Field theories with (2,0) AdS supersymmetry in ${\cal N}=1$ AdS superspace

In three dimensions, it is known that field theories possessing extended $(p,q)$ anti-de Sitter (AdS) supersymmetry with ${\cal N}=p+q \geq 3$ can be realised in (2,0) AdS superspace. Here we present a formalism to reduce every field theory with (2,0) AdS supersymmetry to ${\cal N}=1$ AdS superspace. As nontrivial examples, we consider supersymmetric nonlinear sigma models formulated in terms of ${\cal N}=2$ chiral and linear supermultiplets. The $(2,0) \to (1,0)$ AdS reduction technique is then applied to the off-shell massless higher-spin supermultiplets in (2,0) AdS superspace constructed in [1]. As a result, for each superspin value $\hat s$, integer $(\hat s= s)$ or half-integer $(\hat s= s+1/2)$, with $s=1,2,\dots $, we obtain two off-shell formulations for a massless ${\cal N}=1$ superspin-$\hat s$ multiplet in AdS${}_3$. These models prove to be related to each other by a superfield Legendre transformation in the flat superspace limit, but the duality is not lifted to the AdS case. Two out of the four series of ${\cal N}=1$ supersymmetric higher-spin models thus derived are new. The constructed massless ${\cal N}=1$ supersymmetric higher-spin actions in AdS${}_3$ are used to formulate (i) higher-spin supercurrent multiplets in ${\cal N}=1$ AdS superspace; and (ii) new topologically massive higher-spin off-shell supermultiplets. Examples of ${\cal N}=1$ higher-spin supercurrents are given for models of a complex scalar supermultiplet. We also present two new off-shell formulations for a massive ${\cal N}=1$ gravitino supermultiplet in AdS${}_3$.


Introduction
In three spacetime dimensions, the AdS group is a product of two simple groups, and so are its supersymmetric extensions OSp(p|2; R) × OSp(q|2; R). 1 This implies that N -extended AdS supergravity exists in several incarnations [2], which are known as the (p, q) AdS supergravity theories, where the integers p ≥ q ≥ 0 are such that N = p + q. The so-called (p, q) AdS superspace [3] AdS (3|p,q) = OSp(p|2; R) × OSp(q|2; R) SL(2, R) × SO(p) × SO(q) (1.2) may be interpreted as a maximally symmetric solution of (p, q) AdS supergravity. 2 Within the off-shell formulation for N -extended conformal supergravity which was first sketched in [4] and then fully developed in [5], AdS (3|p,q) originates as a maximally symmetric supergeometry with covariantly constant torsion and curvature generated by a symmetric torsion S IJ = S JI , with the structure-group indices I, J taking values from 1 to N . It turns out that S IJ is nonsingular and can be brought to the form 1 More general AdS supergroups exist for N ≥ 4. 2 In the case of N = 1 AdS supersymmetry, both notations (1,0) and N = 1 are used in the literature.
We will often use the notation AdS 3|2 for N = 1 AdS superspace.
for some positive parameter S of unit dimension. For p = N ≥ 4 and q = 0, there exist more general AdS superspaces [3] than the conformally flat ones defined by (1.2).
In the extended N = p + q ≥ 3 case, general (p, q) supersymmetric field theories in AdS 3 can be realised in (2,0) AdS superspace, AdS (3|2,0) [3,6]. 3 Such realisations are often useful for applications, for instance, in order to study the target space geometry of supersymmetric nonlinear σ-models in AdS 3 [6]. It is worth elaborating on the σ-model story in some more detail. For the N = 3 and N = 4 choices, manifestly (p, q) supersymmetric formulations have been constructed [3] for the most general nonlinear σ-models in AdS 3 (these formulations make use of the curved superspace techniques developed in [5]). This manifestly supersymmetric setting is very powerful since it allows one to generate arbitrary nonlinear σ-models with (p, q) AdS supersymmetry. However, it also has a drawback that the hyperkähler geometry of the σ-model target space is hidden. In order to uncover this geometry, the formulation of the nonlinear σ-model in (2,0) AdS superspace becomes truly indispensable [6]. 4 This work is somewhat similar in spirit to [6,13], however our goals are quite different. Specifically, we develop a formalism to reduce every field theory with (2,0) AdS supersymmetry to N = 1 AdS superspace. This formalism is then applied to carry out the (2, 0) → (1, 0) AdS reduction of the off-shell massless higher-spin supermultiplets in AdS (3|2,0) constructed in [1]. There are at least two motivations for pursuing such an application. Firstly, certain theoretical arguments imply that there exist more general off-shell massless higher-spin N = 1 supermultiplets in AdS 3 than those described in [14]. Secondly, N = 1 supermultiplets of conserved higher-spin currents have never been constructed in AdS 3 (except for the superconformal multiplets of conserved currents in Minkowski superspace [15] which can readily be lifted to AdS 3 ). Both issues will be addressed below. In particular, we will derive new off-shell higher-spin N = 1 supermultiplets in AdS 3 , which will be used to construct new topologically massive higher-spin supermultiplets. 3 General aspects of (2,0) supersymmetric field theory in AdS 3 were studied in [7]. 4 Analogous results exist in four dimensions. The most general N = 2 supersymmetric σ-model in AdS 4 was constructed [8,9] using a formulation in terms of N = 1 covariantly chiral superfields, as an extension of the earlier analysis in the super-Poincaré case [10,11]. One of the main virtues of the N = 1 formulation [8,9] is its geometric character, however the second supersymmetry is hidden. General off-shell N = 2 supersymmetric σ-models in AdS 4 were actually formulated a few years earlier [12] in N = 2 AdS superspace. The latter approach makes N = 2 supersymmetry manifest, but the hyperkähler geometry of the σ-model target space is hidden. The two σ-model formulations are related via the N = 2 → N = 1 AdS superspace reduction [13].
The table of contents reflects the structure of the paper. Our notation and conventions follow [5].

(2,0) → (1,0) AdS superspace reduction
The aim of this section is to elaborate on the details of procedure for reducing the field theories in (2,0) AdS superspace to N = 1 AdS superspace. Explicit examples of such a reduction are given by considering supersymmetric nonlinear σ-models.

Geometry of (2,0) AdS superspace: Complex basis
We begin by briefly reviewing the key results concerning (2,0) AdS superspace, see [6,7] for the details. There are two ways to describe the geometry of (2,0) AdS superspace, which correspond to making use of either a real or a complex basis for the spinor covariant derivatives. We first consider the formulation in the complex basis.
The covariant derivatives of (2,0) AdS superspace are where z M = (x m , θ µ ,θ µ ) are local superspace coordinates, and J is the generator of the R-symmetry group, U(1) R . The generator J is defined to act on the covariant derivatives as follows: The Lorentz connection, Ω A , can be written in several equivalent forms, which are 3) The relations between 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 given in Appendix A.
The covariant derivatives of (2,0) AdS superspace obey the following graded commutation relations: Here the parameter S is related to the AdS scalar curvature as R = −24S 2 .
There exists a universal formalism to determine isometries of curved superspace backgrounds in diverse dimensions [16,17]. This formalism was used in [7] to compute the isometries of (2,0) AdS superspace (as well as supersymmetric backgrounds in off-shell N = 2 supergravity theories [18]). The isometries of (2,0) AdS superspace are generated by the Killing supervector fields ζ A E A , which are defined to solve the master equation and τ and l bc are some real U(1) R and Lorentz superfield parameters, respectively. It follows from eq. (2.5) that the parameters ζ α , τ and l αβ are uniquely expressed in terms of the vector parameter ζ αβ as follows: The vector parameter ζ αβ satisfies the equation This implies the standard Killing equation, One may also prove the following relations The Killing supervector fields prove to generate the supergroup OSp(2|2; R)×Sp(2, R), the isometry group of (2,0) AdS superspace. Rigid supersymmetric field theories in (2,0) AdS superspace are required to be invariant under the isometry transformations. An infinitesimal isometry transformation acts on a tensor superfield U (with suppressed indices) by the rule 2.2 Geometry of (2,0) AdS superspace: Real basis Instead of dealing with the complex basis for the (2,0) AdS spinor covariant derivatives, eq. (2.1), it is more convenient to switch to a real basis in order to carry out reduction to N = 1 AdS superspace AdS 3|2 . Following [3], such a basis is introduced by replacing the complex operators D α andD α with ∇ I α = (∇ 1 α , ∇ 2 α ) defined as follows: In a similar way we introduce real coordinates, z M = (x m , θ µ I ), to parametrise (2,0) AdS superspace. Defining ∇ a = D a , the algebra of (2,0) AdS covariant derivatives (2.4) turns into 5 The action of the U(1) R generator on the spinor covariant derivatives is given by As may be seen from (2.12), the graded commutation relations for the operators ∇ a and ∇ 1 α have the following properties: 1. These (anti-)commutation relations do not involve ∇ 2 α , 2. Relations (2.14) are identical to the algebra of the covariant derivatives of AdS 3|2 , see (2.4).
We thus see that AdS 3|2 is naturally embedded in (2,0) AdS superspace as a subspace. The real Grassmann variables of (2,0) AdS superspace, 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. We also note that no U(1) R curvature is present in the algebra of N = 1 AdS covariant derivatives. These properties make possible a consistent (2, 0) → (1, 0) AdS superspace reduction. Now we will recast the fundamental properties of the (2,0) AdS Killing supervector fields in the real representation (2.11). The isometries of (2,0) AdS superspace are described in terms of those first-order operators which solve the equation for some real parameters τ and l ab = −l ba . Equation (2.15b) is equivalent to and Some nontrivial implications of the above equations which will be important for our subsequent consideration are: Equation (2.17a) implies that ζ a is a Killing vector field, 2.3 Reduction from (2,0) to N = 1 AdS superspace Given a tensor superfield U (x, θ I ) on (2,0) AdS superspace, its N = 1 projection (or bar-projection) is defined by in a special coordinate system to be specified below. By definition, U | depends on the real coordinates z M = (x m , θ µ ), with θ µ := θ µ 1 , which will be used to parametrise N = 1 AdS superspace AdS 3|2 . For the (2,0) AdS covariant derivative its bar-projection is defined as We use the freedom to perform general coordinate, local Lorentz and U(1) R transformations to choose the following gauge condition where denotes the set of covariant derivatives for AdS 3|2 , which obey the following graded commutation relations: In such a coordinate system, the operator ∇ 1 α | contains no partial derivative with respect to θ 2 . As a consequence, ∇ 1 α 1 · · · ∇ 1 α k U = ∇ α 1 · · · ∇ α k U |, for any positive integer k, where U is a tensor superfield on (2,0) AdS superspace. Let us study how the N = 1 descendants of U defined by U α 1 ...α k := ∇ 2 α 1 · · · ∇ 2 α k U transform under the (2,0) AdS isometries, with k a non-negative integer.
The remaining parameters ǫ α and ǫ generate the second supersymmetry and U(1) R transformations, respectively. Using the Killing equations (2.18), it can be shown that they satisfy the following properties These imply that the only independent components of ǫ are ǫ| θ=0 and ∇ α ǫ| θ=0 . They correspond to the U(1) R and second supersymmetry transformations, respectively.
Given a matter tensor superfield U , its (2,0) AdS transformation law It follows from (2.15) and (2.33) that every N = 1 descendant U α 1 ...α k := ∇ 2 α 1 · · · ∇ 2 α k U is a tensor superfield on AdS 3|2 , For the ǫ-transformation we get where q is the U(1) R charge of U defined by JU = qU . In the second term on the right, we have to push ∇ 1 α l to the far left through the (l − 1) factors of ∇ 2 's by making use of the relation {∇ 1 α , ∇ 2 β } = 4ε αβ SJ and taking into account the relation As the next step, the U(1) R generator J should be pushed to the right until it hits U producing on the way insertions of ∇ 1 . Then the procedure should be repeated. As a result, the variation δ ǫ U α 1 ...α k is expressed in terms of the superfields U α 1 ...α k+1 , U α 1 ...α k , · · · U α 1 , U.
So far we have been completely general and discussed infinitely many descendants U α 1 ...α k of U . However only a few of them are functionally independent. Indeed, eq. (2.12a) tells us that and thus every U α 1 ...α k for k > 2 can be expressed in terms of U, U α and U α 1 α 2 . Therefore, it suffices to consider k ≤ 2.
Let us give two examples of matter superfields on (2,0) AdS superspace. We first consider a covariantly chiral scalar superfield φ,D α φ = 0, with an arbitrary U(1) R charge q defined by Jφ = qφ. It transforms under the (2,0) AdS isometries as When expressed in the real basis (2.11), the chirality constraint on φ means As a result, there is only one independent N = 1 superfield upon reduction, We then get the following relations The ǫ-transformation (2.35) is given by Our second example is a real linear superfield L =L ,D 2 L = 0 . The real linearity constraint relates the N = 1 descendants of L as follows: Thus, L is equivalent to two independent, real N = 1 superfields: Here X is unconstrained, while W α obeys the constraint (2.43b) which means that W α is the field strength of an N = 1 vector multiplet. Since L is neutral under the R-symmetry group U(1) R , J L = 0, the second SUSY and U(1) R transformation laws of the N = 1 descendants of L are as follows:

The (2,0) AdS supersymmetric actions in AdS 3|2
Every rigid supersymmetric field theory in (2,0) AdS superspace may be reduced to N = 1 AdS superspace. Here we provide the key technical details of the reduction.
In accordance with [5][6][7]18], there are two ways of constructing supersymmetric actions in (2,0) AdS superspace: (i) either by integrating a real scalar L over the full (2,0) AdS superspace, 6 with E −1 = Ber(E A M ); or (ii) by integrating a covariantly chiral scalar L c over the chiral subspace of the (2,0) AdS superspace, with E being the chiral density. The superfield Lagrangians L and L c are neutral and charged, respectively with respect to the group U(1) R : The two types of supersymmetric actions are related to each other by the rule Instead of reducing the above actions to components, in this paper we need their reduction to N = 1 AdS superspace. We remind the reader that the supersymmetric action in AdS 3|2 makes use of a real scalar Lagrangian L. The superspace and component forms of the action are: (2.51) For the action (2.47) we get with E −1 = Ber(E A M ). The chiral action (2.48) reduces to AdS 3|2 as follows: Making use of the (2,0) AdS transformation law δL = ζL, δL c = (ζ − 2iτ )L c , and the Killing equation (2.15b), it can be checked explicitly that the N = 1 action defined by the right-hand side of (2.52), or (2.53) are invariant under the (2,0) AdS isometry transformations.
Our first example is a general nonlinear σ-model with (2,0) AdS supersymmetry [6,7]. It is described by the action where K(φ i ,φj) is the Kähler potential of a Kähler manifold and W (φ i ) is a superpotential. The U(1) R generator is realised on the dynamical superfields φ i andφ¯i as where J i (φ) is a holomorphic Killing vector field such that for some Killing potential D(φ,φ). The superpotential has to obey the condition in order for the action (2.54) to be invariant under the (2,0) AdS isometry transformations In the real representation (2.11), the chirality condition on φ i turns into It follows that upon N = 1 reduction, φ i leads to just one superfield, In particular, we have the following relations Using the reduction rules (2.52) and (2.53), we obtain where we have made use of the standard notation The action (2.62) is manifestly N = 1 supersymmetric. One may explicitly check that it is also invariant under the second supersymmetry and R-symmetry transformations generated by a real scalar parameter ǫ subject to the constraints (2.31), which are: The family of supersymmetric σ-models (2.54) includes a special subclass which is specified by the two conditions: (ii) all φ's are neutral, Jφ i = 0; and (ii) no superpotential is present, W (φ) = 0. In this case no restriction on the Kähler potential is imposed by eq. (2.56), and the action (2.54) is invariant under arbitrary Kähler transformations with Λ(φ i ) a holomorphic function. The corresponding action in N = 1 AdS superspace is obtained from (2.62) by setting D(ϕ,φ) = 0 and W (ϕ) = 0, and thus the action is manifestly Kähler invariant.
Let us also consider a supersymmetric nonlinear σ-model formulated in terms of several Abelian vector multiplets with action [7] where F (x i ) is a real analytic function of several variables, which is defined modulo linear inhomogeneous shifts with real parameters b i and c. The real linear scalar L i is the field strength of a vector multiplet. Upon reduction to N = 1 AdS superspace, L i generates two different N = 1 superfields: Here the real scalar X i is unconstrained, while the real spinor W i α obeys the constraint which means that W i α is the field strength of an N = 1 vector multiplet. Reducing the action (2.66) to N = 1 AdS superspace gives where we have introduced the target-space metric The vector multiplets in (2.70) can be dualised into scalar ones, which gives with g ij (X) being the inverse metric. Riemannian metrics of the type (2.71) appeared in the literature twenty years ago in the context of N = 4 supersymmetric quantum mechanics [19] and N = 4 superconformal mechanics [20].

Massless higher-spin models: Type II series
There exist two off-shell formulations for a massless multiplet of half-integer superspin (s + 1 2 ) in (2,0) AdS superspace [1], with s =, 2, 3, . . . , which are called the type II and type III series 7 by analogy with the terminology used in [7] for the linearised off-shell formulations for N = 2 supergravity (s = 1). In this section we describe the (2, 0) → (1, 0) AdS superspace reduction of the type II theory. The reduction of the type III theory will be given in section 4.
which is a covariantly linear superfield, It is inert under the gauge transformation (3.4), δ ξ L α(2s−2) = 0. From (3.2b) we can read off the λ-gauge transformation of the field strength: The type II theory is described by the action It is invariant under the gauge transformations (3.2) and (3.4).
is not defined for s = 1. However it comes with the factor (s − 1) and therefore drops out from (3.8) for s = 1. The action (3.8) for s = 1 coincides with the linearised action for (2,0) AdS supergravity, which was originally derived in section 10.1 of [7].

Reduction of the gauge prepotentials to AdS 3|2
Let us turn to reducing the gauge prepotentials (3.1) to N = 1 AdS superspace. 8 Our first task is to work out such a reduction for the superconformal gauge multiplet H α(2s) . In the real representation (2.11), the longitudinal linear constraint (3.3) takes the form It follows that g α(2s) has two independent θ 2 -components, which are The gauge transformation of H α(2s) , eq. (3.2a), allows us to choose two gauge conditions In this gauge we stay with the following unconstrained real N = 1 superfields: There exists a residual gauge freedom which preserves the gauge conditions (3.11). It is described by unconstrained real N = 1 superfields ζ α(2s) and ζ α(2s−1) defined by The gauge transformation laws of the superfields (3.12) are given by Our next step is to reduce the compensator L α(2s−2) to N = 1 AdS superspace. Making use of the representation (2.11), we observe that the chirality condition (3.4) reads The gauge transformation (3.4) allows us to impose a gauge condition Thus, upon reduction to N = 1 superspace, we have the following real superfields . The condition (3.16) is preserved by the residual gauge freedom generated by a real unconstrained N = 1 superfield η α(2s−2) defined by We may now determine how the η-transformation acts the superfields (3.17a) and (3.17b). We obtain where we have used the chirality constraint (3.15) and the expression (3.18) for the residual gauge transformation.
Next, we analyse the λ-gauge transformation and reduce the N = 2 field strength L α(2s−2) to AdS 3|2 . In the real basis for the covariant derivatives, the real linearity constraint (3.6) is equivalent to two constraints: These constraints imply that the resulting N = 1 components of L α(2s−2) are given by of which the former is unconstrained and the latter is a constrained N = 1 superfield that proves to be a gauge-invariant field strength, as we shall see below. The relation between L α(2s−2) and the prepotential L α(2s−2) is given by (3.5), which can be expressed as We now compute the bar-projection of (3.22) in the gauge (3.16) and make use of the definition (3.17b) to obtain Making use of (3.22) and (3.17a), the bar-projection of i∇ 2 β L α(2s−2) leads to the N = 1 field strength , and is a descendant of the real unconstrained prepotential Ψ β; α(2s−2) defined modulo gauge transformation (3.19a). The field strength proves to be gauge invariant under (3.19a), and it satisfies the condition as a consequence of (3.20b) and the identity (A.7b). Let us express the gauge transformation of L α(2s−2) , eq. (3.7) in terms of the real basis for the covariant derivatives. This leads to In a similar way, one should also rewrite ∇ 2 β δL α(2s−2) in the real basis. This allows us to derive the gauge transformations for L α(2s−2) and W β; α(2s−2) We can then read off the transformation law for the prepotential Ψ β; α(2s−2) where we have also taken into account the η-gauge freedom (3.19a).
Applying the N = 1 reduction rule (2.52) to the type II action (3.8), we find that it becomes a sum of two actions, Explicit expressions for these N = 1 actions will be given in the next subsection.
Given a positive integer n > 0, we say that a supersymmetric gauge theory describes a multiplet of superspin n/2 if it is formulated in terms of a superconformal gauge prepotential H α(n) and possibly a compensating multiplet. The gauge freedom of the real tensor superfield H α(n) is with the gauge parameter ζ α(n−1) being real but otherwise unconstrained.

Longitudinal formulation for massless superspin-(s + 1 2 ) multiplet
One of the two N = 1 theories provides an off-shell formulation for the massless superspin-(s + 1 2 ) multiplet. It is formulated in terms of the real unconstrained gauge superfields which are defined modulo gauge transformations where the parameter ζ α(2s) is unconstrained real. The gauge-invariant action is where Q is the quadratic Casimir operator of the 3D N = 1 AdS supergroup (A.9) . The action (3.32) coincides with the off-shell N = 1 supersymmetric action for massless half-integer superspin in AdS in the form given in [14]. This supersymmetric gauge theory in AdS 3|2 was described in [14]. Its flat-superspace limit was presented earlier in [26]. In what follows, we will refer to the above theory as the longitudinal formulation for the massless superspin-(s .32) is not defined for s = 1. However it comes with the factor (s − 1) and drops out from (3.32) for s = 1. The resulting action is the linearised action for N = 1 AdS supergravity. In the flat-superspace limit, the action is equivalent to the one given in [27].
The structure W β; βα(2s−3) W γ; γα(2s−3) in (3.36a) is not defined for s = 1. However it comes with the factor (s − 1) and drops out from (3.36a) for s = 1. The resulting gauge-invariant action provides an off-shell realisation for a massless gravitino multiplet in AdS 3 . In the flatsuperspace limit, this model reduces to the one described in [26].
In the s > 1 case, the gauge freedom of the prepotential Ψ β; α(2s−2) (3.35) allows us to impose a gauge condition for some field ϕ α(2s−3) . Since we gauge away the symmetric part of Ψ β; α(2s−2) , the two gauge parameters ζ α(2s−1) and η α(2s−2) are related. The theory is now realised in terms of the following dynamical variables with the gauge freedom It follows that in the flat-superspace limit, S = 0, and in the gauge (3.38), the action (3.36) coincides with eq. (B.25) of [21]. The component structure of this model will be discussed in Appendix B.1.

Massless higher-spin models: Type III series
In this section we carry out the N = 1 AdS superspace reduction of the type III theory [1] following the procedure employed in section 3.

The type III theory
We fix a positive integer s > 1. In accordance with [1], the massless type III multiplet of superspin (s + 1 2 ) is described in terms of two unconstrained real tensor superfields The dynamical superfields are defined modulo gauge transformations of the form where the gauge parameter λ α(2s−1) is unconstrained complex, and the longitudinal linear parameter g α(2s) is defined as in (3.3). As in the type II case, H α(2s) is the superconformal gauge multiplet, while V α(2s−2) is a compensating multiplet. The only difference from the type II case occurs in the gauge transformation law for the compensator V α(2s−2) .
The compensator V α(2s−2) also possesses its own gauge freedom of the form with the gauge parameter ξ α(2s−2) being covariantly chiral, but otherwise arbitrary.
Associated with V α(2s−2) is the real field strength which is inert under (4.3), δ ξ V α(2s−2) = 0. It is not difficult to see that V α(2s−2) is covariantly linear, It varies under the λ-gauge transformation as Modulo normalisation, there exists a unique action being invariant under the gauge transformations (4.2) and (4.3). It is given by Although the structure D β V βα(2s−3)D2 D γ V γα(2s−3) in (4.7) is not defined for s = 1, it comes with the factor (s − 1) and drops out from (4.7) for the s = 1 case. In this case the action coincides with the type III supergravity action in (2,0) AdS superspace, which was originally derived in section 10.2 of [7].

Reduction of the gauge prepotentials to AdS 3|2
The reduction of the superconformal gauge multiplet H α(2s) to AdS 3|2 has been carried out in the previous subsection. We saw that in the gauge (3.11), H α(2s) is described by the two unconstrained real superfields H α(2s+1) and H α(2s) defined according to (3.12), with their gauge transformation laws given by eqs. (3.14a) and (3.14b), respectively. Now it remains to reduce the prepotential V α(2s−2) to N = 1 AdS superspace, following the same approach as outlined in the type II series. The gauge transformation (4.3) allows us to choose a gauge condition V α(2s−2) | = 0 . (4.8) The compensator V α(2s−2) is then equivalent to the following real N = 1 superfields, which we define as follows: The residual gauge freedom, which preserves the gauge condition (4.8) is described by a real unconstrained N = 1 superfield η α(2s−2) defined by As a result, we may determine how (4.9a) and (4.9b) vary under η-transformation Next, we analyse the λ-gauge transformation and reduce the field strength V α(2s−2) to AdS 3|2 . In the real basis for the covariant derivatives, the real linearity constraint (4.5) turns into: This tells us that V α(2s−2) is equivalent to two real N = 1 superfields The relation between the field strength V α(2s−2) and the prepotential V α(2s−2) is given by (4.4), which can be expressed as (4.14) We now compute the bar-projection of (4.14) in the gauge (4.8) and make use of the definition (4.9b) to obtain The bar-projection of i∇ 2 β V α(2s−2) leads to the N = 1 field-strength which is a real superfield, Ω β; α(2s−2) =Ω β; α(2s−2) , and is a descendant of the real unconstrained prepotential Υ β; α(2s−2) defined modulo gauge transformation (4.11a). One may check that the field strength is invariant under (4.11a) and obeys the condition Let us express the gauge transformation of V α(2s−2) , eq. (4.6) in terms of the real basis for the covariant derivatives. This leads to One should also express its corollary ∇ 2 β δV α(2s−2) in the real basis for the covariant derivatives. We determine the gauge transformations law for V α(2s−2) and Ω β; α(2s−2) to be δV α(2s−2) = 1 2s ∇ β ζ βα(2s−2) , (4.19a) From (4.19b) we read off the transformation law for the prepotential Υ β; α(2s−2) : where we have also taken into account the η-gauge freedom (4.11a).
Performing N = 1 reduction to the original type III action (4.7), we arrive at two decoupled N = 1 actions Upon reduction to N = 1 superspace, the type III theory leads to two N = 1 supersymmetric gauge theories.

Longitudinal formulation for massless superspin-s multiplet
One of the two N = 1 theories provides an off-shell realisation for massless superspin-s multiplet described in terms of the real unconstrained superfields which are defined modulo gauge transformations of the form where the gauge parameter ζ α(2s−1) is unconstrained real. The gauge-invariant action is given by Modulo an overall normalisation factor, (4.24) coincides with the off-shell N = 1 supersymmetric action for massless superspin-s multiplet in the form given in [14]. In the flat-superspace limit it reduces to the action derived in [26].
The structure Ω β; βα(2s−3) Ω γ; γα(2s−3) in (4.28a) is not defined for s = 1. However it comes with the factor (s − 1) and hence drops out from (4.28a) for s = 1. The resulting gauge-invariant action provides an off-shell formulation for a linearised supergravity multiplet in AdS 3 . In the flat-superspace limit, it reduces to the linearised supergravity model proposed in [26].

Analysis of the results
Let s > 0 be a positive integer. For each superspin value, integer (s) or half-integer (s+ 1 2 ), we have constructed two off-shell formulations which have been called longitudinal and transverse. Now we have to explain this terminology.
Consider a field theory in AdS 3|2 that is described in terms of a real tensor superfield V α(n) . We assume the action to have the form It is natural to call ∇ β V α(n) a longitudinal superfield, by analogy with a longitudinal vector field. This theory possesses a dual formulation that is obtained by introducing a first-order action where Σ β;α(n) is unconstrained and the Lagrange multiplier is for some unconstrained prepotential Ψ γ; α(n) . Varying (5.2) with respect to Ψ γ; α(n) gives and then S first-order reduces to the original action (5.1). On the other hand, we may start from S first-order and integrate Σ β;α(n) out. This will lead to a dual action of the form This is a gauge theory since the action is invariant under gauge transformations δΨ γ; α(n) = i n+1 ∇ γ η α(n) .

(5.6)
The gauge-invariant field strength W β; α(n) can be called a transverse superfield, due to the constraint (5.3) it obeys.
It is natural to call the dual formulations (5.1) and (5.5) as longitudinal and transverse, respectively. Now, let us consider the transverse and longitudinal formulations for the massless superspin-s models, which are given by eqs. The above discussion can be illustrated by considering the model for linearised gravity in AdS 3 . It is described by the action which is invariant under gauge transformations In the flat-space limit, S = 0, the model possesses a dual formulation in which the scalar compensator y is replaced with a gauge one-form. 9 However, such a duality transformation cannot be lifted to AdS 3 .

Non-conformal higher spin supercurrents
In the previous sections, we have shown that there exist two different off-shell formulations for the massless higher-spin N = 1 supermultiplets. Massless half-integer superspin theory can be realised in terms of the dynamical variables (3.30) and (4.26), while the models (3.34) and (4.22) define massless multiplet of integer superspin s, with s > 1. These models lead to different N = 1 higher-spin supercurrent multiplets. Our aim in this section is to describe the general structure of N = 1 supercurrent multiplets in AdS.

N = 1 supercurrents: Half-integer superspin case
Our half-integer supermultiplet in the longitudinal formulation (3.30) can be coupled to external sources The condition that the above action is invariant under the gauge transformations (3.31) gives the conservation equation For the transverse theory (4.26) described by the prepotentials {H α(2s+1) , Υ β; α(2s−2) }, we construct an action functional of the form Requiring that the action is invariant under the gauge transformations (4.27) leads to From the above consideration, it follows that the most general conservation equation in the half-integer superspin case takes the form 6.2 N = 1 supercurrents: Integer superspin case In complete analogy with the half-integer superspin case, we couple the prepotentials (4.22) in terms of which the integer superspin-s is described, to external sources For such an action to be invariant under the gauge freedom (4.23), the sources must be conserved Next, we turn to the transverse formulation (3.34) characterised by the prepotentials {H α(2s+1) , Ψ β; α(2s−2) } and construct an action functional Demanding that the action be invariant under the gauge transformations (3.35), we derive the following conditions From the above consideration, the most general conservation equation for the multiplet of currents in the integer superspin case is given by

From N = 2 supercurrents to N = 1 supercurrents
In our recent paper [1], we constructed the general conservation equation for the N = 2 higher-spin supercurrent multiplets in (2,0) AdS superspace, which takes the form Here J α(2s) denotes the higher-spin supercurrent, while the trace supermultiplets Y α(2s−2) and Z α(2s−2) are covariantly linear. The explicit form of this multiplet of currents was presented by considering simple N = 2 supersymmetric models for a chiral scalar superfield. Unlike in 4D N = 1 supergravity where every supersymmetric matter theory can be coupled to only one of the off-shell supergravity formulations (either old-minimal or new-minimal), here in the (2,0) AdS case our trace multiplets require both type II and type III compensators to couple to.
The general conservation equation (6.11) naturally gives rise to the N = 1 higher-spin supercurrent multiplets discussed in the previous subsection. One may show that in the real basis, (6.11) turns into: The real linearity constraints on the trace supermultiplets are equivalent to It follows from (6.12) and (6.13) that J α(2s) contains two independent real N = 1 supermultiplets: Making use of (6.13), one may readily show that ∇ β U β; α(2s−2) = 0 , (6.16a) On the other hand, eq. (6.12) implies that the N = 1 superfields obey the following conditions Indeed, the right-hand side of eq. (6.17a) coincides with (6.5a). Therefore, eqs. (6.16a) and (6.17a) define the N = 1 higher-spin current multiplets associated with the massless half-integer superspin formulations (3.30) and (4.26). In a similar way, it can be observed that eqs. (6.16b) and (6.17b) correspond to the N = 1 higher-spin supercurrents for the two integer superspin models (3.34) and (4.22).

Examples of N = 1 higher-spin supercurrents
In this section we give an explicit realisation of the N = 1 multiplet of higher-spin supercurrent introduced earlier.
Consider a massless chiral scalar multiplet in (2,0) AdS superspace with action [1] The chiral superfield is charged under the R-symmetry group U(1) R , This action is a special case of the supersymmetric nonlinear sigma model studied in subsection (2.4) with a vanishing superpotential, W (Φ) = 0 . Making use of (2.62), the reduction of the action (7.1) to N = 1 AdS superspace is where we have denoted ϕ := Φ| . This action is manifestly N = 1 supersymmetric. It also possesses hidden second supersymmetry and U(1) R invariance. These transformations are where ǫ α is given in terms or ǫ according to (2.31a), and the real parameter ǫ is constrained by (2.31b). It can be shown that on the mass shell it holds that It was shown in [1] that by using the massless equations of motion, D 2 Φ = 0, the N = 2 higher-spin supercurrent multiplet associated with the theory (7.1) is described by the conservation equation Here the real supercurrent J (2s) =J (2s) is given by while the trace multiplet T (2s−2) has the form T (2s−2) = 2iS(1 − 2r)(2s + 1)(s + 1) One may check that T (2s−2) is covariantly linear, As is seen from (7.6c), T (2s−2) vanishes for r = 1/2, in which case Φ is an N = 2 superconformal multiplet.
The complex trace multiplet T (2s−2) may split it into its real and imaginary parts: Φ . (7.7c) Note that we make use of a condensed notation [1] throughout this section. In this notation, we introduce auxiliary real variables ζ α ∈ R 2 , such that any tensor superfield U α(m) can be associated with the following field which is a homogeneous polynomial of degree m in ζ α . Let us introduce operators which increase the degree of homogeneity in ζ α : In addition, we also have two operators that decrease the degree of homogeneity in ζ α The operators associated with the real spinor covariant derivatives may be defined in a similar way: In accordance with (6.14), the supercurrent J (2s) reduces to two different multiplets upon projection to N = 1 superspace: of which the former corresponds to the integer superspin current and the latter half-integer superspin current.
In the case of half-integer superspin, the conservation equation (6.5) is satisfied provided we impose (7.5): ϕ , (7.14b) It may also be verified that the N = 1 supercurrent multiplet for integer superspin obeys the conditions (6.10) on-shell: ϕ , (7.15b) The above technique can also be used to construct N = 1 higher-spin supercurrents for the Abelian vector multiplets model described by the action (2.66). We will not elaborate on a construction in the present work.

Applications and open problems
Let us briefly summarise the main results obtained in this paper. In section 2 we developed a formalism to reduce every field theory with (2,0) AdS supersymmetry to N = 1 AdS superspace. In sections 3 and 4 we applied this reduction procedure to the off-shell massless higher-spin supermultiplets in AdS (3|2,0) constructed in [1]. For each superspin value, integer (s) or half-integer (s + 1 2 ), we constructed two off-shell gauge formulations for massless N = 1 supermultiplets in AdS 3 . In section 5, we proved that these two theories, known as the longitudinal and transverse, are dually equivalent only in the flat superspace limit. In section 6 we formulated the non-conformal higher-spin supercurrent in N = 1 AdS superspace. In 7 we provided the explicit examples of these supercurrents for models of a chiral scalar superfield.
There are several interesting applications of the results obtained in this paper. In particular, the massless higher-spin N = 1 supermultiplets in AdS 3 , which were derived in sections 3 and 4, can be used to construct off-shell topologically massive supermultiplets in AdS 3 by extending the approaches advocated in [14,22,26]. Such a massive supermultiplet is described by a gauge-invariant action being the sum of massless and superconformal higher-spin actions, following the philosophy of topologically massive theories [34][35][36][37].
Given a positive integer n, the conformal superspin-n 2 action [14,26,38] is where W α(n) (H) denotes the higher-spin super-Cotton tensor. The latter is a unique descendant of H α(n) with the properties where δ ζ H α(n) is the gauge transformation (3.29). These properties imply the gauge invariance of (8.1). In a flat superspace, W α(n) has the form [38] The construction of W α(n) in arbitrary conformally flat backgrounds is described in [39]. The dynamical system (8.4a) was introduced in [14], while its flat-superspace counterpart appeared earlier in [26]. The other theory, eq. (8.4b), is a new formulation for massive superspin-s multiplet in AdS 3 .
In the Minkowski superspace limit, the dynamical systems (8.4a) and (8.4b) are equivalent, since they are related to each other by the superfield Legendre transformation described in section 5. On the mass shell, dynamics can be recast in terms of the gaugeinvariant field strength W α(2s) which obeys the equations [26] where the mass m and helicity parameter σ are determined by µ. 10 It is an interesting open problem to understand whether the AdS models (8.4a) and (8.4b) lead to equivalent dynamics, modulo a redefinition of the mass parameter µ.
There are two off-shell gauge-invariant formulations for a topologically massive superspin-(s + 1 2 ) multiplet in AdS 3 . The corresponding actions are: The theory defined by (8.6a) was introduced in [14], while its flat-superspace counterpart appeared earlier in [26]. The other model, eq. (8.6b), is a new formulation for a massive superspin-(s + 1 2 ) multiplet in AdS 3 . In the Minkowski superspace limit, the dynamical systems (8.6a) and (8.6b) are equivalent, since they are related to each other by the superfield Legendre transformation described in section 5. It is also an interesting open problem to understand whether the models (8.6a) and (8.6b) in AdS 3 generate equivalent dynamics.
We now present two off-shell formulations for the massive N = 1 gravitino supermultiplet in AdS 3 and analyse the corresponding equations of motion. 11 The massive extension of the longitudinal theory (4.25) is described by the action with µ a real mass parameter. The massive gravitino action is thus constructed from the massless one by adding mass-like terms. In the limit µ → 0, the action reduces to (4.25).
The equations of motion for the dynamical superfields H αβ and V are Making use of the identity (A.7b), it immediately follows from (8.12) that and then (8.12) is equivalent to Therefore, we have demonstrated that the model (8.7) leads to the following conditions on the mass shell: Such conditions are required to describe an irreducible on-shell massive gravitino multiplet in 3D N = 1 AdS superspace [40].
In the transverse formulation (3.37), the action for a massive gravitino multiplet is defined by In the limit µ → 0, the action reduces to (3.37). One may check that the equations of motion for this model imply that The actions (8.7) and (8.16) can be made into gauge-invariant ones using the Stueckelberg construction.
In the Minkowski superspace limit, the massive models (8.7) and (8.16) lead to the identical equations of motion described in terms of H αβ : In the AdS case, the equations (8.15) and (8.17) lead to equivalent dynamics modulo a redefinition of µ. It is an interesting open problem to understand whether there exists a duality transformation relating these models.
It should be emphasised that there is an alternative construction of gauge-invariant Lagrangian formulations for massive higher-spin supermultiplets in R 2,1 and AdS 3 , which were developed in [53,54]. It is obtained by combining the massive bosonic and fermionic higher-spin actions [55,56], and therefore this construction is intrinsically on-shell. The formulations given in [53][54][55][56] are based on the gauge-invariant approaches to the dynamics of massive higher-spin fields, which were advocated by Zinoviev [57] and Metsaev [58]. It is an interesting open problem to understand whether there exists an off-shell uplift of these models.
All off-shell higher-spin N = 2 supermultiplets in AdS 3 , both with (2,0) and (1,1) AdS supersymmetry [1,21], are reducible gauge theories (in the terminology of the Batalin-Vilkovisky quantisation [59]), similar to the massless higher-spin supermultiplets in AdS 4 [25]. The Lagrangian quantisation of such theories is nontrivial. In the four-dimensional case, the quantisation of the theories proposed in [25] was carried out in [60]. All off-shell higher-spin N = 1 supermultiplets in AdS 3 , which we have constructed in this paper, are irreducible gauge theories that can be quantised using the Faddeev-Popov procedure [61] as in the non-supersymmetric case, see e.g. [62,63]. This opens the possibility to develop heat kernel techniques for higher-spin theories in AdS 3|2 , as an extension of the fourdimensional results [16,64,65].

Acknowledgements:
We are very grateful to Dmitri Sorokin for fruitful discussions and to Darren Grasso for comments on the manuscript. The work of JH is supported by an Australian Government Research Training Program (RTP) Scholarship. The work of SMK is supported in part by the Australian Research Council, project No. DP160103633.
We make use of real Dirac γ-matrices, γ a := (γ a ) α β defined by They obey the algebra where the Levi-Civita tensor is normalised as ε 012 = −ε 012 = 1. Some useful relations involving γ-matrices are Given a three-vector A a , it can equivalently be described as a symmetric rank-2 spinor A αβ = A βα , The relationship between the Lorentz generators with two vector indices (M ab = −M ba ), one vector index (M a ) and two spinor indices (M αβ = M βα ) is as follows: 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 by the rules: We collect some useful identities for N = 1 AdS covariant derivatives, which we denote by ∇ A = (∇ a , ∇ α ). Making use of the (anti)-commutation relation (2.12a) and (2.12b), we obtain the following identities where ∇ 2 = ∇ α ∇ α and ✷ = ∇ a ∇ a = − 1 2 ∇ αβ ∇ αβ . An important corollary of (A.7a) and (A.7c) is The left-hand side of (A.7d) can be expressed in terms of the quadratic Casimir operator of the 3D N = 1 AdS supergroup [14]: We also note the following commutation relation Given an arbitrary superfield F and its complex conjugateF , the following relation holds where ǫ(F ) denotes the Grassmann parity of F . B Component structure of N = 1 higher-spin actions In this appendix we will discuss the component structure of the two new off-shell N = 1 supersymmetric higher-spin theories: the transverse massless superspin-s multiplet (3.36), and the transverse massless superspin-(s + 1 2 ) multiplet (4.28a). For simplicity we will carry out our analysis in flat Minkowski superspace. In accordance with (2.51), the component form of an N = 1 supersymmetric action is computed by the rule

B.1 Massless superspin-s action
Let us first work out the component structure of the massless integer superspin model (3.36). In the flat-superspace limit, the transverse action (3.36) takes the form . (B.2) As described in (3.38), it is possible to choose a gauge condition Ψ (α 1 ; α 2 ···α 2s−1 ) = 0, such that the above action turns into It is invariant under the following gauge transformations where the gauge parameter η α(2s−2) is a real unconstrained superfield.
The gauge freedom (B.4) can be used to impose a Wess-Zumino gauge In order to preserve these gauge conditions, the residual gauge freedom has to be constrained by These imply that there remain two independent, real components of η α(2s−2) : In the gauge (B.5), the independent component fields of ϕ α(2s−3) can be chosen as Applying the reduction rule (B.1) to the N = 1 action (B.3), we find that it splits into bosonic and fermionic parts: The bosonic action is given by 16s(s − 1) 2 ∂ δλ y δλα(2s−6) ∂ βγ y βγα(2s−6) . (B.13) Integrating out the auxiliary field F α(2s−2) leads to The gauge transformations for the fields h α(2s) and y α(2s−4) can be easily read off from the gauge transformations of the superfields H α(2s) and ϕ α(2s−3) , respectively. Modulo an overall normalisation factor, (B.14) corresponds to the massless Fronsdal spin-s action S (2s) F described in [14].
We have thus proved that at the component level and upon elimination of the auxiliary field, the transverse theory (B.3) is equivalent to a sum of two massless models: the bosonic Fronsdal spin-s model and the fermionic Fang-Fronsdal spin-(s + 1 2 ) model.
In the bosonic sector, we have the following set of fields: The above fermionic fields correspond to the dynamical variables of the Fang-Fronsdal spin-(s + 1 2 ) model. As follows from (B.33a), (B.33b) and (B.33c), their gauge freedom is equivalent to that of the massless spin-(s + 1 2 ) gauge field. Indeed, direct calculations of the component action give the standard massless gauge-invariant spin-(s + 1 2 ) action S (2s+1) F F . The component structure of the obtained supermultiplets is a three-dimensional counterpart of so-called (reducible) higher-spin triplet systems. In AdS D an action for bosonic higher-spin triplets was constructed in [66] and for fermionic triplets in [67,68]. Our superfield construction provides a manifestly off-shell supersymmetric generalisation of these systems. It might be of interest to extend it to AdS 4 .