Higher spin supercurrents in anti-de Sitter space

We propose higher spin supercurrent multiplets for N=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N}=1 $$\end{document} 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 N=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N}=1 $$\end{document} supersymmetric multiplet of integer superspin s in AdS, where s = 2, 3, . . . , 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 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 [1,20].
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., [3,21] 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 [22]. 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 [22], 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 JHEP09(2018)027 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 Making use of the gauge off-shell formulations for massless higher spin supermultiplets in AdS [22], one can define consistent higher spin supercurrent multiplets in AdS superspace (i.e. higher spin extensions of the supercurrent) that contain ordinary bosonic and fermionic conserved currents in AdS. One can then look for explicit realisations of such higher spin supercurrents in concrete supersymmetric theories in AdS, for instance models for massless and massive chiral scalar superfields. Such a program is a natural extension of the flat-space results obtained in recent papers [25,26] in which two of us built on the structure of higher spin supercurrent multiplets in models for superconformal chiral superfields [27]. In accordance with the standard Noether method (see, e.g., [28] 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 [22], 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 [23,24] were constructed recently in [29][30][31], 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 [27].
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 [32] and Makeenko [33], while the conserved higher spin currents for scalar fields were first obtained in [33][34][35] (see also [36,37]). The conserved higher spin currents for scalar fields in AdS were studied, e.g., in [38][39][40][41][42]. 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. [38,39] studied only the conformal scalar, and only the first order correction to the flat-space expression was given explicitly. The construction presented in [42] is more complete in the sense that all conserved higher spin currents were computed exactly for a 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 JHEP09(2018)027 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.
Various aspects of supersymmetric field theories on AdS 4 have been studied in detail over the last forty years, see, e.g., [1,9,10,17,18,20,[43][44][45][46][47][48][49] and references therein. The energy-momentum tensor of such a theory belongs to the Ferrara-Zumino supercurrent (or, equivalently, to the non-minimal AdS supercurrent which is related to the Ferrara-Zumino supermultiplet by a well-defined improvement transformation [12].). In this paper we present, for the first time, higher spin extensions of the AdS supercurrents and derive their explicit realisations for various supersymmetric theories on AdS, including a model of N massive chiral scalar superfields with an arbitrary mass matrix. Our results have numerous applications. For instance, the conserved higher spin supercurrents computed in section 5 and 6 can readily be reduced to component fields. This will give closedform expressions for conserved higher spin bosonic and fermionic currents in models with massive scalar and spinor fields, thus leading to more general results than those known in the literature [38][39][40][41][42]. Another applications of the results obtained are consistent cubic coupling between chiral scalar supermultiplets and massless higher spin supermultiplets. Our results also make it possible to develop an effective action approach to massless higher spin supermultiplets along the lines advocated in [50][51][52] and more recently in [27]. We also refine some statements given recently in the literature, see section 7.
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 [22] 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 [32][33][34][35].

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 [22]. 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

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where E A M is the inverse superspace vielbein, and is the Lorentz connection. The Lorentz generators M bc ⇔ (M βγ ,Mβγ) 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 constraintsDβ 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

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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 (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 [53]. Given a complex tensor superfield V α(m)α(n) with n = 0, it can be represented as a sum of transverse linear and longitudinal linear multiplets, Choosing V α(m)α(n) to be transverse linear (Γ α(m)α(n) ) or longitudinal linear (G α(m)α(n) ), the above relation gives for some prepotentials Φ α(m)α(n+1) and Ψ α(m)α(n−1) . The constraints (2.7) hold for unconstrained Φ α(m)α(n+1) and Ψ α(m)α(n−1) . These prepotentials are defined modulo gauge transformations of the form: with the gauge parameters ξ α(m)α(n+2) and ζ α(m)α(n−2) being unconstrained. 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

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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 [54,55] 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 [22] 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) ,

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The gauge freedom postulated in [22] 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 [27]. The new formulation will be an extension of the one given in [26] in the flatsuperspace case.

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Our next task is to derive an AdS extension of the gauge-invariant action in Minkowski superspace (given by eq. (2.8) in [26]). 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 [26].
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.
The above procedure leads to the following action in AdS, which is invariant under the gauge transformations (3.7) and (3.6) The above action is real due to the identity (2.6). In the limit of vanishing curvature of the AdS superspace (µ → 0), (3.10) reduces to the action constructed in [26]. The V-gauge freedom (3.7) allows us to gauge away Σ α(s−1)α(s−2) , In this gauge, the action (3.10) reduces to that describing the longitudinal formulation for the massless superspin-s multiplet [22]. The gauge condition (3.11) does not fix completely the V-gauge freedom. The residual gauge transformations are generated by with L α(s)α(s−1) being an unconstrained superfield. With this expression for V α(s−1)α(s−1) , the gauge transformations (3.7a) and (3.7b) coincide with (3.3). Thus, the action (3.10) indeed provides an off-shell formulation for the massless superspin-s multiplet in the AdS superspace.
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 The former makes the gauge symmetry (3.6) manifestly realised, while the latter turns the ζ-transformation (3.7a) into a manifest symmetry.

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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 [26], the theory with action (3.16) can be reformulated in terms of a transverse linear superfield by applying the duality transformation introduced in [22]. 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(s)α(s) = G α(s)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) and

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U 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

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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 [22]. The gauge condition (3.23) is preserved by residual local Vand 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 [22].

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

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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 [56]. Imposing the gauge condition Φ = 0 reduces the action (3.26) to the original longitudinal formulation for the massless gravitino multiplet in AdS [22]. 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 [57]. In Minkowski superspace, there exists one more dual realisation for the massless gravitino multiplet model [26] 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

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and its conjugate using the scheme described in [22]. 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 [22].

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 [22].

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For completeness, we also give the conjugate equation Similar considerations for the transverse formulation lead to the following nonconformal supercurrent multiplet Thus, the trace multipletF α(s−1)α(s) is longitudinal linear. In the flat-superspace limit, the higher spin supercurrent multiplets (4.8) and (4.9) reduce to those described in [25].
As in [25], 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 U (m,n) (ζ,ζ) := ζ α 1 . . . ζ αmζα 1 . . .ζα n U α 1 ...αmα 1 ...αn , (4.10) 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 (4.12b) Making use of the above notation, the transverse linear condition (4.8a) and its conjugate becomeD  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.
Since the trace multiplet T α(s−1)α(s−2) is transverse, eq. (4.8a), there exists a welldefined complex tensor operator X α(s−1)α(s−1) such that (4.17) Let us introduce the real U α(s−1)α(s−1) and imaginary V α(s−1)α(s−1) parts of X α(s−1)α(s−1) , Then it may be checked that the operators 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.
There exists an improvement transformation for the supercurrent multiplet (4.8). Given a chiral scalar superfield Ω, introduce The operators J (s,s) and T (s−1,s−2) prove to obey the conservation equation (4.8).

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 [26].
Taking the sum of (4.24a) and (4.24b) leads to  In the flat-superspace limit, the higher spin supercurrent multiplet described by eqs. (4.22) and (4.25) reduces to the one proposed in [26].

Improvement transformation
There exist an improvement transformation for the supercurrent multiplet (4.24). Given a chiral scalar superfield Ω, we introduce

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 [25,27] J (s,s) = s k=0 (−1) k s k The calculation of (5.3) in AdS is much more complicated than in flat superspace due to the fact that the algebra of covariant derivatives (2.4) is nontrivial. Let us sketch the main steps in evaluating the left-hand side of eq. (5.3) with J (s,s) given by (5.2). We start with the obvious relations ∂ ∂ζ α D (1,1) = 2iζαD αα , To simplify eq. (5.4b), we may pushζαD αα , say, to the left provided that we take into account its commutator with D (1,1) : [ζαD αα , D (1,1) ] = −4iμµ ζ αζαζβMαβ . 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 [25] 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

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.

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The above results can be recast in terms of the matrix conformal supercurrent J (s,s) = J ij (s,s) with components

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

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As in the single field case we consider a general ansatz Then for k = 1, 2, . . . s − 2, condition (i) implies that the coefficients c k must satisfy while (ii) gives the following equation The above conditions coincide with eqs.
Starting with a general ansatz and imposing conditions (i) and (ii) yield the following equations for the coefficients d k

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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 (5.30c) is positive, while the left-hand side is negative, (s − 1)d s−2 − d 0 < 0. Therefore, our solution (5.31) is only consistent for s = 2n, n = 1, 2, . . . . Finally, we consider A ij = −A ji with the corresponding J (s,s) given by (5.18). The analysis in this case is similar to the one presented above and we will simply state the results. If s is odd the non-conformal higher spin supercurrents exist if {A,M } = 0. The trace supercurrentT (s−2,s−1) is given by (5.26) with the coefficients c k given by If s is even the non-conformal higher spin supercurrents exist if [A,M ] = 0. The trace supercurrentT (s−2,s−1) is given by (5.29) with the coefficients d k given by 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 means that for even s we can define a more general supercurrent (5.34), where H ij is a generic matrix which we can split as before into the symmetric and antisymmetric parts, H ij = S ij + iA ij . From the above consideration it then follows that the corresponding more general solution forT (s−2,s−1) reads

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 [27] and is given by

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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: (6.9c)

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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 [26,27].

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 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.

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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. (4.31) becomes

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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 25) 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 In the rest of this section, we will discuss several applications of the results obtained in the paper.

Higher spin supercurrents for a massive chiral multiplet: 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 [29] 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 [29]. The point is that the authors of [29] 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,

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This theory is known to be dual to a tensor multiplet model [62] S[L] = − 1 2 d 4 xd 2 θd 2θ E L 2 , (7.5) 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 ( The corresponding trace multiplet proves to be

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 [30]. 12 It was claimed in [30] that the flatsuperspace model 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 [30] because only a particular ansatz for the Noether procedure was studied in [30].

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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 [31] as an extension of the non-supersymmetric approach [67].
As a generalisation of the conserved supercurrents J α(n)α(n) = W α(n)Wα(n) , one can introduce and can be viewed as Noether currents for the generalised superconformal higher spin multiplets introduced in [27]. Starting from the conserved supercurrents (7.26), one can construct a generalisation of (7.23). We will not elaborate on a construction here.

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., [3,21] JHEP09(2018)027 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

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. Given an integer s ≥ 2, the massless spin-s field [63] is described by real potentials h α(s)α(s) and h α(s−2)α(s−2) with the gauge freedom 16 for an arbitrary real gauge parameter λ α(s−1)α(s−1) . The field h α(s)α(s) may be interpreted as a conformal spin-s field [64,65]. 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 [63] given in section 6.9 of [3].

(B.2)
Requiring that S Our derivation of (B.3) is analogous to that given in [36]. Let us introduce the following operators 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  .14b), respectively. Similarly, the coefficients d k and g k in (C.20a) and (C.20b) differ from similar coefficients in (C.18a) and (C.18b) by a factor of i. This means that for odd s we can define a more general supercurrent (C.21), where C ij is a generic matrix which we can split as before into the symmetric and antisymmetric parts, C ij = S ij + iA ij . From the above consideration it then follows that the corresponding more general solution for t (s−2,s−1) reads t Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.