Higher spin supercurrents in anti-de Sitter space

We propose higher spin supercurrent multiplets for ${\cal N}=1$ supersymmetric field theories in four-dimensional anti-de Sitter space (AdS). Their explicit realisations are derived for various supersymmetric theories, including a model of $N$ massive chiral scalar superfields with an arbitrary mass matrix. We also present new off-shell gauge formulations for the massless ${\cal N}=1$ supersymmetric multiplet of integer superspin $s$ in AdS, where $s =2,3,\dots$, as well as for the massless gravitino multiplet (superspin $s=1$) which requires special consideration.

The structure of consistent supercurrent multiplets in N = 1 AdS supersymmetry [11,12] considerably differs from that in the N = 1 super-Poincaré case, see e.g. [13,14]. Specifically, there exist three minimal supercurrents with 12+12 degrees of freedom in M 4|4 [14], and only one in AdS 4|4 [11], the latter being the AdS extension of the Ferrara-Zumino supercurrent [15]. Furthermore, the so-called S-multiplet advocated by Komargodski and Seiberg [16] does not admit a minimal extension to AdS. 2 These differences between the supercurrent multiplets in M 4|4 and AdS 4|4 have nontrivial dynamical implications. For instance, since every N = 1 supersymmetric field theory in AdS should have a well-defined 1 The classification by Festuccia and Seiberg [1] was given purely at the component level, with some results stated without proof. It was re-derived, in a completely rigorous way, in [2] using the superspace formalism developed in the mid-1990s [3]. As curved N = 1 superspaces, all maximally supersymmetric backgrounds were described in [4] (see also [5] for a new derivation of the results in [2,4], which works equally well for all known off-shell formulations for N = 1 supergravity). 2 The consistent supergravity extension of the S-multiplet was given in [11].
Ferrara-Zumino supercurrent [11,12], the Kähler target space of every supersymmetric nonlinear σ-model in AdS must be non-compact and possess an exact Kähler two-form, in accordance with the analysis of Komargodski and Seiberg [16]. 3 The same conclusion was also obtained by direct studies of the most general N = 1 supersymmetric nonlinear σ-models in AdS [1,19].
It should be pointed out that the consistent AdS supercurrents [11,12] are closely related to two classes of supersymmetric gauge theories: (i) the known off-shell formulations, minimal (see, e.g., [3,20] for reviews) and non-minimal [12], for N = 1 AdS supergravity; and (ii) the two dually equivalent series of massless higher spin supermultiplets in AdS proposed in [21]. More specifically, as discussed in [12], there are only two irreducible AdS supercurrents, with (12 + 12) and (20 + 20) degrees of freedom. 4 The former is naturally associated with the so-called longitudinal action S || (3/2) for a massless superspin-3/2 multiplet in AdS [21], which is formulated in terms of a real vector prepotential H αα and a covariantly chiral superfield σ. The latter is associated with a unique dual formulation S ⊥ (3/2) where the chiral superfield is replaced by a complex linear superfield Γ. The functional S || (3/2) proves to be the linearised action for minimal N = 1 AdS supergravity. The dual action S ⊥ (3/2) results from the linearisation around the AdS background of non-minimal N = 1 AdS supergravity [12]. 5 Both actions represent the lowest superspin limits of two infinite series of dual models, S || (s+ 1 2 ) and S ⊥ (s+ 1 2 ) , for off-shell massless gauge supermultiplets in AdS of half-integer superspin (s+ 1 2 ), where s = 1, 2 . . . , constructed in [21]. Off-shell formulations for massless gauge supermultiplets in AdS of integer superspin s, with s = 1, 2 . . . , were also constructed in [21]. In the flat-superspace limit, the supersymmetric higher spin theories of [21] reduce to those proposed in [22,23].
superfields [26]. In accordance with the standard Noether method (see, e.g., [27] for a review), the construction of conserved higher spin supercurrents for various supersymmetric theories in AdS is equivalent to generating consistent cubic vertices of the type HJ, where H denotes some off-shell higher spin gauge multiplet [21], and J = D p ΦD q Ψ is the higher spin current which is constructed in terms of some matter multiplets Φ and Ψ and the AdS covariant derivatives D. This is one of the important applications of the results presented in the present paper. In the flat-superspace case, several cubic vertices involving the off-shell higher spin multiplets of [22,23] were constructed recently in [28,29,30], as an extension of the superconformal cubic couplings between a chiral scalar superfield and an infinite tower of gauge massless multiplets of half-integer superspin given in [26].
It should be pointed out that conserved higher spin currents for scalar and spinor fields in Minkowski space have been studied in numerous publications. To the best of our knowledge, the spinor case was first described by Migdal [31] and Makeenko [32], while the conserved higher spin currents for scalar fields were first obtained in [32,33,34] (see also [35,36]). The conserved higher spin currents for scalar fields in AdS were studied, e.g., in [37,38,39,40,41]. Since the curvature of AdS space is non-zero, explicit calculations of conserved higher spin currents are much harder than in Minkowski space. This is one of the reasons why Refs. [37,38] studied only the conformal scalar, and only the first order correction to the flat-space expression was given explicitly. The construction presented in [41] is more complete in the sense that all conserved higher spin currents were computed exactly for any free massive scalar field. This was achieved by making use of a somewhat unorthodox formulation in the so-called ambient space. All these works dealt with integer spin currents. The important feature of supersymmetric theories is that they also possess half-integer spin currents. They belong to the higher spin supercurrent multiplets we construct in this work. Another nice feature of the supersymmetric case is that the calculation of higher spin supercurrent multiplets in AdS superspace is considerably simpler than the problem of computing the ordinary conserved higher spin currents in AdS space. This paper is organised as follows. Section 2 contains a summary of the results concerning supersymmetric field theory in AdS superspace. Section 3 is devoted to a novel formulation for the massless integer superspin multiplets in AdS. This formulation is shown to reduce to that proposed in [21] upon partially fixing the gauge freedom. We also describe off-shell formulations (including a novel one) for the massless gravitino multiplet in AdS. In section 4 we introduce higher spin supercurrent multiplets in AdS and describe improvement transformations for them. Sections 5 and 6 are devoted to the explicit construction of higher spin supercurrents for N massive chiral multiplets. Several nontrivial applications of the results obtained are given in section 7. The main body of the paper is accompanied by three technical appendices. Appendix A reviews the irreducible supercurrent multiplets in AdS following [11,12]. Appendices B and C review the conserved higher spin currents for N scalars and spinors, respectively, with arbitrary mass matrices. These results are scattered in the literature, including [31,32,33,34].

Field theory in AdS superspace
In this section we give a summary of the results which are absolutely essential when doing N = 1 supersymmetric field theory in AdS in a manifestly OSp(1|4)-invariant way. We mostly follow the presentation in [21]. Our notation and two-component spinor conventions agree with [3], except for the notation for superspace integration measures.
Let z M = (x m , θ µ ,θμ) be local coordinates for N = 1 AdS superspace, AdS 4|4 . The geometry of AdS 4|4 may be described in terms of covariant derivatives of the form where E A M is the inverse superspace vielbein, and is the Lorentz connection. The Lorentz generators M bc ⇔ (M βγ ,Mβ˙γ) act on twocomponent spinors as follows: The covariant derivatives of AdS 4|4 satisfy the following algebra with µ = 0 being a complex parameter, which is related to the scalar curvature R of AdS space by the rule R = −12|µ| 2 .
In our calculations, we often make use of the following identities, which can be readily derived from the covariant derivatives algebra (2.4): where D 2 = D α D α , andD 2 =DαDα. These relations imply the identity which guarantees the reality of the action functionals considered in the next sections.
Complex tensor superfields Γ α(m)α(n) := Γ α 1 ...αmα 1 ...αn = Γ (α 1 ...αm)(α 1 ...αn) and G α(m)α(n) are referred to as transverse linear and longitudinal linear, respectively, if the constraints are satisfied. For n = 0 the latter constraint coincides with the condition of covariant chirality,DβG α(m) = 0. With the aid of (2.5a), the relations (2.7) lead to the linearity conditions The transverse condition (2.7a) is not defined for n = 0. However its corollary (2.8a) remains consistent for the choice n = 0 and corresponds to complex linear superfields Γ α(m) constrained by In the family of constrained superfields Γ α(m) introduced, the scalar multiplet, m = 0, is used most often in applications. One can define projectors P ⊥ n and P || n on the spaces of transverse linear and longitudinal linear superfields respectively: with the properties P ⊥ n 2 = P ⊥ n , P || n 2 = P || n , P ⊥ n P || n = P || n P ⊥ n = 0 , P ⊥ n + P || n = ½ . (2.11) Superfields (2.7) were introduced and studied by Ivanov and Sorin [10] in their analysis of the representations of the AdS supersymmetry. A nice review of the results of [10] is given in the book [42].
The isometry group of N = 1 AdS superspace is OSp(1|4). The isometries transformations of AdS 4|4 are generated by the Killing vector fields Λ A E A which are defined to solve the Killing equation for some Lorentz superfield parameter ω bc = −ω cb . As shown in [3], the equations in (2.15) are equivalent to The solution to these equations is given in [3]. If T is a tensor superfield (with suppressed indices), its infinitesimal OSp(1|4) transformation is In Minkowski space, there are two ways to generate supersymmetric invariants, one of which corresponds to the integration over the full superspace and the other over its chiral subspace. In AdS superspace, every chiral integral can be always recast as a full superspace integral. Associated with a scalar superfield L is the following OSp(1|4) invariant where E denotes the chiral integration measure. 6 Let L c be a chiral scalar,DαL c = 0. It generates the supersymmetric invariant d 4 xd 2 θ E L c . The specific feature of AdS superspace is that the chiral action can equivalently be written as an integral over the full superspace [43,44] Unlike the flat superspace case, the integral on the right does not vanish in AdS.

Massless integer superspin multiplets
Let s be a positive integer. The longitudinal formulation for the massless superspin-s multiplet in AdS was realised in [21] in terms of the following dynamical variables Here, H α(s−1)α(s−1) is an unconstrained real superfield, and G α(s)α(s) is a longitudinal linear superfield. The latter is a field strength associated with a complex unconstrained prepotential Ψ α(s)α(s−1) , The gauge freedom postulated in [21] is given by where the gauge parameter is L α(s)α(s−1) is unconstrained.
In this section we propose a reformulation of the longitudinal theory that is obtained by enlarging the gauge freedom (3.3) at the cost of introducing a new purely gauge superfield variables in addition to H α(s−1)α(s−1) , Ψ α(s)α(s−1) andΨ α(s−1)α(s) . In such a setting, the gauge freedom of Ψ α(s)α(s−1) coincides with that of a superconformal multiplet of superspin s [26]. The new formulation will be an extension of the one given in [25] in the flat-superspace case.
The gauge freedom of Ψ α 1 ...αsα 1 ...α s−1 is chosen to coincide with that of the superconformal superspin-s multiplet [26], which is with unconstrained gauge parameters V α(s−1)α(s−1) and ζ α(s)α(s−2) . The V-transformation is defined to act on the superfields H α(s−1)α(s−1) and Σ α(s−1)α(s−2) as follows The longitudinal linear superfield defined by (3.2) is invariant under the ζ-transformation (3.7a) and varies under the V-transformation as Our next task is to derive an AdS extension of the gauge-invariant action in Minkowski superspace (given by eq. (2.8) in [25]). The geometry of AdS superspace is completely determined by the algebra (2.4). We start with the following action functional in AdS superspace, which is a minimal AdS extension of the action constructed in [25].
The gauge-invariant action in AdS is expected to differ from (3.9) by some µ-dependent terms, which are required to ensure invariance under the gauge transformations (3.7) and, by construction, (3.6). We compute the variation of (3.9) under (3.7) and then add certain µ-dependent terms to achieve an invariant action. The identities (2.5) prove to be useful in carrying out such calculations.
One can impose an alternative gauge fixing In accordance with (3.7b), in this gauge the residual gauge freedom is described by 14) The action (3.10) includes a single term which involves the 'naked' gauge field Ψ α(s)α(s−1) and not the field strength G α(s)α(s) , the latter being defined by (3.2) and invariant under the ζ-transformation (3.7a). This is actually a BF term, for it can be written in two different forms 1 s The former makes the gauge symmetry (3.6) manifestly realised, while the latter turns the ζ-transformation (3.7a) into a manifest symmetry.
Making use of (3.15) leads to a different representation for the action (3.10). It is

Dual formulation
As in the case of the flat superspace [25], the theory with action (3.16) can be reformulated in terms of a transverse linear superfield by applying the duality transformation introduced in [21].
We now associate with our theory (3.16) the following first-order action 7 Varying (3.17) with respect to the Lagrange multiplier and taking into account the constraint (3.18) yields U α(t)α(s) = G α(t)α(s) ; then, S first-order turns into the original action (3.16). On the other hand, we can eliminate the auxiliary superfields U α(s)α(s) andŪ α(s)α(s) from (3.17) using their equations of motion. This leads to the dual action where we have defined The first-order model introduced is equivalent to the original theory (3.16). The action (3.17) is invariant under the gauge ξ-transformation (3.6) which acts on U α(s)α(s) and Γ α(s)α(s) by the rule is invariant under the gauge transformations (3.6) and (3.21b).
The first-order action (3.17) is also invariant under the gauge V-transformation (3.7b) and (3.7c), which acts on U α(s)α(s) and Γ α(s)α(s) as In accordance with (3.7c), the V-gauge freedom may be used to impose the condition In this gauge the action (3.19) reduces to the one defining the transverse formulation for the massless superspin-s multiplet [21]. The gauge condition (3.23) is preserved by residual local V-and ξ-transformations of the form Making use of the parametrisation (3.12), the residual gauge freedom is which is exactly the gauge symmetry of the transverse formulation for the massless superspin-s multiplet [21].

Models for the massless gravitino multiplet in AdS
The massless gravitino multiplet (i.e., the massless superspin-1 multiplet) was excluded from the above consideration. Here we will fill the gap.
The (generalised) longitudinal formulation for the gravitino multiplet is described by the action where Φ is a chiral scalar superfield,DαΦ = 0, and This action is invariant under gauge transformations of the form This is one of the two models for the massless gravitino multiplet in AdS introduced in [11]. In a flat-superspace limit, the action reduces to that given in [45]. Imposing the gauge condition Φ = 0 reduces the action (3.26) to the original longitudinal formulation for the massless gravitino multiplet in AdS [21].
The action (3.26) involves the chiral scalar Φ and its conjugate only in the combination (ϕ +φ), where ϕ = Φ/µ. This means that the model (3.26) possesses a dual formulation realised in terms of a real linear superfield L, (3.28) The dual model is described by the action [11] This action is invariant under the gauge transformations (3.27a), (3.27b) and In a flat-superspace limit, the action (3.29) reduces to that given in [46].
In Minkowski superspace, there exists one more dual realisation for the massless gravitino multiplet model [25] which is obtained by performing a Legendre transformation converting Φ into a complex linear superfield. This formulation cannot be lifted to the AdS case, the reason being the fact that the action (3.26) involves the chiral scalar Φ and its conjugate only in the combination (ϕ +φ), where ϕ = Φ/µ.
The dependence on Ψ α andΨα in the last term of (3.26) can be expressed in terms of G αα andḠ αα if we introduce a complex unconstrained prepotential U for Φ in the standard way Then making use of (2.5d) gives Since the resulting action depends on G αα andḠ αα , we can introduce a dual formulation for the theory that is obtained turning G αα andḠ αα into a transverse linear superfield and its conjugate using the scheme described in [21]. The resulting action is where we have defined The action (3.34) is invariant under the following gauge transformations Imposing the gauge condition U = 0 reduces the action (3.34) to the original transverse formulation for the massless gravitino multiplet in AdS [21].

Higher spin supercurrents
In this section we introduce higher spin supercurrent multiplets in AdS. First of all, we recall the structure of the gauge superfields in terms of which the massless half-integer superspin multiplets are described [21].
As in [24], it is useful to introduce auxiliary complex variables ζ α ∈ C 2 and their conjugatesζα. Given a tensor superfield U α(m)α(n) , we associate with it the following field on C 2 which is homogeneous of degree (m, n) in the variables ζ α andζα. We introduce operators that increase the degree of homogeneity in the variables ζ α andζα, We also introduce two operators that decrease the degree of homogeneity in the variables ζ α andζα, specifically Making use of the above notation, the transverse linear condition (4.8a) and its conjugate becomeD The conservation equations (4.8b) and (4.8c) turn into SinceD 2 (0,−1) J (s,s) = 0, the conservation equation (4.14a) is consistent provided This is indeed true, as a consequence of the transverse linear condition (4.13a).

Improvement transformations
The conservation equations (4.8) and (4.9) define two consistent higher spin supercurrents in AdS. Similar to the two irreducible AdS supercurrents [12], with (12 + 12) and (20 + 20) degrees of freedom, the higher spin supercurrents (4.8) and (4.9) are equivalent in the sense that there always exists a well defined improvement transformation that converts (4.8) into (4.9). Such an improvement transformation is constructed below.
In accordance with the result obtained, for all applications it suffices to work with the longitudinal supercurrent (4.8). This is why in the integer superspin case, which will be studied in section 4.4, we will introduce only the longitudinal supercurrent.

Non-conformal supercurrents: Integer superspin
We now make use of the new gauge formulation (3.10), or equivalently (3.16), for the integer superspin-s multiplet to derive the AdS analogue of the non-conformal higher spin supercurrents constructed in [25].
Taking the sum of (4.24a) and (4.24b) leads to The equations (4.22), (4.23) and (4.26) describe the conserved current supermultiplet which corresponds to our theory in the gauge (3.13). As a consequence of (4.23), the conservation equation In the flat-superspace limit, the higher spin supercurrent multiplet described by eqs.

Improvement transformation
There exist an improvement transformation for the supercurrent multiplet (4.24). Given a chiral scalar superfield Ω, we introduce 5 Higher spin supercurrents for chiral superfields:

Half-integer superspin
In the remainder of this paper we will study explicit realisations of the higher spin supercurrents introduced above in various supersymmetric field theories in AdS.

Superconformal model for a chiral superfield
Let us consider the superconformal theory of a single chiral scalar superfield where Φ is covariantly chiral,DαΦ = 0. We can define the conformal supercurrent J (s,s) in direct analogy with the flat superspace case [26,24]  From the above consideration, it follows that We also state some other properties which we often use throughout our calculations The above identities suffice to prove that the supercurrent (5.2) does obey the conservation equation (5.3).

Non-superconformal model for a chiral superfield
Let us now add the mass term to (5.1) and consider the following action with m a complex mass parameter. In the massive case J (s,s) satisfies a more general conservation equation (4.14a) for some superfield T (s−1,s−2) . Making use of the equations of motion we obtainD where we have denoted We now look for a superfield T (s−1,s−2) such that (i) it obeys the transverse linear constraint (4.13a); and (ii) it satisfies the equation Our analysis will be similar to the one performed in [24] in flat superspace. We consider a general ansatz with some coefficients c k which have to be determined. For k = 1, 2, ...s − 2, condition (i) implies that the coefficients c k must satisfy while (ii) gives the following equation Condition (ii) also implies that It turns out that the equations (5.14) lead to a unique expression for c k given by If the parameter s is odd, s = 2n + 1, with n = 1, 2, . . . , one can check that the equations (5.14a)-(5.14c) are identically satisfied. However, if the parameter s is even, s = 2n, with n = 1, 2, . . . , there appears an inconsistency: the right-hand side of (5.14c) is positive, while the left-hand side is negative, (s − 1)c s−2 + c 0 < 0. Therefore, our solution (5.15) is only consistent for s = 2n + 1, n = 1, 2, . . . . Relations (5.2), (5.13), (5.14d) and (5.15) determine the non-conformal higher spin supercurrents in the massive chiral model (5.9). Unlike the conformal higher spin supercurrents (5.2), the non-conformal ones exist only for the odd values of s, s = 2n + 1, with n = 1, 2, . . . . In the flat-superspace limit, the above results reduce to those derived in [24] and in a revised version (v3, 26 Oct.) of Ref. [28] (which appeared a few days before [24]).

Superconformal model with N chiral superfields
We now generalise the superconformal model (5.1) to the case of N covariantly chiral scalar superfields Φ i , i = 1, . . . N, The novel feature of the N > 1 case is that there exist two different types of conformal supercurrents, which are: Here S and A are arbitrary real symmetric and antisymmetric constant matrices, respectively. We have put an overall factor √ −1 in eq. (5.18) in order to make J − (s,s) real. One can show that the currents (5.17) are (5.18) are conserved on-shell: The above results can be recast in terms of the matrix conformal supercurrent J (s,s) = J ij (s,s) with components which is Hermitian, J (s,s) † = J (s,s) . The chiral action (5.16) possesses rigid U(N) symmetry acting on the chiral column-vector Φ = (Φ i ) by Φ → gΦ, with g ∈ U(N), which implies that the supercurrent (5.20) transforms as J (s,s) → gJ (s,s) g −1 .

Massive model with N chiral superfields
Now let us consider a theory of N massive chiral multiplets with action where M ij is a constant symmetric N × N mass matrix. The corresponding equations of motion are We now look for a superfieldT (s−2,s−1) such that (i) it obeys the transverse antilinear constraint Starting with a general ansatz and imposing conditions (i) and (ii) yield the following equations for the coefficients d k The equations (5.30) lead to a unique expression for d k given by If the parameter s is even, s = 2n, with n = 1, 2, . . . , one can check that the equations (5.30a)-(5.30d) are identically satisfied. However, if the parameter s is odd, s = 2n + 1, with n = 1, 2, . . . , there appears an inconsistency: the right-hand side of (5. Note that the coefficients c k in (5.32) differ from similar coefficients in (5.15) by a factor of −i. This means that for odd s we can define a more general supercurrent where H ij is a generic matrix which can be split into the symmetric and antisymmetric parts H ij = S ij + iA ij . Here both S and A are real and we put an i in front of A because J (s,s) must be real. From the above consideration it then follows that the corresponding more general solution forT (s−2,s−1) reads  6 Higher spin supercurrents for chiral superfields:

Integer superspin
In this section we provide explicit realisations for the fermionic higher spin supercurrents (integer superspin) in models described by chiral scalar superfields.

Massive hypermultiplet model
Consider a free massive hypermultiplet in AdS 9 where the superfields Ψ ± are covariantly chiral,DαΨ ± = 0 and m is a complex mass parameter. By a change of variables it is possible to make m real. Let us introduce another set of fields Φ ± ,DαΦ ± = 0, related to Ψ ± by the following transformations Under the transformations (6.2), the action (6.1) turns into where the mass parameter M is now real. In the massless case, M = 0, the conserved fermionic supercurrent J α(s)α(s−1) was constructed in [26] and is given by Making use of the massless equations of motion, − 1 4 (D 2 − 4μ) Φ ± = 0, one may check that J (s,s−1) obeys, for s > 1, the conservation equations We will now construct fermionic higher spin supercurrents corresponding to the massive model (6.3). Making use of the massive equations of motion we obtain It can be shown that the massive supercurrent J (s,s−1) also obeys (4.28).
We now look for a superfield T (s−1,s−1) such that (i) it obeys the longitudinal linear constraint (4.29); and (ii) it satisfies (4.31), which is a consequence of the conservation equation (4.30). For this we consider a general ansatz Condition (i) implies that the coefficients must be related by while for k = 1, 2, . . . s − 2, condition (ii) gives the following recurrence relations: Condition (ii) also implies that The above conditions lead to simple expressions for c k and d k : where k = 1, 2, . . . s − 1.

Superconformal model with N chiral superfields
In this subsection we will generalise the above results for N chiral superfields Φ i , i = 1, . . . N. We first consider the superconformal model (5.16). Let us construct the following fermionic supercurrent where C ij is a constant complex matrix. By changing the summation index it is not hard to show that J (s,s−1) = 0 if (i) s is odd and C ij is symmetric; and (ii) s is even and C ij is antisymmetric, that is In the case of a single chiral superfield, the supercurrent (6.11) exists for even s, 14) The flat-superspace version of (6.14) can be extracted from the results of [25,26].

Massive model with N chiral superfields
Now we move to the massive model (5.21). As was discussed in previous subsection, to construct the conserved currents we first have to calculate D (−1,0) J (s,s−1) using the equations of motion in the massive theory. The calculation depends on whether C ij is symmetric or antisymmetric.

Symmetric C
If C ij is a symmetric matrix, using the massive equation of motion, we obtain Here we have two cases to consider: To find T (s−1,s−1) we consider a general ansatz It is possible to show that no solution for T (s−1,s−1) can be found unless we impose 10 Furthermore, condition (i) implies that the coefficients must be related by while for k = 1, 2, . . . s−2, while condition (ii) and eq. (6.19) gives the following recurrence relations Condition (ii) also implies that The above conditions lead to simple expressions for d k : where k = 1, 2, . . . s − 1 and s is even.
Case 2: If we take CM to be antisymmetric, a similar analysis shows that no solution for T (s−1,s−1) exists for even values of s.

Antisymmetric C
If C ij is antisymmetric we get: As in the symmetric C case, there are also two cases to consider: Then it follows that eq.
Note that it is the equation same as eq. (6.17) which means that the solution for T (s−1,s−1) is the same as in Case 1. That is, the matrices C and M must satisfy CM =CM, T (s−1,s−1) is given by eq. (6.18) and the coefficients (c k ) ij , (d k ) ij are given by eqs. (6.20).
Case 2: If we take CM to be antisymmetric, a similar analysis shows that no solution for T (s−1,s−1) exists for odd values of s.

Massive hypermultiplet model revisited
As a consistency check of our general method, let us reconsider the case of a hypermultiplet studied previously. For this we will take N = 2, the mass matrix in the form and denote Φ i = (Φ + , Φ − ). If s is even we will take C in the form Note that C commutes with M. The condition CM =CM is equivalent to arg(c) = arg(m)+nπ. For simplicity, let us choose both c and m to be real. Under these conditions eq. (6.11) for J (s,s−1) becomes Introducing a new summation variable k ′ = s − 1 − k for the second and fourth terms, we obtain We see that for even s it coincides with the hypermultiplet supercurrent given by (6.4) up to an overall coefficient 2c. If s is odd we have to choose C to be antisymmetric Note that C now anticommutes with M. For simplicity, we again choose c and m to be real. Now the expression (6.11) for J (s,s−1) becomes We see that for odd s it coincides with the hypermultiplet supercurrent given by (6.4) up to an overall coefficient 2c. To summarise, we reproduced the hypermultiplet supercurrent (6.4) for both even and odd values of s. However, for even s it came from a symmetric matrix (6.26) and for odd s it came from an antisymmetric matrix (6.29).
Let us now consider T (s−1,s−1) . First, we will note that the product CM is given by This means that T (s−1,s−1) is given by the following expression valid for all values of s where the matrix (d k ) ij is given by It is easy to see that this expression for T (s−1,s−1) coincides with the one obtained for the hypermultiplet in the previous subsections in eqs. (6.8), (6.9a), (6.10) up to an overall factor 2c.

Summary and applications
In this paper, we have proposed higher spin conserved supercurrents for N = 1 supersymmetric theories in four-dimensional anti-de Sitter space. We have explicitly constructed such supercurrents in the case of N chiral scalar superfields with an arbitrary mass matrix M. The structure of the supercurrents depends on whether the superspin is integer or half-integer, as well as on the value of the superspin, and the mass matrix. Let us summarise our results.
In the case of half-integer superspin s + 1/2, the supercurrent has the structure J (s,s) = H ij J ij In the rest of this section, we will discuss several applications of the results obtained in the paper.

Integer superspin
Let us return to the model (5.9) describing the dynamics of a single massive chiral multiplet in AdS. It proves to possess conserved fermionic higher spin supercurrents. For even integer superspin, s = 2, 4, . . . , the supercurrent J (s,s−1) is given by (6.14). The corresponding trace multiplet is where the coefficients c k and d k are given by (6.10). As an example, for s = 2 we obtain It was claimed in [28] that the chiral model in Minkowski superspace does not possess any conserved fermionic supercurrents J (s,s−1) , for any value of the mass parameter m. Here we have demonstrated that they, in fact, do exist when s is even.
There is a simple explanation for why the conserved fermionic supercurrents were overlooked in the analysis of [28]. The point is that the authors of [28] considered only a particular ansatz for the Noether procedure to construct cubic vertices, δ g Φ = AΦ, where A is a higher-derivative operator containing infinitely many local parameters. However, in order to generate the conserved fermionic supercurrents we constructed, it is necessary to deal with a more general ansatz δ g Φ = AΦ +D 2 BΦ, with B another higher-derivative operator. 11

Higher spin supercurrents for a tensor multiplet
Let us consider a special case of the non-superconformal chiral model (5.9) with the mass parameter m = µ, This theory is known to be dual to a tensor multiplet model [50] which is realised in terms of a real linear superfield L =L, constrained by (D 2 −4µ)L = 0, which is the gauge-invariant field strength of a chiral spinor superfield We recall that the duality between (7.4) and (7.5) follows, e.g., from the fact the off-shell constraint The corresponding trace multiplet proves to be The coefficient c k is given by eq. (5.15), s is odd. The Ferrara-Zumino supercurrent (s = 1) for the model (7.5) in an arbitrary supergravity background was derived in section 6.3 of [3]. Modulo normalisation, the AdS supercurrent is J αα =DαLD α L + L D α ,Dα L , (7.11a) and the corresponding trace multiplet is The supercurrent obeys the conservation equation (A.1).

Higher spin supercurrents for a complex linear multiplet
The superconformal non-minimal scalar multiplet in AdS is described by the action where Γ is a complex linear scalar, (D 2 − 4µ)Γ = 0. This is a dual formulation for the superconformal chiral model (5.1). As is well known, the duality between (5.1) and (7.12) follows from the fact that the off-shell constraint (D 2 − 4μ)Γ = 0 (7.13a) and the equation of motion for ΓDαΓ = 0 (7.13b) are equivalent to the equation of motion forΦ, (D 2 −4μ)Φ = 0, and the off-shell constraint DαΦ = 0, respectively. In other words, on the mass shell we can identifyΓ with Φ.
The higher spin supercurrents, J (s,s) and J (s,s−1) , for the model (7.12) are obtained from (5.2) and (6.14), respectively, by replacing Φ withΓ. The fermionic supercurrent J (s,s−1) exists for even values of s. In the flat-superspace limit, the expression for J (s,s) obtained coincides with the main result of [29]. 12 It was claimed in [29] that the flat-superspace model S[Γ,Γ] = − d 4 xd 2 θd 2θΓ Γ ,D 2 Γ = 0 (7.14) does not possess any conserved fermionic supercurrents J (s,s−1) . Here we have demonstrated that they, in fact, do exist when s is even. Just like in the case of a massive chiral multiplet, the fermionic supercurrents were overlooked in [29] because only a particular ansatz for the Noether procedure was studied in [29].
Now, for any positive integer n > 0, we can try to generalise the higher spin supercurrent (5.2) as follows: This demonstrates that J (s+n,s+n) is not conserved in AdS 4|4 .
In the flat-superspace limit, µ → 0, the right-hand side of (7.25) vanishes and J (s+n,s+n) becomes conserved. In Minkowski superspace, the conserved supercurrent J (s+n,s+n) was recently constructed in [30] as an extension of the non-supersymmetric approach [55].

A AdS supercurrents
There are only two irreducible AdS supercurrents, with (12 + 12) and (20 + 20) degrees of freedom [11]. 14 The former is associated with minimal AdS supergravity (see, e.g., [3,20] for reviews) and the corresponding conservation equation is The latter corresponds to non-minimal AdS supergravity [12], and the conservation equation isDα The vector superfields J a and J a are real.
The non-minimal supercurrent (A.2) is equivalent to the Ferrara-Zumino multiplet (A.1) in the sense that there always exists a well-defined improvement transformation that turns (A.2) into (A.1), as demonstrated in [12]. In AdS superspace, the constraint on the longitudinal linear compensator ζ α is equivalent to for well-defined real operators V and U. If we now introduce then the operators J αα and T prove to satisfy the conservation equation (A.1).
For the Ferrara-Zumino supercurrent (A.1), there exists an improvement transformation that is generated by a chiral scalar operator Ω. Specifically, using the operator Ω allows one to introduce new supercurrent J αα and chiral trace multiplet T defined by The operators J αα and T obey the conservation equation (A.1) for arbitrary Ω. 15

B Conserved currents for free real scalars
In this appendix we will consider higher spin currents in free scalar field theory in flat space. Similar analysis for free fermions will be done in the next appendix.
To construct non-conformal higher spin currents, we couple h α(s)α(s) and h α(s−2)α(s−2) to external sources 15 Extension of the improvement transformation (A.5) to the case of supergravity is discussed in section 6.3 of [3]. 16 We follow the description of Fronsdal's theory [51] given in section 6.9 of [3].
Requiring that S (s) source be invariant under the λ-transformation in (B.1) gives the conservation equation Our derivation of (B.3) is analogous to that given in [35].
Let us introduce the following operators The conservation equation (B.3) then becomes Note that both j (s,s) and t (s−2,s−2) are real.
Let us now consider the model for N massless real scalar fields φ i , with i = 1, . . . N, in Minkowski space which admits conserved higher spin currents of the form where C ij is a constant matrix. It can be shown that j (s,s) = 0 if s is odd and C ij is symmetric. Similarly, j (s,s) = 0 if s is even and C ij is antisymmetric. Thus, we have to consider two separate cases: the case of even s with symmetric C and, the case of odd s with antisymmetric C. Using the massless equation of motion ✷φ i = 0 , one may show that j (s,s) satisfies the conservation equation We now turn to the massive model where M = (M ij ) is a real, symmetric N × N mass matrix. In the massive theory, the conservation equation is described by (B.5) and so we first need to compute ∂ (−1,−1) j (s,s) using the massive equations of motion For symmetric C, we obtain If C ij is antisymmetric, we get Thus, in the massive real scalars there are four cases to consider: 1. Both C and CM 2 are symmetric ⇐⇒ [C, M 2 ] = 0, s even. There are four cases to consider: g k−1 + g k = −4 (s + 1)(s + 2) s − 1 (−1) k s k s k + 1 k 1 k + 2 − k + 1 (s − k + 2)(s − k + 1) . and (C.13a)-(C.13f) are identically satisfied. However, when s is odd, there appears an inconsistency: the right-hand side of (C.13d) is positive, while the left-hand side is negative, c s−2 + c 0 < 0. Therefore, our solution (C.14) is only consistent for s = 2n, n = 1, 2, . . . . We observe that the coefficients c k and g k in eq. (C.19a) and (C.19b), respectively differ from similar coefficients in (C.14a) and (C.14b) by a factor of i. Hence, for even s we may define a more general supercurrent j (s,s) = C ij s−1 k=0 (−1) k s k s k + 1 ∂ k (1,1) ζ α ψ i α ∂ s−k−1 (1,1)ζαψ jα , (C. 21) where C ij is a generic matrix which can be split into the symmetric and antisymmetric parts: C ij = S ij + iA ij . Here both S and A are real and we put an i in front of A because j (s,s) must be real. From the above consideration it then follows that the corresponding more general solution for t