there is no gauge symmetry present for min(m + 1, n + 1) < t ≤ max(m + 1, n + 1).
In the previous section it was observed that on-shell, (4.6) is the only pseudo-mass value compatible with the massless gauge symmetry (4.7a). For true partially massless supermultiplets the story is considerably more complicated. This is because the correct gauge symmetry at the superspace level is not yet known, much less the gauge invariant actions and field strengths. However, by making use of the superprojectors proposed in section 3, it is possible to systematically derive the most general gauge symmetry compatible with the on-shell conditions (4.1). In appendix B this procedure is carried out in detail for the real supermultiplet H αα . The results of this analysis, and that of the supermultiplet H α(2)α(2) obtained by analogy, are summarised below. In table 3, 9 the most general gauge symmetries for the aforementioned supermultiplets are given. In both cases it turns out that a gauge symmetry is present only when the pseudo-mass takes the values specified in (4.13). The gauge parameters are on-shell in the sense that they each possess the same pseudo-mass as its parent gauge field and that they satisfy certain irreducibility constraints. Specifically, in the super-depth t = 1 case, the gauge parameters ζ α and ζ α(2)α in table 3 are LTAL and TLAL, respectively, and obey the reality conditions (4.9b) and (4.9c). The gauge parameter ξ α , which corresponds to H α(2)α(2) with super-depth t = 2, is LTAL and obeys the reality condition Dα β ξ β = 3iμξα .
(4. 16) In the case of maximal super-depth, the gauge parameters σ and η are chiral and obey the reality conditions The method used to derive the above results (see appendix B.2 for the details) is quite general in that it deduces all types of gauge symmetry an on-shell supermultiplet can possess, and at which mass values they appear. However, for higher-rank multiplets it quickly becomes infeasible due to the computational expense. Nevertheless, from the results of the above analysis, we are now in a position to make an ansatz for the gauge transformations of the half-integer superspin-(s + 1 2 ) multiplets H (t) α(s)α(s) . For true partially-massless multiplets with super-depth 2 ≤ t ≤ s + 1, they take the form 10 The system of equations (4.1) and (4.13) is invariant under the transformations (4.18a) as long as the gauge parameters are TLAL and satisfy the reality conditions The same is true for the transformations (4.18b) given that σ is chiral,Dασ = 0, and that it satisfies the equations It may be shown that (4.19) and (4.20) imply that each gauge parameter satisfies the same mass-shell equation as its corresponding gauge field, as required. We note that upon substituting t = 1 into (4.18a) and (4.19), one recovers the massless gauge transformations (4.9) with ξ α(s)α(s−1) ≡ ζ α(s)α(s−1) .

Massive supermultiplets
We define a massive superfield φ α(m)α(n) to be one satisfying the on-shell conditions (4.1) and whose pseudo-mass satisfies but is otherwise arbitrary. By virtue of (4.4) this definition ensures that the physical mass is positive and that the corresponding representation of OSp(1|4) is unitary.

Component analysis
We now turn to the component analysis of the on-shell superfield φ α(m)α(n) satisfying the conditions (4.1) with arbitrary M 2 . For this we make use of the bar-projection of a tensor superfield V = V(x, θ,θ) (with suppressed indices) which is defined as usual: The supermultiplet of fields associated with V is defined to consist of all independent fields contained in V|, D α V|,DαV|, . . . . The covariant derivative of AdS 4 , ∇ a , is related to D a according to the rule In practice, we assume a Wess-Zumino gauge condition to be imposed on the geometric objects in (3.1) such that the background geometry is purely bosonic, The supersymmetry transformation of the component fields of V is computed according to the rule (A.14) Below we make use of the non-supersymmetric AdS quadratic Casimir operator (2.8) which is related to the Casimir (3.4) via QV = Q − 1 4 µD 2 +μD 2 V .

Massive supermultiplets
In general there are four non-vanishing independent complex component fields, They have each been defined in such a way that they are transverse which may be shown by making use of the on-shell conditions (4.1). Furthermore, one may show that the bottom and top components satisfy the mass-shell equations whilst the two middle component fields satisfy the differential equations The latter imply that only one of the two fields B α(m+1)α(n) and C α(m)α(n+1) is independent, and lead to the higher derivative mass-shell equations where we have denoted Using the relation (2.12), one may confirm that the above component results are in agreement with the decomposition (2.1), as dictated by representation theory. Here s = 1 2 (m + n + 1) and E 0 ≡ E 0 (M) is defined according to (4.5).

Partially massless supermultiplets
In this section we restrict our attention to on-shell true partially massless supermultiplets with super-depth 2 ≤ t ≤ min(m + 1, n + 1). The massless case with t = 1 will be considered separately in the next section. Upon specifying the pseudo-mass to be given by (4.13), one can show that the mass-shell equations (5.6) and (5.8) become Here τ (t,m,n) are the non-supersymmetric partially massless values (2.17), which are related to the supersymmetric ones (3.30c) through In accordance with the discussion of on-shell partially massless fields given in section 2.3, the following remarks hold: • The pair of equations (5.5a) and (5.11a) admits a depth t A = t − 1 gauge symmetry δ ζ A α(m)α(n) = ∇ (α 1 (α 1 · · · ∇ αt Aα t A ζ α t A +1 ...αm)α t A +1 ...αn) (5.13) when 2 ≤ t ≤ min(m, n).
• The above gauge symmetries hold only for gauge parameters which are transverse and which satisfy the same mass-shell equation as the parent gauge field.
One can reverse the logic and ask the question: for what values of M 2 does there appear a gauge symmetry (of any depth) at the component level? The answer is precisely those values which appear in the superprojectors (3.30c) (though, as above, special care must be taken when deducing the upper and lower bounds on the range of possible depths).
From the above component analysis one can see that the true partially massless representation P t, m, n of OSp(1|4) decomposes into so(3, 2) subrepresentations as follows P t, m, n = P t − 1, m, n ⊕ P t − 1, m + 1, n ⊕ P t, m + 1, n ⊕ P t, m + 1, n + 1 .

Massless supermultiplets
To study the component structure of an on-shell massless supermultiplet of superspin s > 0, it is advantageous to work with the gauge-invariant field strength W α(2s) defined by (4.10) instead of the prepotential φ α(m)α(n) , where 2s = m + n + 1. The fundamental properties of W α(2s) , imply the following equations: The chiral field strength W α(2s) has two independent component fields Their properties follow from the equations (5.18) and (5.19) These relations tell us that the component field strengths C α(2s) and C α(2s+1) furnish the massless representation S s + 1, s = D(s + 1, s) ⊕ D s + 3 2 , s + 1 2 .

Wess-Zumino supermultiplet
Various aspects of the Wess-Zumino model in AdS 4 were studied in the 1980s, see [15,63,[74][75][76][77] and references therein. Here we will only discuss its group-theoretic aspects for completeness. In superspace, the Wess-Zumino model is formulated in terms of a chiral scalar superfield Φ and its conjugateΦ. Without self-coupling, the model is described by the action Here λ is a dimensionless complex parameter. 11 The equations of motion corresponding to this model are Using (5.23), it can be shown that the superfield Φ satisfies the mass-shell equation The on-shell chiral scalar Φ contains two independent component fields, which are ϕ : = Φ| , (5.25a) By making use of the superfield mass-shell equation (5.24), it can be shown that the component fields (5.25) satisfy the mass-shell equations where the pseudo-masses ρ 2 (i) are In the massless limit λ → 0, the mass-shell conditions (5.27) take the form which is consistent with the on-shell massless field conditions (2.20). We wish to show that the model (5.22) describes two Wess-Zumino representations (2.2). First, let us point out that the equation for ϕ, (5.26a), is factorised into the product of two second-order differential equations and, hence, describes two spin-0 modes with masses ρ (1) and ρ (2) . This means that the Wess-Zumino model describes three representations of so (3,2). To see that the structure of these representations is consistent with (2.2) it is necessary compute the minimal energies E (i) 0 associated with the pseudo-masses ρ (i) . Using the prescription advocated in section 2.2, we find We see that there exist two branches of minimal energy solutions which furnish two Wess-Zumino representations S E 0 , 0 , eq. (2.2). We will call these branches the positive branch and the negative branch. These solutions are given in table 4. These branches Component fields E 0 (positive branch) E 0 (negative branch) furnish the Wess-Zumino representations S 1 + |λ|, 0 and S 1 − |λ|, 0 , eq. (2.2). In order for these representations to be unitary, we require E 0 > 1 2 . Therefore, the positive branch describes unitary representations for all values of λ. The negative branch describes unitary representations for |λ| < 1 2 . In the massless case λ = 0 the two branches coincide and we have two massless scalars with energies E 0 = 1 and E 0 = 2 and a massless fermion with energy E 0 = 3 2 . 12 There are three special values of λ. The choice λ = 0 corresponds to the superconformal or massless model. For λ = 1 the model has a dual formulation in terms of a tensor supermultiplet [78]. In this case the equations of motion (5.23) can be recast in term of a real superfield L = Φ +Φ to take form while the chirality of Φ gives The equations (5.30) imply that L satisfies the mass-shell equation which together define an on-shell scalar superfield. Finally for λ = 1/2 the on-shell chiral scalar Φ and its conjugateΦ contain a sub-representation describing the super-singleton, as follows from table 4.

Massive vector multiplet
The off-shell massive vector multiplet in a supergravity background was formulated in [78,79]. In the superspace setting, it is naturally described in terms of a real scalar prepotential V . Its action functional in AdS 4|4 is given by with M a non-zero real parameter. The corresponding equation of motion As pointed out in section 2.1, the massless spin-0 representations of osp(1|4) correspond to two possible values of the minimal energy E 0 = 1 or E 0 = 2, and the massless spin-1 2 representation corresponds to E 0 = 3 2 .
implies that V is a linear superfield, Then making use of (A.4a) gives The equations (5.34a) and (5.34b) define an on-shell irreducible supermultiplet. 13 It is worth pointing out that the above model has a dual formulation, which is the massive tensor multiplet model [78].
On the mass shell, the independent component fields of V are: The corresponding equations of motion are: The equations (5.36b) and (5.36c) lead to the quartic equation where (5.38) The above results imply that the component fields furnish the massive superspin-1 2 representation of osp(1, 4) It is instructive to compare the system of equations (5.30) with (5.33) and its corollaries (5.34). The former describes a non-conformal tensor multiplet in AdS 4 , which reduces to the free massless tensor multiplet in the flat-superspace limit.
where E 0 = 1 + 1 2 4 M 2 µμ + 1. The spinor equation (5.36b) and its conjugate (5.36c) are second-order partial differential equations. First-order equations for the component spinor fields can be obtained if one makes use of a Stückelberg reformulation of the above model. It is obtained through the replacement for some chiral superfield Φ. This leads to the action which is invariant under gauge transformations of the form The corresponding equations of motion are given by Using the gauge freedom (5.42), one may impose the gauge condition Φ = 0 whereupon the original model (5.32) is recovered. On the other hand, one can instead choose the following Wess-Zumino gauge which exhausts the gauge freedom. The remaining non-zero component fields are The spinor fields ψ α and χ α and their conjugates prove to satisfy the following first-order differential equations Making use of the equations (5.46b) allows us to express the fields χ α andχα in terms of ψ α andψα. Then the latter fields prove to satisfy the equations (5.36b) and (5.36c). As regards the bosonic fields (5.45b) and (5.45c), they may be seen to obey the equations (5.36a) and (5.36d).

Massive gravitino supermultiplet
As an illustration of our general discussion in section 4, here we present an off-shell model for the massive gravitino (superspin-1) multiplet in AdS 4 , The unitarity bound for the superspin-1 case is E 0 ≥ 2, with the E 0 = 2 value corresponding to the massless gravitino multiplet. We point out that on-shell models (i.e. without auxiliary fields) for the the massive gravitino multiplet in AdS 4 have appeared in the literature [80] (see also [81]).
Two off-shell formulations for the massless gravitino in AdS 4 were introduced in [16]. One of them is described by the action 14 where H is a real scalar superfield, and G αα is a longitudinal linear superfield constructed in terms of an unconstrained prepotential Ψ α , We propose a massive extension of the action ( We propose the following mass term: where m is a real parameter of unit mass dimension. The equations of motion corresponding to S massive [H, Ψ,Ψ], eq. (6.4), are given by where we have introduced the field strength W α = − 1 4 D 2 − 4µ D α H. Acting on (6.6a) and (6.6b) withD 2 gives Next, acting on (6.6b) with D α yields Using relations (6.6a), (6.7) and (6.8), one can show that Substituting (6.9) into (6.8), it immediately follows that H = 0 (6.10) on the mass shell. Now, the above relations imply that the spinor Ψ α obeys the following mass-shell conditions: These are the on-shell irreducibility constraints for a massive superspin-1 multiplet, as defined in section 4. The unitarity bound for the massive gravitino multiplet is These conditions have been used in the above derivation. One may think of the first Dirac-type equation in (6.11b) as the reality condition relating Ψ α andΨα.
In conclusion, we give the explicit expression for the massive gravitino action (6.4) Note that in the flat superspace limit, µ → 0, the action reduces to that describing the massive gravitino model derived in [83].
In the massless limit, m → 0, the second mass-shell condition (6.11b) becomes This agrees with the definition of on-shell massless supermultiplets given in section 4.
On the mass shell, the independent component fields of Ψ α are: Here A αα is a complex vector field. The component equations of motion, which follow from the first equation in (6.11b), are: 0 = i∇ βα χ β + m +μ χα , (6.16a) 0 = i∇ βα B αβ +μA αα − m +μ Ā αα , (6.16b) These equations can be viewed as reality conditions expressing the component fields ofΨα in terms of those contained in Ψ α . Making use of the relations (6.16) and their conjugates leads to the following equations: It can be shown that the equations (6.17b) and (6.17c) imply the quartic equation The analysis above indicates that the component fields furnish the massive superspin-1 representation (6.1), where E 0 = 1 + κ 2 µμ . Additionally, the mass-shell conditions are consistent with the results in section 5.1, for a superfield with index structure m − 1 = n = 0, upon the redefinition κ 2 = M 2 − µμ.

Factorisation of superconformal higher-spin actions
In section 3.2 we introduced the gauge invariant actions for superconformal higher-spin multiplets and used them to motivate our ansatz for the TLAL projectors. Let us come back to this topic and demonstrate that these actions factorise into products of minimal second order differential operators.
We begin with the familiar case when m = n = s and impose the reality condition (3.45). It is clear that for these values the higher-spin Bach tensor (3.27a) and the descendant (3.29a) coincide, so that we have the relations Here H α(s)α(s) is TLAL, whilst ζ α(s)α(s−1) is complex and unconstrained.
By virtue of the gauge invariant and TLAL nature of the higher-spin Bach tensor, the longitudinal parts give vanishing contribution to the action since they correspond to the pure gauge sector. Therefore only the TLAL component of the prepotential remains and, since the projector Π (s,s) acts as the identity on this subspace, using (7.1) we obtain the following factorisation S (s,s) This process is equivalent to fixing the gauge freedom (3.21) by imposing the gauge condition H α(s)α(s) ≡ H α(s)α(s) , since the action (7.3) no longer possesses any gauge symmetry.
Next we consider the complex supermultiplet Φ α(m)α(n) with n > m. In contrast to (7.1), one may show that in this case the following relation holds which upon inverting yields In terms of the TLAL projector this reads Employing the same trick as in the previous case, one obtains the following factorisation of the SCHS action (3.26a) where φ α(m)α(n) is the TLAL part of Φ α(m)α(n) . We see that, due to the mismatch of m and n, the SCHS action does not factorise wholly into products of second-order operators. However, upon taking appropriate derivatives of the equation of motion resulting from varyingφ α(m)α(n) in (7.7), one arrives at the following fully factorised on-shell equation According to our definitions in section 4, we see that some extra (non-unitary) massive modes, corresponding to the values of λ (t,m,n) with m + 1 < t ≤ n + 1 enter the spectrum of the wave equation (7.8).
Finally, if m > n, then one may show that the following relation holds Upon inverting and expressing in terms of the TLAL projector, one may write where we have used the identities (3.33) and B α(n)α(m) (Φ) = B α(n)α(m) (Φ). In this case the factorised action takes the form In the non-supersymmetric case, the factorisation of the conformal operators was observed long ago in [31,84,85] for the lower-spin values s = 3/2 and s = 2. The factorisation of the higher-spin conformal operators was conjectured in [86,87] (see also [88]), and later proved by several groups [29,[89][90][91]. In this section we have provided the first derivation of the factorisation of the superconformal higher-spin actions in N = 1 superspace. Just as it is for the non-supersymmetric case, the latter factor into products of minimal second-order differential operators involving all partial mass values.

Conclusion
In conclusion we briefly summarise the main results of this paper. We constructed the superspin projection operators Π (m,n) in AdS 4|4 . They are given by eq. (3.30) for m ≥ n > 0, and by eq. (3.35) for m > n = 0 (the other cases are obtained by complex conjugation). The operator Π (m,n) maps an unconstrained superfield Φ α(m)α(n) into one possessing the properties of a conserved conformal supercurrent, see the equations (3.31) and (3.36). Making use of the superprojectors, we obtained a new representation for the superconformal higher-spin gauge actions in AdS 4 given by eq. (3.40), which was used to demonstrate their factorisation, as discussed in section 7.
We provided a systematic discussion of how to realise the unitary (both massive and massless) and the partially massless representations of the N = 1 AdS 4 superalgebra osp(1|4) in terms of on-shell superfields. In particular, we established a one-to-one correspondence between the on-shell partially massless supermultiplets φ α(m)α(n) with superdepth t and the poles of Π (m,n) determined by λ (t,m,n) µμ belonging to the range (4.13). The corresponding gauge transformations were derived in the m = n case.
Our results make it possible to address a number of interesting open problems including the computation of partition functions for the superconformal higher-spin gauge theories in AdS 4 . This will be discussed elsewhere.
Acknowledgements: SMK is grateful to I. L. Buchbinder
The following identities hold: We often make use of the following identities, which can be readily derived from the algebra of covariant derivatives (2.7): where D 2 = D α D α , andD 2 =DαDα. Other useful identities are: Of special importance are the relations: The isometry group of AdS superspace, AdS 4|4 , is OSp(1|4). The isometry transformations of AdS 4|4 are generated by the Killing supervector fields ξ A E A which are defined to solve the Killing equation for some Lorentz parameter ξ βγ = ξ γβ . Given a supersymmetric field theory in AdS 4 formulated in terms of superfield dynamical variables V (with suppressed indices), its action is invariant under the isometry transformations with ξ B being an arbitrary Killing supervector field.
The Killing equation (A.6) implies the following [66]: It is seen that the parameters ξ β , ξ βγ and their conjugates are expressed in terms of the vector parameter ξ ββ , and the latter obeys the equation It also follows from (A.8) that, for every element of the set of parameters Υ = {ξ B , ξ βγ ,ξβ˙γ}, its covariant derivative D A Υ is a linear combination of the elements of Υ. Therefore, all information about the superfield parameters Υ is encoded in their bar-projections, Υ|.

B Partially massless gauge symmetry
This appendix is devoted to the derivation of partially massless gauge transformations both in the non-supersymmetric and supersymmetric cases.

B.1 The non-supersymmetric case
Given two integers m and n such that m ≥ n > 0, let h α(m)α(n) be an on-shell field, We would like to determine those values of ρ for which the above system of equations is compatible with a gauge symmetry.

B.2 The supersymmetric case
We first consider the real supermultiplet H αα =H αα which is on-shell Once again we would like to determine those values of M for which the above system of equations is compatible with a gauge symmetry. We will see that this occurs only at partially massless values. For the supermultiplet H αα there are only two such values, M 2 (1) ≡ λ (1,1,1) µμ = 5µμ , (B.11a) M 2 (2) ≡ λ (2,1,1) µμ = 3µμ , (B.11b) corresponding to the super-depths t = 1 (massless) and t = 2 respectively.
We begin by positing a gauge transformation of the form δH αα =DαΛ α − D αΛα (B.12) for an unconstrained gauge parameter Λ α . From this we must deduce which constraints must be placed upon Λ α in order for δH αα to be a solution to the equations (B.10).
Gauge invariance of (B.10a) requires Λ α to also have pseudo-mass M, Next we decompose the gauge parameter into irreducible parts. Performing this procedure in accordance with the discussion in section 3.3, we find that (B.12) takes the form δH αα =Dαζ α − D αζα + D αα ξ + D α ,Dα ζ + D αα σ +σ . (B.14) In (B.14), the gauge parameter ζ α is LTAL By conducting a similar analysis on the real supermultiplet H α(2)α (2) , one arrives at the gauge transformations presented in table 3. In principle this procedure may be carried out for any on-shell superfield φ α(m)α(n) with fixed m and n, but in practice this procedure is quite time consuming. In section 4.2 we use the above analysis to motivate an ansatz for the case arbitrary half-integer superspin.