${\cal N}=2$ supersymmetric higher spin gauge theories and current multiplets in three dimensions

We describe several families of primary linear supermultiplets coupled to three-dimensional ${\cal N}=2$ conformal supergravity and use them to construct topological $BF$-type terms. We introduce conformal higher-spin gauge superfields and associate with them Chern-Simons-type actions that are constructed as an extension of the linearised action for ${\cal N}=2$ conformal supergravity. These actions possess gauge and super-Weyl invariance in any conformally flat superspace and involve a higher-spin generalisation of the linearised ${\cal N}=2$ super-Cotton tensor. For massless higher-spin supermultiplets in (1,1) anti-de Sitter (AdS) superspace, we propose two off-shell Lagrangian gauge formulations, which are related to each other by a dually transformation. Making use of these massless theories allows us to formulate consistent higher-spin supercurrent multiplets in (1,1) AdS superspace. Explicit examples of such supercurrent multiplets are provided for models of massive chiral supermultiplets. Off-shell formulations for massive higher-spin supermultiplets in (1,1) AdS superspace are proposed.


Introduction
In four spacetime dimensions (4D), there exist two off-shell formulations for pure N = 1 anti-de Sitter (AdS) supergravity: minimal (see, e.g., [1,2] for pedagogical reviews) and non-minimal [3]. 1 These supergravity theories are related to each other by a superfield duality transformation [3] and possess a single maximally supersymmetric solution, the famous N = 1 AdS superspace [4,5,6], which is the simplest member of the family of N -extended AdS superspaces AdS 4|4N = OSp(2|4) SO(3, 1) × SO(N ) . (1.1) These supergravity theories are also intimately related to the two dually equivalent series of massless gauge supermultiplets of half-integer superspin s + 1 2 ≥ 3 2 (describing two ordinary massless spin-(s+ 1 2 ) and spin-(s+1) fields on the mass shell) in AdS 4 , which were proposed in [7] as a natural extension of the formulations in Minkowski space constructed earlier in [8,9]. Specifically, for the lowest superspin value corresponding to s = 1, one series yields the linearised action for minimal AdS supergravity, while the other leads to the linearised non-minimal AdS supergravity.
In the 3D case, the AdS group is reducible, SO(2, 2) ∼ = SL(2, R) × SL(2, R) /Z 2 , and so are its simplest supersymmetric extensions, OSp(p|2; R)×OSp(q|2; R). This implies that N -extended AdS supergravity exists in several incarnations [10]. These are known as the (p, q) AdS supergravity theories, where the non-negative integers p ≥ q are such that N = p + q. For any allowed values of p and q, the pure (p, q) AdS supergravity was constructed in [10] as a Chern-Simons theory with the gauge group OSp(p|2; R)×OSp(q|2; R). The Chern-Simons construction is not particularly useful when one is interested in coupling AdS supergravity to supersymmetric matter. This is one of the reasons why off-shell formulations for 3D N -extended conformal supergravity have been developed [11,12,13].
Within the off-shell supergravity framework of [12], (p, q) AdS superspace AdS (3|p,q) = OSp(p|2; R) × OSp(q|2; R) SL(2, R) × SO(p) × SO(q) (1.2) originates as a maximally symmetric conformally flat supergeometry with covariantly constant torsion and curvature generated by a tensor S IJ = S JI [14], with the SO(N ) indices I, J taking values from 1 to N . It turns out that the symmetric matrix S = (S IJ ) is nonsingular, and the parameters p and q = N −p determine its signature. The ordinary AdS space is the bosonic body of AdS (3|p,q) . The curvature of AdS 3 is proportional to tr(S 2 ). The Killing vector fields of AdS (3|p,q) can be shown to generate the isometry group OSp(p|2; R) × OSp(q|2; R), see [14] for the technical details.
The 3D N = 2 supersymmetry is a natural cousin of the 4D N = 1 one. This is the lowest value of N for which there are at least two inequivalent AdS superspaces, AdS (3|1, 1) and AdS (3|2,0) , which were thoroughly studied in [15]. The former is the 3D counterpart of the 4D N = 1 AdS superspace, while the latter has no 4D analogue. The superspaces AdS (3|1, 1) and AdS (3|2,0) are maximally symmetric solutions of the known off-shell N = 2 AdS supergravity theories presented in [15]. AdS (3|1,1) is the unique maximally symmetric solution of the two dually equivalent (1,1) AdS supergravity theories, minimal and nonminimal ones. AdS (3|2,0) is the unique maximally symmetric solution of the (2,0) AdS supergravity, which was originally formulated in [11] in the component setting. The early superspace descriptions of the minimal (1,1) supergravity were given in [16,17].
Since there are three off-shell N = 2 AdS supergravity theories [15], one might expect existence of three series of massless higher-spin gauge supermultiplets. In this paper we present two series of massless higher-spin actions, which are associated with the minimal and the non-minimal (1,1) AdS supergravity theories, respectively, and which generalise similar constructions in the super-Poincaré case [18]. Off-shell higher-spin actions with (2,0) AdS supersymmetry will be described in a separate work [19].
Similar to the pure gravity and simple supergravity theories in three dimensions, pure N = 2 supergravity (massless superspin-3/2 multiplet) and its higher-spin extensions have no propagating degrees of freedom. Nevertheless, there are at least two nontrivial applications of the massless higher-spin gauge supermultiplets. Firstly, one can follow the pattern of topologically massive (super)gravity [20,21,22,23] and construct massive higher-spin supermultiplets by combining a massless action with a higher-spin extension of the action for linearised conformal supergravity. This has been achieved in [18] in the N = 2 super-Poincaré case, and similar ideas have been implemented in the frameworks of N = 1 Poincaré and AdS supersymmetry [24,25]. Secondly, making use of the off-shell formulations for massless higher-spin supermultiplets in AdS 3 , 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 3 . One can then look for explicit realisations of such higher-spin supercurrents in concrete supersymmetric theories in AdS 3 . Such a program in the 4D N = 1 Poincaré and AdS supersymmetric cases has been described in a series of papers [26,27,28,29]. Alternatively, one can develop a 3D extension of the approach advocated in [30,31,32] and based on the use of superfield Noether procedures [33,34].
Before we turn to the main body of this work, a few comments are in order about maximally supersymmetric backgrounds in the off-shell N = 2 supergravity theories, since the superspaces AdS (3|1,1) and AdS (3|2,0) are special examples of such supermanifolds. The most general maximally supersymmetric backgrounds are characterised by several conditions [35,36] on the torsion superfields R, S and C a , which determine the superspace geometry of N = 2 conformal supergravity (see section 2 for the technical details). These requirements are as follows: The (1,1) AdS superspace is singled out by the conditions S = 0 and C a = 0, with R and its conjugateR having non-zero constant values. The (1,1) AdS superspace is characterised by the following algebra of covariant derivatives [15]: where M ab denotes the Lorentz generator. The (2,0) AdS superspace belongs to the family of all maximally supersymmetric backgrounds with R = 0. These backgrounds are characterised by the following algebra of covariant derivatives [35,36]: Here J is the generator of the N = 2 R-symmetry group, U(1) R , and M a := 1 2 ε abc M bc . The solution with C a = 0 and S = 0 corresponds to (2,0) AdS superspace [15]. It may be shown that the U(1) R connection is flat if and only if S = 0 [12]. The non-vanishing U(1) R curvature is the main reason why the structure of massless higher-spin gauge supermultiplets in (2,0) AdS superspace [19] considerably differs from their counterparts with (1,1) AdS supersymmetry. This paper is organised as follows. In section 2, primary linear supermultiplets coupled to N = 2 conformal supergravity are described and then used to construct topological BF -type terms. Given a positive integer n > 0, we introduce a conformal gauge superfield H α(n) and show that, for every conformally flat superspace, there exists a unique primary gauge-invariant descendant W α(n) (H) of H α(n) with the properties (2.25). In terms of H α(n) and W α(n) (H) we construct a higher-spin extension of the linearised action for N = 2 conformal supergravity. Section 3 provides a brief summary of the key results concerning the (1,1) AdS superspace and superfield representations of the corresponding isometry group. In sections 4 and 5, we present two dually equivalent off-shell Lagrangian formulations for every massless higher-spin supermultiplet in (1,1) AdS superspace. Making use of these massless theories allows us to formulate, in section 6, consistent higher-spin supercurrent multiplets. Explicit examples of such supercurrents are provided in sections 7 and 8 for models described by chiral supermultiplets. In section 9 we discuss several extensions of the constructions obtained. The paper is concluded with two appendices. Appendix A describes our notation, conventions and several important identities involving the spinor covariant derivatives of (1,1) AdS superspace. Appendix B describes the N = 2 → N = 1 superspace reduction of the massless integer superspin model (5.6) in Minkowski superspace.

Superconformal higher-spin multiplets
Before presenting superconformal higher-spin multiplets, we give a succinct review of 3D N = 2 conformal supergravity following [11,12]. There exists more general formulation for conformal supergravity [13] known as the N = 2 conformal superspace. For our purposes it suffices to use the formulation of [12], which is obtained from the N = 2 conformal superspace by partially fixing the gauge freedom.
The superspace covariant derivatives have the form (2.1) Here E A and Ω A denote the inverse supervielbein and the Lorentz connection, respectively, The Lorentz generators with two vector indices (M ab = −M ba ), with one vector index (M a ) and with two spinor indices (M αβ = M βα ) are defined in Appendix A. The U(1) R generator J in (2.1) is defined to act on the covariant derivatives as follows: In order to describe N = 2 conformal supergravity, the torsion has to obey the covariant constraints proposed in [11]. Solving the constraints leads to the following algebra of covariant derivatives [12,15] where the U(1) R charges of the torsion superfields R,R and C αβ are −2, +2 and 0, respectively. They also satisfy the Bianchi identities Throughout this paper, we make use of the definitions D 2 := D α D α andD 2 :=D αD α .
The algebra of covariant derivatives given by (2.4) does not change under the super-Weyl transformation [12,15] which induces the following transformation of the torsion tensors: where the parameter σ is an arbitrary real scalar superfield. The super-Weyl invariance (2.6) is intrinsic to conformal supergravity. For every supergravity-matter system, its action is required to be a super-Weyl invariant functional of the supergravity multiplet coupled to certain conformal compensators, see [12,15] for more details.
The N = 2 supersymmetric extension of the Cotton tensor [37] is given by It may be checked that W αβ transforms homogeneously, The curved superspace is conformally flat if and only if W αβ = 0 [13].

Primary superfields
Let T α(n) := T α 1 ...αn = T (α 1 ...αn) be a symmetric rank-n spinor superfield of U(1) R charge q, JT α(n) = qT α(n) . (2.10) The T α(n) is said to be super-Weyl primary of dimension d if its infinitesimal super-Weyl transformation law is As follows from (2.8), the super-Cotton tensor is super-Weyl primary of dimension +2. We now introduce several types of primary superfields that will be important for our subsequent consideration.
A symmetric rank-n spinor superfield G α(n) is called longitudinal linear if it obeys the following first-order constraintD which implies If G α(n) is super-Weyl primary, then the constraint (2.12) is consistent provided the dimension d G (n) and U(1) R charge q G (n) of G α(n) are related to each other as follows: (2.14) In the scalar case, n = 0, the constraint (2.12) becomes the condition of covariant chirality, D α G = 0. The dimension d G and U(1) R charge q G of any primary chiral scalar superfield G are related as d G + q G = 0, in accordance with [12].
Given a positive integer n, a symmetric rank-n spinor superfield Γ α(n) is called transverse linear if it obeys the first-order constraint which implies If Γ α(n) is super-Weyl primary, then the constraint (2.15) is consistent provided the dimension d Γ (n) and U(1) R charge q Γ (n) of Γ α(n) are related to each other as follows: In the n = 0 case, the constraint (2.15) is not defined. However its corollary (2.16) is perfectly consistent, and defines a covariantly linear scalar superfield Γ. The dimension d Γ and U(1) R charge q Γ of any primary linear scalar Γ are related as d Γ + q Γ = 1, in accordance with [12].
In the case of 4D N = 1 AdS supersymmetry, longitudinal linear and transverse linear superfields were pioneered by Ivanov and Sorin [6] who studied the superfield representations of the AdS isometry group OSp(1|4). In the framework of 4D N = 1 conformal supergravity, primary longitudinal linear and transverse linear supermultiplets were introduced for the first time by Kugo and Uehara [38]. Such superfields were used in [7,8,9,18,29] for the description of off-shell massless gauge theories in four and three dimensions.
In the n = 0 case, the prepotential solution (2.19b) is still valid. The prepotential Φ α can be chosen to be unconstrained complex provided the constraint (2.18) is the only condition imposed on Γ. However, if we are dealing with a real linear superfield, then the constraints are solved [12] in terms of an unconstrained real prepotential V , which is defined modulo gauge transformations of the form: If L is super-Weyl primary, then eq. (2.17) tells us that that the dimension of L is +1. In this case it is consistent to consider the gauge prepotential V to be inert under the super-Weyl transformations [12], δ σ V = 0.
Let us assume that the background curved superspace allows the existence of a real transverse linear superfield W α(n) =W α(n) , Then it is automatically conserved, in accordance with (2.4b). The super-Cotton tensor W αβ is an example of such supermultiplets. If W α(n) is super-Weyl primary, then its dimension is equal to (1 + n/2), in accordance with (2.17). As will be shown in the next subsection, a solution to (2.25) in terms of an unconstrained prepotential exists for every conformally flat superspace.

Conformal gauge superfields
Let n be a positive integer. A real symmetric rank-n spinor superfield H α(n) is said to be a conformal gauge supermultiplet if (i) it is super-Weyl primary of dimension (−n/2), and (ii) it is defined modulo gauge transformations of the form with the gauge parameter λ α(n−1) being unconstrained complex. The dimension of H α(n) in (2.27) is uniquely fixed by requiring the longitudinal linear superfield g α(n) =D (α 1 λ α 2 ...αn) in the right-hand side of (2.28) to be super-Weyl primary. Indeed, the gauge parameter g α(n) must be neutral with respect to the R-symmetry group U(1) R since H α(n) is real, and then the dimension of g α(n) is equal to (−n/2), in accordance with (2.14).
Starting with H α(n) one can construct its real descendant W α(n) (H) = AH α(n) , where A is a linear differential operator involving D A , the torsion superfields and their covariant derivatives, with the following the properties: 2. The gauge variation of W α(n) vanishes if the superspace is conformally flat, where W (2) is the super-Cotton tensor (2.7).
3. W α(n) is divergenceless if the superspace is conformally flat, Here O W (2) stands for contributions containing the super-Cotton tensor and its covariant derivatives.
In general, W α(n) (H) is uniquely defined modulo a normalisation and contributions involving the super-Cotton tensor (2.7).
Suppose that the background curved superspace M 3|4 is conformally flat, and obeys the conservation equations (2.25). These properties and the super-Weyl transformation laws (2.27) and (2.29) imply that the action 2 is gauge and super-Weyl invariant, In accordance with the results of [37,45], it is natural to think of H αβ and W αβ (H) as the linearised prepotential for N = 2 conformal supergravity and the linearised super-Cotton tensor respectively. It is worth recalling that (2.32) is the equation of motion for conformal supergravity. The functional (2.34) is proportional to the linearised action for conformal supergravity, which is obtained by linearising the nonlinear action for N = 2 conformal supergravity [39,40] around a stationary point defined by (2.32). We can interpret W α(n) to be a linearised higher-spin super-Cotton tensor. We now turn to constructing W α(n) on a conformally flat superspace.
In Minkowski superspace, the linearised higher-spin super-Cotton tensors were constructed in [18], and here we reproduce these results. Associated with a real prepotential H α(n) = H α 1 ...αn is the following real symmetric rank-n spinor descendant and D A = (∂ a , D α ,D α ) are the covariant derivatives for Minkowski superspace, The field strength (2.36) is invariant, under the gauge transformations where the gauge parameter λ α(n−1) is unconstrained complex. The field strength (2.36) is conserved, Making use of W α(n) allows us to construct the higher-spin super-Cotton tensor W α(n) in any conformally flat superspace M 3|4 .
In accordance with (2.6), for a conformally flat superspace M 3|4 we can choose a local frame in which the covariant derivatives have the form for some real scale factor σ. Then, in accordance with (2.29), the higher-spin super-Cotton tensor W α(n) in M 3|4 is related to the flat-space one, eq. (2.36), by the rule Similarly, eq. (2.27) tells us that the prepotentials H α(n) and H α(n) can be chosen to be related to each other by In general, it is a difficult technical problem to express W α(n) in terms of the covariant derivatives D A and the gauge prepotential H α(n) , for arbitrary n.
There exists a refined version of the representation (2.42) for those conformally flat superspaces which are characterised by the condition This family includes the (1,1) AdS superspace defined by the (anti)commutation relations (1.5). If (2.45) holds, then eq. (2.6d) tells that the scale factor in (2.42) is constrained, with the chiral scalar η being, in principle, arbitrary. Now, applying a local R-symmetry transformation leads to covariant derivatives without U(1) R connection. The resulting covariant derivatives are In the case of (1,1) AdS superspace, the scale factor η was computed in [15].

(1,1) AdS superspace
In this section we give a brief summary of the key results concerning the (1,1) AdS superspace [15], as well as elaborate on superfield representations of the (1,1) AdS isometry group. The covariant derivatives of AdS (3|1,1) satisfy the following algebra [15]: with µ = 0 being a complex parameter. As compared with (1.5), we have denoted R = µ. This notation will be used in the remainder of this paper.
The covariantly transverse linear and longitudinal linear superfields on an arbitrary supergravity background were described in the previous section. In the case of (1,1) AdS superspace, such superfields play an important role. One can define projectors P ⊥ n and P || n on the spaces of transverse linear and longitudinal linear superfields, respectively. The projectors are with the properties Given a complex tensor superfield V α(n) with n = 0, it can be represented as a sum of transverse linear and longitudinal linear multiplets, Choosing V α(n) to be longitudinal linear (G α(n) ) or transverse linear (Γ α(n) ), the above identity gives the relations (2.19a) and (2.19b) for some prepotentials Ψ α(n−1) and Φ α(n+1) , respectively.
In accordance with the general formalism of [2], the isometries of AdS (3|1,1) are generated by those real supervector fields λ A E A which obey the Killing equation and l ab is some local Lorentz parameter. As demonstrated in [15], this equation implies that the parameters λ α and l ab are uniquely expressed in terms of the vector λ a , and the vector parameter obeys the equation In comparison with the 3D N = 2 Minkowski superspace, the specific feature of AdS (3|1,1) is that any two of the three parameters {λ αβ , λ α , l αβ } are expressed in terms of the third parameter, in particular From (3.7) and (3.9) we deduceD Every solution λ A of the above relations is called is a Killing supervector field of AdS (3|1,1) . These supervector fields can be shown to generate the isometry group of AdS (3|1,1) , In Minkowski superspace M 3|4 , 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 (1,1) AdS superspace, every chiral integral can always be recast as a full superspace integral. Associated with a scalar superfield L is the following supersymmetric invariant where E denotes the chiral integration measure. Let L c be a covariantly chiral scalar Lagrangian,D α L c = 0. It generates a supersymmetric invariant of the form d 3 xd 2 θ E L c . The specific feature of (1,1) AdS superspace is that the chiral action can equivalently be written as an integral over the full superspace [15] Unlike the flat superspace case, the integral on the right does not vanish in AdS.
Supersymmetric invariant (3.11) can be reduced to component fields by the rule [35]  (3.14) In general, the θ,θ-independent component, T | θ=θ=0 , of a superfield T (x, θ,θ) is denoted T |. To complete the formalism of component reduction, we only need the following relation In what follows, we will work with full superspace integrals only and make use of the notation d 3|4 z := d 3 xd 2 θd 2θ .
4 Massless half-integer superspin gauge theories in (1,1) AdS superspace The superconformal higher-spin action (2.34) in a conformally flat superspace is formulated in terms of the conformal gauge superfields H α(n) . The same gauge superfield, at least for n = 2s, with s = 1, 2, . . . , can be used to construct massless actions in two of the three N = 2 maximally symmetric backgrounds, which are Minkowski superspace and (1,1) AdS superspace. Such actions, however, involve not only H α(n) but also some compensators.
In Minkowski space, there are two off-shell formulations for the massless N = 2 multiplet of half-integer superspin (s + 1/2), with s = 2, 3, . . ., which are dual to each other [18]. They are referred to as transverse and longitudinal. Here we extend these gauge theories to (1,1) AdS superspace.
The dynamical superfields H α(2s) and Γ α(2s−2) are postulated to be defined modulo gauge transformations of the form where the complex gauge parameter λ α(2s−1) is unconstrained. The gauge transformation of H α(2s) coincides with (2.28) for n = 2s. From δ λ Γ α(2s−2) we read off the gauge transformation of the prepotential Φ α(2s−1) defined by eq. (4.2), which is Modulo an overall normalisation factor, there is a unique quadratic action which is invariant under the gauge transformations (4.4). It is given by . (4.6) In the flat superspace limit, this action reduces to the one derived in [18].
The s = 1 choice was excluded from the above consideration, since the constraint (2.15) is not defined for n = 0. However, as discussed in section 2, the corollary (2.16) of (2.15) is perfectly consistent for n = 0 and defines a covariantly transverse linear scalar superfield (2.18), We therefore postulate Γ and its conjugateΓ to be the compensators in the s = 1 case. Choosing s = 1 in the gauge transformation law (4.4) gives The variation δ λ Γ is compatible with the constraint (4.7), that is (D 2 −µ)δ λ Γ = 0. Finally, choosing s = 1 in (4.6) gives the linearised action for non-minimal (1,1) AdS supergravity, which was originally derived in section 9.2 of [15].

Longitudinal formulation
The longitudinal formulation for the massless superspin-(s + 1 2 ) multiplet is described in terms of the following variables: (4.9) Here H α(2s) is the same as in (4.1), and the complex superfield G α(2s−2) is longitudinal linear, eq. (2.12). In accordance with (2.19a), the constraint (2.12) can be solved in terms of an unconstrained complex prepotential Ψ α(2s−3) , which is defined modulo gauge transformations of the form with the gauge parameter ζ α(2s−4) being unconstrained complex.

(4.15)
This action is invariant under the gauge transformations In the flat superspace limit, this action reduces to the one derived in [18].
In the s = 1 case, the compensator G becomes covariantly chiral,D α G = 0. Choosing s = 1 in (4.15) gives the linearised action for minimal (1,1) AdS supergravity, which was originally derived in section 9.1 of [15], provided we identify G = 3σ. Choosing s = 1 in the gauge transformation law (4.16) gives It is clear that the variation δ λ G is covariantly chiral.

AdS superspace
When attempting to develop a Lagrangian formulation for a massless multiplet of superspin s, where s = 1, 2, . . . , a naive expectation is that the dynamical variables of such a theory should consist of a conformal gauge superfield H α(2s−1) =H α(2s−1) , introduced in subsection 2.3, in conjunction with some compensator(s). Instead, our approach in this section will be based on developing 3D N = 2 analogues of the two dually equivalent off-shell formulations, the so-called longitudinal and transverse ones, for the massless N = 1 multiplets of integer superspin in AdS 4 [7]. Then we will provide a reformulation of the longitudinal formulation derived in the next subsection in a way similar to the one proposed in the 4D N = 1 AdS case [29]. Such a reformulation naturally leads to the appearance of a conformal gauge superfield H α(2s−1) .

Longitudinal formulation
Given an integer s ≥ 1, the longitudinal formulation for the massless superspin-s multiplet is realised in terms of the following dynamical variables: Here, U α(2s−2) is an unconstrained real superfield, and the complex superfield G α(2s) is longitudinal linear, eq. (2.12). In accordance with (2.19a), the constraint (2.12) can be solved in terms of an unconstrained complex prepotential Ψ α(2s−1) , which is defined modulo gauge transformations of the form with the gauge parameter ζ α(2s−2) being unconstrained complex.
We postulate the dynamical superfields U α(2s−2) and Γ α(2s) to be defined modulo gauge transformations of the form Here the gauge parameter L α(2s−1) is an unconstrained complex superfield, and γ α(2s−2) := D βL βα(2s−2) is transverse linear. From (5.4b) we read off the gauge transformation law of the prepotential, Modulo an overall normalisation factor, there is a unique quadratic action which is invariant under the gauge transformations (5.4). The action is The special s = 1 case, which corresponds to the massless gravitino multiplet, will be studied in more detail in subsection 5.4.

Transverse formulation
The transverse formulation for the massless superspin-s multiplet is realised in terms of the following dynamical variables: Here, U α(2s−2) is the same as in (5.1), and the complex superfield Γ α(2s) is transverse linear, eq. (2.15). In accordance with (2.19b), the constraint on Γ α(2s) is solved in terms of an unconstrained prepotential Φ α(2s+1) , which is defined modulo gauge transformations of the form with the gauge parameter ξ α(2s+2) being unconstrained.
The action (5.18) includes a single term which involves the 'naked' gauge fieldΨ α(2s−1) and not the field strengthḠ α(2s) , the latter being defined by (5.2) and invariant under the ζ-transformation (5.16a). This is actually a BF term, for it can be written in two different forms The former makes the gauge symmetry (5.15) manifestly realised, while the latter turns the ζ-transformation (5.16a) into a manifest symmetry.
Making use of (5.21) leads to a different representation for the action (5.18). It is Before concluding this section, it is worth discussing the structure of the dynamical variable Ψ α(2s−1) . This superfield is unconstrained complex, and its gauge transformation law is given by eq. (5.16a). Comparing (5.16a) with the gauge transformation law (2.28) n = 2s − 1, which corresponds to the conformal gauge superfield H α(2s−1) , we see that Ψ α(2s−1) may be interpreted as a complex conformal gauge superfield.

Massless gravitino multiplet
The massless gravitino multiplet, which corresponds to the s = 1 case, was excluded from our consideration of the previous subsection. Here we will fill the gap.
The (generalised) longitudinal formulation for the gravitino multiplet is described by the action where Φ is a covariantly chiral scalar superfield,D α Φ = 0, and This action is invariant under gauge transformations of the form where the gauge parameters V and ζ are unconstrained complex superfields.
The gauge V-freedom (5.25) allows us to impose the condition Φ = 0. In this gauge the action (5.23) turns into (5.6) with s = 1, and the residual gauge V-freedom is described by V = D β L β , where the spinor gauge parameter L α is unconstrained complex.
The action (5.23) involves the chiral scalar Φ and its conjugate only in the combination (ϕ +φ), where ϕ = Φ/µ. This means that the model (5.23) possesses a dual formulation realised in terms of a real linear superfield subject to the constraint (2.22).

Higher-spin supercurrents
Inspired by the analysis of Dumitrescu and Seiberg [41], the most general supercurrent multiplets for theories with (1,1) AdS or (2,0) AdS supersymmetry were introduced in [15], with the (1,1) AdS case being a natural extension of the 4D N = 1 AdS supercurrents classified in [3,42]. Here we will formulate higher-spin supercurrents in (1,1) AdS superspace by making use of the off-shell formulations for massless supersymmetric higher-spin gauge theories in (1,1) AdS superspace, which have been constructed in the previous two sections. Our analysis will be analogous to the one recently given in the 4D N = 1 case [29].

Non-conformal supercurrents: Half-integer superspin
The two off-shell formulations for the massless supers[in-(s + 1 2 ) multiplet, which we reviewed in sections 4.1 and 4.2, lead to different higher-spin supercurrent multiplets. In this subsection we first described the explicit structure of these supermultiplets and then show how they are related to each other.

Longitudinal supercurrent
In the framework of the longitudinal formulation (4.15), let us couple the prepotentials H α(2s) , Ψ α(2s−3) andΨ α(2s−3) , to external sources . source to be invariant under the gauge transformations (4.4a) and (4.14) gives the following conservation equation:D For completeness, we also give the conjugate equation As in [29], it is useful to introduce auxiliary real variables ζ α . Given a tensor superfield U α(m) , we associate with it the following field which is homogeneous of degree m in the variables ζ α . We introduce operators that increase the degree of homogeneity in the variable ζ α , We also introduce two operators that decrease the degree of homogeneity in the variable ζ α , specifically Making use of the above notation, the transverse linear condition (6.2a) and its conjugate becomeD The conservation equations (6.2b) and (6.2c) turn into Since (D (−1) ) 2 J (2s) = 0, the conservation equation (6.7a) is consistent provided This is indeed true, as a consequence of the transverse linear condition (6.6a).

Improvement transformation
We now construct a well-defined improvement transformation which converts the higher-spin supercurrent (6.2) to (6.11), thus showing that they are indeed equivalent.
The transverse linearity condition (6.2a) implies that there exists a well-defined complex tensor operator X α(2s−2) such that Let us split X α(2s−2) into its real and imaginary parts, Then one may check that the operators satisfy the conservation equation (6.11b) and the longitudinal linear condition (6.11a).
The improvement transformation (6.14) turns the higher-spin supercurrent (6.2) to (6.11) It is also not difficult to construct an inverse improvement transformation converting the higher-spin supercurrent (6.11) to (6.2). Therefore the higher-spin supercurrents (6.2) and (6.11) are equivalent, and it is suffices to work with one of them, say, the longitudinal supermultiplet (6.2). The situation proves to be analogous in the integer superspin case, for which we will formulate in the next subsection a higher-spin supercurrent associated with the new gauge formulation (5.18).

Non-conformal supercurrents: Integer superspin
We now make use of the new gauge formulation (5.18), or equivalently (5.22), for the integer superspin-s multiplet to derive the 3D analogue of the non-conformal higher-spin supercurrents proposed in [29].

Superconformal model for a chiral superfield
Let us consider the superconformal theory of a single chiral scalar superfield where Φ is covariantly chiral,D α Φ = 0. We construct the following conformal supercurrent J (2s) , which is a minimal extension of the conserved supercurrent constructed in flat N = 2 Minkowski superspace [43].
Making use of the massless equations of motion, (D 2 − 4μ) Φ = 0, one may check that J (2s) satisfies the conservation equation 3) The calculation of (7.3) in AdS is much more complicated than in flat superspace due to the fact that the algebra of covariant derivatives (3.1) is nontrivial. Let us sketch the main steps in evaluating the left-hand side of eq. (7.3) with J (2s) given by (7.2). We start with the obvious relations To simplify eq. (7.4b), we may push ζ β D αβ , say, to the left provided that we take into account its commutator with D (2) : Associated with the Lorentz generators are the operators where M (2) appears in the right-hand side of (7.5). These operators annihilate every superfield U (m) (ζ) of the form (6.3), 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 (7.2) does obey the conservation equation (7.3).

Non-superconformal model for a chiral superfield
Let us now add the mass term to (7.1) and consider the following action with m a complex mass parameter. In the massive case J (2s) satisfies a more general conservation equation (6.7b) for some superfieldT (2s−3) . Making use of the equations of motion we obtain where we have denoted We now look for a superfieldT (2s−3) such that (i) it obeys the transverse antilinear constraint (6.6b); and (ii) it satisfies the equation Our analysis will be similar to the one performed in [29] in the case of four-dimensional AdS. 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 It turns out that the equations (7.15) 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 (7.15a)-(7.15c) 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 (7.15c) is positive, while the left-hand side is negative, (s − 1)c s−2 + c 0 < 0. Therefore, our solution (7.16) is only consistent for s = 2n + 1, n = 1, 2, . . . . Relations (7.2), (7.14), (7.15d) and (7.16) determine the non-conformal higher-spin supercurrents in the massive chiral model (7.10). Unlike the conformal higher-spin supercurrents (7.2), the non-conformal ones exist only for the odd values of s, s = 2n + 1, with n = 1, 2, . . . .

Superconformal model with N chiral superfields
In this subsection we will generalise the superconformal model (7.1) to the case of N covariantly chiral scalar superfields Φ i , i = 1, . . . N, 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. (7.19) in order to make J − (2s) real. One can show that the currents (7.18) and (7.19) are conserved on-shell: The above results can be recast in terms of the matrix conformal supercurrent J (2s) = J ij (2s) with components which is Hermitian, J (2s) † = J (2s) . The chiral action (7.17) 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 (7.21) transforms as J (2s) → gJ (2s) g −1 .
8 Higher-spin supercurrents for chiral matter: Integer superspin In this section we provide explicit realisations for the fermionic higher-spin supercurrents (integer superspin) in a model of a single massive chiral scalar superfield.
We start by considering the massive action Then the action (8.1) turns into We emphasise that the mass parameter M is now real.
In the massless case, M = 0, the conserved fermionic supercurrent J α(2s−1) is given by We will now construct fermionic higher-spin supercurrents corresponding to the massive model (8.3). Making use of the massive equation of motion we obtain It can be shown that the massive supercurrent J (2s−1) also obeys (6.21).
We now look for a superfield T α(2s−2) such that (i) it obeys the longitudinal linear constraint (6.22); and (ii) it satisfies (6.24), which is a consequence of the conservation equation (6.23). 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 a simple expression for d k : where k = 1, 2, . . . s − 1 and the parameter s is even for J (2s−1) to be non-zero.

Concluding comments
The constructions presented in this paper have several interesting extensions, some of which are briefly discussed below.
Our results can be used to construct off-shell formulations for massive higher-spin supermultiplets in (1,1) AdS superspace. 3 This is readily achieved in the case of a halfinteger superspin by considering two dually equivalent gauge-invariant actions 2 ) and S (s+ 1 2 ) are given by eqs. (4.6) and (4.15), respectively. In the flat-superspace limit, the actions (9.1a) and (9.1b) reduce to those proposed in [18].
We expect that the equations of motion in the topologically massive models (9.1a) and (9.1b) describe a subclass of the irreducible on-shell massive supermultiplets in (1,1) AdS superspace proposed in [45]. This is indeed the case in Minkowski superspace, as demonstrated in [18]. However, analysis of the equations of motion in (1,1) AdS superspace is more complicated since we still do not have a closed-form expression for the higher-spin super-Cotton tensor W α(n) , eq. (2.43), in terms of the prepotential H α(n) and the covariant derivatives D A of (1,1) AdS superspace. Here we simply recall the explicit structure of irreducible on-shell massive higher-spin supermultiplets in (1,1) AdS superspace [45]. Given a positive integer n > 0, such a supermultiplet is realised in terms of a real symmetric rank-n spinor T α(n) constrained by It can be shown that New duality transformations were introduced in [46] for theories formulated in terms of the linearised higher-spin super-Cotton tensor W α(n) in Minkowski superspace, eq. (2.36).
These duality transformations can readily be generalised to arbitrary conformally flat backgrounds by replacing W α(n) with W α(n) given by eq. (2.43).
Following [14], we can introduce a real basis for the spinor covariant derivatives which is obtained by replacing the complex operators D α andD α with ∇ I α , where I = 1, 2, defined by where we have represented µ = − i e 2iϕ |µ|. The new covariant derivatives can be shown to obey the following algebra: The graded commutation relations for the operators ∇ a and ∇ 1 α have the following properties: (i) they do not involve ∇ 2 α ; and (ii) they are identical to those defining the N = 1 AdS superspace, AdS 3|2 , see [14] for the details. These properties mean that AdS 3|2 is naturally embedded in (1,1) AdS superspace as a subspace. The Grassmann variables θ µ I = (θ µ 1 , θ µ 2 ) may be chosen in such a way that AdS 3|2 corresponds to the surface defined by θ µ 2 = 0. Every supersymmetric field theory in (1,1) AdS superspace may be reduced to AdS 3|2 . Such N = 2 → N = 1 AdS superspace reduction may be carried out for all the higher-spin supersymmetric theories constructed in this paper. Implementation of this program will be described elsewhere. Here we only point out that reducing the longitudinal model for the massless superspin-s multiplet (presented in subsection 5.1) to AdS 3|2 leads to a new massless higher-spin gauge theory that was not described in [25].
In the 3D case, an antisymmetric tensor F ab = −F ba is Hodge-dual to a three-vector F a , specifically Then, the symmetric spinor F αβ = F βα , which is associated with F a , can equivalently be defined in terms of F ab : These three algebraic objects, F a , F ab and F αβ , are in one-to-one correspondence to each other, F a ↔ F ab ↔ F αβ . The corresponding inner products are related to each other as follows: The Lorentz generators with two vector indices (M ab = −M ba ), one vector index (M a ) and two spinor indices (M αβ = M βα ) are related to each other by the rules: M a = 1 2 ε abc M bc and M αβ = (γ a ) αβ M a . These generators act on a vector V c and a spinor Ψ γ as follows: The covariant derivatives of (1,1) AdS superspace obey various identities, which can be readily derived from the covariant derivatives algebra (3.1). We have made use of the following identities: where D 2 = D α D α , andD 2 =D αD α . These relations imply the identity which guarantees the reality of the actions considered in the main body of the paper.
B N = 2 → N = 1 superspace reduction In this appendix we carry out the N = 2 → N = 1 superspace reduction [24] of the massless integer-superspin model (5.6). For simplicity our analysis is restricted to flat superspace. An extension to the AdS case will be discussed elsewhere.
The action (B.25) defines a new N = 1 supersymmetric higher-spin theory which did not appear in the analysis of [24]. It may be shown that at the component level it reduces, upon imposing a Wess-Zumino gauge and eliminating the auxiliary fields, to a sum of two massless actions, one of which is the bosonic Fronsdal-type spin-s model and the other is the fermionic Fang-Fronsdal-type spin-(s + 1 2 ) model.