Higher-spin gauge models with (1,1) supersymmetry in AdS${}_3$: Reduction to (1,0) superspace

In three dimensions, there are two types of ${\cal N}=2$ anti-de Sitter (AdS) supersymmetry, which are denoted (1,1) and (2,0). They are characterised by different supercurrents and support different families of higher-spin gauge models (massless and massive) which were constructed in arXiv:1807.09098 and arXiv:1809.00802 for the (1,1) and (2,0) cases, respectively, using superspace techniques. It turns out that the precise difference between the (1,1) and (2,0) higher-spin supermultiplets can be pinned down by reducing these gauge theories to (1,0) AdS superspace. The present paper is devoted to the $(1,1) \to (1,0)$ AdS superspace reduction. In conjunction with the outcomes of the $(2,0) \to (1,0)$ AdS superspace reduction carried out in arXiv:1905.05050, we demonstrate that every known higher-spin theory with (1,1) or (2,0) AdS supersymmetry decomposes into a sum of two off-shell (1,0) supermultiplets which belong to four series of inequivalent higher-spin gauge models. The latter are reduced to components.


Introduction
To study N -extended supersymmetric theories in d dimensions, it is advantageous to deal with a formulation that permits some amount of supersymmetry to be realised manifestly. In general there exist two superspace settings to achieve this. One of them makes use of the standard N -extended superspace (sometimes endowed with additional commuting variables [4][5][6]) and provides a manifestly supersymmetric formulation. The other employs a smaller N -extended superspace, with N < N , to keep manifest only N supersymmetries. Both approaches have found numerous applications in the literature. For instance, it is known that the geometric properties of general N = 2 supersymmetric nonlinear σ-models in Minkowski space M 4 [7] are remarkably transparent if these theories are realised in N = 1 superspace [8,9]. One of the two supersymmetries is manifest and off-shell in this setting, while the second supersymmetry is hidden and on-shell (the commutator of the first and the second supersymmetries closes only on-shell). On the other hand, in order to construct general off-shell N = 2 supersymmetric nonlinear σ-models, manifestly supersymmetric techniques are indispensable, and there exist two powerful N = 2 superspace approaches: (i) the harmonic superspace [5,10]; and (ii) the projective superspace [6,11,12]. One of the conceptual virtues of these manifestly supersymmetric formulations is the possibility to generate N = 2 nonlinear σ-model actions (and thus hyperkähler metrics) from Lagrangians of arbitrary functional form. 1 Analogous results exist for general N = 2 supersymmetric nonlinear σ-models in four-dimensional anti-de Sitter space (AdS 4 ), which were originally formulated as off-shell theories in N = 2 AdS superspace [14]. This approach makes N = 2 supersymmetry manifest, but the hyperkähler geometry of the σ-model target space is hidden. Some time later, the most general N = 2 supersymmetric σ-models in AdS 4 were constructed using a formulation in terms of N = 1 covariantly chiral superfields [15,16]. One of the two supersymmetries is hidden in this approach, but the geometric properties of the N = 2 supersymmetric σ-models in AdS 4 become transparent. Specifically, the target space must be a non-compact hyperkähler manifold endowed with a Killing vector field which generates an SO(2) group of rotations on the two-sphere of complex structures. The two σ-model formulations developed in [14] and [15,16] are related via the N = 2 → N = 1 AdS 4 superspace reduction worked out in [17].
Extended supersymmetric theories in AdS 3 possess exceptionally many superspace realisations. We recall that the (connected) AdS group in three dimensions is not a simple group, SO 0 (2, 2) ∼ = SL(2, R) × SL(2, R) /Z 2 , (1.1) and so are its supersymmetric extensions OSp(p|2; R) × OSp(q|2; R). This means that there are several species of N -extended AdS supersymmetry [18], which are known as the (p, q) AdS supersymmetry types, where the integers p ≥ q ≥ 0 are such that N = p + q.
In principle, field theories possessing the (p, q) AdS supersymmetry may be realised in the so-called (p, q) AdS superspace [19], AdS (3|p,q) , which may be viewed as a maximally supersymmetric solution of the (p, q) AdS supergravity [18], with anti-de Sitter space AdS 3 being the bosonic body of the superspace. 2 More specifically, within the supergravity framework of [19,20] the superspace AdS (3|p,q) originates as a maximally symmetric supergeometry with covariantly constant torsion and curvature generated by a symmetric real 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 the parameters p and q = N − p determine its signature. Since the isometry group of AdS (3|p,q) is OSp(p|2; R) × OSp(q|2; R) and since S IJ is invariant under its compact subgroup SO(p) × SO(q), the global realisation of the superspace is In fact, starting from the superspace geometry of N -extended conformal supergravity [20] and restricting the torsion to be covariantly constant and Lorentz invariant, a general AdS superspace solution for N ≥ 4 includes not only the torsion S IJ , but also a completely antisymmetric curvature X IJKL = X [IJKL] . It turns out that the latter may be nonzero only if S IJ = Sδ IJ , which means p = N and q = 0. Such solutions correspond to exotic AdS superspaces, AdS of off-shell massless higher-spin N = 1 supersymmetric models in AdS 3 constructed in [3,37]. A formalism to reduce every field theory with (1,1) AdS supersymmetry to N = 1 AdS superspace is developed in section 4. This formalism is then applied throughout sections 5−8 to carry out the (1,1) → (1,0) AdS superspace reduction of massless higherspin models with (1,1) AdS supersymmetry. These are detailed in sections 5−6 for the half-integer (s + 1 2 ) superspin case, and in sections 7−8 for the integer (s) superspin case. Our results are summarised and discussed in section 9, where we also analyse the difference between the (2,0) and (1,1) higher-spin (massless and massive) supermultiplets. Our spinor notation and conventions are collected in appendix A. In appendix B, we study the component structure of the off-shell massless higher-spin N = 1 supersymmetric models in AdS 3 reviewed in section 3.

N = 1 supersymmetric field theory in AdS 3
In this section we provide salient facts about the geometry of N = 1 AdS superspace, AdS 3|2 , and its isometries following [19]. We also recall certain duality transformations in AdS 3|2 , following [3].
Let z M = (x m , θ µ ) be local coordinates parametrising AdS 3|2 . The geometry of AdS 3|2 is encoded in the set of covariant derivatives of the form where E A M is the inverse vielbein and ω A bc the Lorentz connection. The explicit relation between Lorentz generators with two vector indices (M ab ), with one vector index (M a ) and with two spinor indices (M αβ ) is described in appendix A. The covariant derivatives obey the following algebra [24] {∇ α , ∇ β } = 2i∇ αβ − 4i|µ|M αβ , (2.2b) In spinor notation, eqs. (2.2b) take the form Here |µ| > 0 is a constant parameter, which determines the curvature of AdS 3 .
The parameter |µ| in (2.2) was denoted S in [19]. However, in this paper we prefer to make use of the notation |µ| which is more appropriate in the context of the (1, 1) → (1, 0) superspace reduction, which will be considered later. It should be remarked that the geometry of N = 1 AdS superspace can also be described by the graded commutation relations which are obtained from (2.2) by replacing |µ| → −|µ|. The two choices, S = |µ| and S = −|µ|, correspond to the so-called (1, 0) and (0, 1) AdS superspaces [19], which are different realisations of N = 1 AdS superspace. The (1, 0) and (0, 1) AdS superspaces are naturally embedded in (1, 1) AdS superspace 6 and are related to each other by a parity transformation.
One may derive several useful identities: where we have denoted ∇ 2 = ∇ α ∇ α and ✷ = ∇ a ∇ a = − 1 2 ∇ αβ ∇ αβ . In particular, it follows from the algebra of the covariant derivatives that This differential operator can be expressed in terms of the quadratic Casimir operator of the N = 1 AdS supergroup [37] Given an arbitrary superfield F and its complex conjugateF , the following relation holds where ǫ(F ) denotes the Grassmann parity of F .
According to the general formalism of [30], the isometry transformations of AdS 3|2 are generated by the Killing supervector fields, which by definition solve the master equation [19] [ξ + 1 2 ζ bc M bc , ∇ A ] = 0 , (2.8) for some Lorentz parameter ζ bc = −ζ cb . It can be shown that the Killing equation (2.8) is equivalent to the set of relations [19] (see also [37]) Thus, ξ a is a Killing vector. Given a tensor superfield U(x, θ) (with suppressed indices) on AdS 3|2 , its infinitesimal isometry transformation law is given by To study the dynamics of N = 1 supersymmetric field theories in AdS 3 , a manifestly supersymmetric action principle is required. Such an action is associated with a real scalar Lagrangian L and has the form In what follows, we make use of the notation d 3|2 z := d 3 xd 2 θ.
The component form of the action (2.12) is To conclude this section, we review the duality transformation described in [3]. Consider a field theory in AdS 3|2 which is formulated in terms of a real tensor superfield V α(n) with the action (2.14) We call ∇ β V α(n) a longitudinal superfield, by analogy with a longitudinal vector field. The theory (2.14) has a dual formulation that is obtained by introducing a first-order action Here Σ β;α(n) is unconstrained and the Lagrange multiplier is for some unconstrained prepotential Ψ γ; α(n) . Varying (2.15) with respect to Ψ γ; α(n) gives thus S first-order reduces to the original action (2.14). On the other hand, integrating out Σ β;α(n) from S first-order leads to a dual action of the form It is natural to call the gauge-invariant field strength W β; α(n) a transverse superfield, due to constraint (2.16). The dual formulations (2.14) and (2.18) are referred to as longitudinal and transverse, respectively. In the n = 1 case, the duality transformation corresponds to the standard duality between the N = 1 scalar and vector multiplets in three dimensions, see [38].
The above duality transformation naturally extends to Minkowski superspace M 3|2 in the limit |µ| → 0. Therefore, given two dually equivalent theories in AdS 3|2 , their flat superspace counterparts are also dually equivalent. The opposite is not always true. For instance, the flat superspace counterparts of the massless half-integer superspin models (3.6) and (3.10) are dual. However, these models are not dual in AdS 3 . The requirement of gauge invariance uniquely fixes the actions (3.6) and (3.10) in AdS 3|2 , including the presence of certain |µ|-dependent terms which are incompatible with duality invariance.
3 Massless higher-spin N = 1 supermultiplets According to [3], for each superspin valueŝ ≥ 1, whereŝ is either integer (ŝ = s) or half-integer (ŝ = s + 1 2 ), there exist two off-shell formulations for a massless N = 1 superspin-ŝ multiplet in AdS 3 . In the flat superspace limit, these models prove to related to each other via a superfield Legendre transformation. However, when uplifted to N = 1 AdS superspace, it becomes apparent that these duality relations no longer hold. In this section, we review the explicit formulation of these respective theories, which were derived in [3,37].
As is known, any massless multiplet of superspinŝ > 1 has no propagating degrees of freedom, and the notion of superspin is purely kinematical. It is used by analogy with massive supermultiplets. The concept of superspin is well defined in the massive case, and we follow the definition used in [39] in the super-Poincaré case. Specifically, if n is a positive integer, an on-shell massive multiplet of superspin n/2 is described by a real symmetric rank-n spinor superfield, H α 1 ···αn =H α 1 ...αn = H (α 1 ···αn) , which obeys the differential conditions [40] Here, m is a real constant of unit mass dimension, D α is the spinor covariant derivative of N = 1 Minkowski superspace, and D 2 = D α D α . The superfield H α(n) contains two ordinary on-shell massive fields, which are Their helicity values are n 2 σ and n+1 2 σ, respectively, see [39] for the technical details. An off-shell massless superspin-n/2 gauge theory in AdS 3 can be realised in terms of two superfields, of which one is universally a superconformal gauge prepotential H α(n) and the other is a compensating multiplet. Here H α(n) := H α 1 ....αn = H (α 1 ...αn) is a real symmetric rank-n spinor which is defined modulo gauge transformations of the form 7 where the gauge parameter ζ α(n−1) is a real unconstrained superfield, and ⌊x⌋ stands for the floor function denoting the integer part of a real number x ≥ 0.

Massless half-integer superspin theories
There exist two off-shell formulations for the massless superspin-(s + 1 2 ) multiplet, which are referred to as longitudinal and transverse.

Longitudinal formulation
The longitudinal formulation is realised in terms of the real unconstrained dynamical variables which are defined modulo gauge transformations of the form where the real gauge parameter ζ α(2s) is unconstrained. The unique gauge-invariant action formulated in terms of the superfields (3.4) takes the following form where Q is the quadratic Casimir operator (2.5). The action (3.6) was derived in [37]. In the flat superspace limit, |µ| → 0, the action (3.6) coincides with the model derived in [39].

Transverse formulation
The transverse formulation is constructed in terms of the real superfields where Υ β;α(2s−2) is a reducible superfield pertaining to the representation 2 ⊗ (2s − 1) of SL(2, R). The superfield Υ β;α(2s−2) can be decomposed into irreducible components by the following rule The dynamical variables (3.7) possess the following gauge freedom where the gauge parameters ζ α(2s) and η α(2s−2) are real unconstrained. The unique quadratic action which is invariant under gauge transformations (3.9) takes the form where Ω β;α(2s−2) corresponds to the real N = 1 field strength The above theory was introduced in [3].

Massless integer superspin theories
Analogous to the half-integer case, there exist two off-shell formulations for the massless superspin-s multiplet, which are referred to as longitudinal and transverse.

Longitudinal formulation
The longitudinal formulation is realised in terms of the real unconstrained variables which are defined modulo gauge transformations of the form where the gauge parameter ζ α(2s−1) is real unconstrained. The unique action which is invariant under gauge transformations (3.13) assumes the form Up to normalisation, action (3.14) coincides with the off-shell N = 1 supersymmetric action for the massless superspin-s multiplet derived in [37]. Taking the flat-superspace limit, action (3.14) reduces to the model derived in [39].

Transverse formulation
The transverse formulation is constructed in terms of the real unconstrained superfields which have the following gauge freedom where the gauge parameters ζ α(2s−1) and η α(2s−2) are real unconstrained. The gaugeinvariant action is given by where W β; α(2s−2) denotes the real N = 1 field strength For s > 1, the ζ-gauge freedom (3.16b) can be used to impose the gauge condition for some superfield ϕ α(2s−3) . The residual gauge freedom is characterised by 20) which means that η α(2s−2) is the only independent gauge parameter. As a consequence, the model can be reformulated in terms of the following gauge superfields which are defined modulo gauge transformations of the form The corresponding gauge-invariant action assumes the following form (3.24) The above theory was proposed in [3]. In the flat-superspace limit, the action (3.24) coincides with the N = 1 model (B.25) presented in [1].

(1,1) → (1,0) AdS superspace reduction
In this section, we begin by reviewing the necessary aspects of (1, 1) AdS superspace which are then utilised to develop a consistent reduction procedure for field theories with (1,1) AdS supersymmetry to N = 1 AdS superspace.
The covariant derivatives of AdS (3|1,1) have the form where z M = (x m , θ µ ,θ µ ) are local complex superspace coordinates, while E A and Ω A denote the inverse supervielbein and Lorentz connection, The covariant derivatives satisfy the following algebra with µ = 0 being a complex parameter determining the constant curvature of AdS (3|1,1) . The phase of µ = |µ|e iϕ can be given any fixed value by a re-definition D α → e iρ D α and D α → e −iρD α , with ρ constant.
Let G α(n) be a symmetric rank-n spinor superfield. It is said to be longitudinal linear if it obeys the following first-order constraint In the scalar case, n = 0, the constraint (4.4a) becomes the condition of covariant chirality, D α G = 0.
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 P ⊥ n 2 = P ⊥ n , P || n 2 = P || n , P ⊥ n P || n = P || n P ⊥ n = 0 . (4.10) These projectors are three-dimensional (3D) cousins of those introduced by Ivanov and Sorin [42] in the case of N = 1 AdS supersymmetry in four dimensions.
Let V α(n) (n = 0) be an arbitrary complex tensor superfield. It can be represented as a sum of transverse linear and longitudinal linear multiplets [1] where indices placed between vertical bars (for example |γ|) are not subject to symmetrisation. If we choose V α(n) to be either longitudinal linear (G α(n) ) or transverse linear (Γ α(n) ), then the above identity produces the relations (4.7a) and (4.7b) for some prepotentials Ψ α(n−1) and Φ α(n+1) , respectively.
The isometry transformations of AdS (3|1,1) are generated by real supervector fields λ A E A which solve the Killing equation and l ab is some local Lorentz parameter. As shown in [21], this equation implies that the parameters λ α and l ab can be uniquely determined in terms of the vector λ a , 14) 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 } can be expressed in terms of the third parameter. In particular, From (4.14) and (4.16) we deduceD The solution to these equations is given in [21].

(1,1) AdS superspace: Real basis
It proves beneficial to realise the (1,1) AdS covariant derivatives in a real basis when performing the reduction procedure to N = 1 AdS superspace. In accordance with [19], the algebra of covariant derivative (4.3) can be converted to the real basis by (i) making a convenient choice µ = −i|µ|; and (ii) replacing the complex operators D α ,D α with Furthermore, we introduce the real coordinates z M = (x m , θ µ I ) which are used to parametrise (1, 1) AdS superspace. Choosing to define ∇ a = D a , it can be shown that the algebra of (1, 1) AdS covariant derivatives assumes the following form in the real basis (4.18) It is apparent from (4.19) that the operators ∇ a and ∇ 1 α possess the following properties: 1. These operators form a closed algebra given by These properties imply that AdS 3|2 is naturally realised as a surface embedded in (1, 1) AdS superspace. One can make an appropriate choice in the real Grassmann variables θ µ I = (θ µ 1 , θ µ 2 ) such that AdS 3|2 can be identified as the surface defined by θ µ 2 = 0 in AdS (3|1,1) . These properties enable the consistent reduction of any field theory with (1,1) AdS supersymmetry to N = 1 AdS superspace.
We now wish to recast the fundamental properties of the Killing supervector fields of (1,1) AdS superspace (4.13) in the real representation (4.18). The isometries of (1,1) AdS superspace are generated by the (1,1) AdS Killing supervector fields, which are defined to satisfy the Killing equation for some real Lorentz parameter l ab = −l ba . It can be shown that equation (4.22) is equivalent to the set of equations and Equations (4.23) and (4.24) can be recast in the equivalent form It follows from (4.24a) that the parameter λ a is a Killing vector field and relations (4.24b) are Killing spinor equations.

Reduction from (1,1) to N = 1 AdS superspace
Given a tensor superfield U(x, θ I ) on (1,1) AdS superspace, where indices have been suppressed, we define its bar-projection to N = 1 AdS superspace by the rule in a special coordinate system which will be described below. Given the (1, 1) covariant derivative in the the real representation (4.18) we define its N = 1 projection by the rule We use the freedom to perform general coordinate and local Lorentz transformations to impose the gauge where ∇ A = (∇ a , ∇ α ) is the set of covariant derivatives for AdS 3|2 , see eq. (2.1). In the chosen coordinate system, the operator ∇ 1 α does not involve any partial derivatives with respect to θ 2 . Thus for any positive integer k, it follows that (∇ 1 We now consider the N = 1 projection of the (1, 1) AdS Killing supervector (4.21) where we have introduced the N = 1 superfields ξ a := λ a | , ξ α := λ α 1 | , ǫ α := λ α 2 | .  It is important to note that the superfields (ξ a , ξ α , ζ ab ) parametrise the infinitesimal isometries of AdS 3|2 . Such transformations are generated by the Killing supervector fields, ξ = ξ a ∇ a + ξ α ∇ α , satisfying the N = 1 Killing equation, eq. (2.8). Indeed, the relations (2.9) and (2.10) automatically follow from the (1, 1) AdS Killing equations (4.25), upon projection. The parameter ǫ α , which generates the second supersymmetry transformation, has the property Given the transformation law of a tensor superfield U(x, θ I ) on (1, 1) AdS superspace we find its projection to N = 1 AdS superspace to be The first transformation (4.37a) coincides with the infinitesimal transformation generated by a Killing supervector in AdS 3|2 , eq. (2.11). Thus, U| can be identified as a tensor superfield on N = 1 AdS superspace. The other transformation (4.37b) corresponds to the second supersymmetry transformation, which is generated by ǫ α .

The (1,1) AdS supersymmetric actions in AdS 3|2
Every supersymmetric field theory in (1, 1) AdS superspace can be reduced to N = 1 AdS superspace. In the following subsection, we explore the necessary mathematical framework which will be employed to develop such a reduction procedure. As presented in [20][21][22]25], manifestly supersymmetric actions in (1,1) AdS superspace can be constructed by either 1. Integrating a real scalar Lagrangian L over the full (1, 1) AdS superspace, 2. Integrating a covariantly chiral Lagrangian L c over the chiral subspace, where E is the chiral density. The two supersymmetric invariants are related by the rule In (1, 1) AdS superspace, every chiral action can always be recast as an integral over the full superspace We will use the notation d 3|4 z := d 3 x d 2 θd 2θ for the full superspace measure.
Instead of reducing the above supersymmetric actions to components, we wish to obtain a prescription which allows for their reduction to N = 1 AdS superspace. The supersymmetric action in AdS 3|2 is described by a real scalar Lagrangian L The action (4.38) reduces to AdS 3|2 as follows In the remainder of this paper, we will carry out the (1, 1) → (1, 0) AdS superspace reduction of the massless higher-spin supermultiplets, and show that the reduced actions coincide with those presented in section 3.

Massless half-integer superspin: Transverse formulation
In (1, 1) AdS superspace, there exist two off-shell formulations for the massless multiplet of half-integer superspin-(s + 1 2 ), with s ≥ 2 [1]. These two theories, which are called transverse and longitudinal, prove to be dual to each other. In the following section, we develop the (1, 1) → (1, 0) reduction procedure for the transverse formulation.

Reduction of gauge prepotentials to AdS 3|2
We wish to reduce the gauge prepotentials (5.1) to N = 1 AdS superspace. We start by reducing the superconformal gauge multiplet H α(2s) . Converting the longitudinal linear constraint of g α(2s) (4.4a) to the real representation (4.18) yields Performing a Taylor expansion of g α(2s) (θ I ) about θ 2 , and using (5.5), we find the independent θ 2 -components of g α(2s) to be The gauge transformation (5.2a) allows us to impose the gauge conditions In this Wess-Zumino (WZ) gauge, we stay with the unconstrained real N = 1 superfields The residual gauge freedom which preserves gauge conditions (5.7) are described by the unconstrained real N = 1 superfields From (5.9), we can readily determine the gauge transformations of (5.8) Next, we wish to reduce Γ α(2s−2) to N = 1 AdS superspace. The superfield Γ α(2s−2) obeys the transverse linear constraint (4.5a), which takes the following form in the real representation (4.18) Utilising the gauge transformation of Γ α(2s−2) (5.2b) and the real representation (4.18), we find From (5.13), we can immediately read off the gauge transformations of the complex N = 1 superfields (5.12) Let us express the N = 1 superfields (5.12) in terms of their real and imaginary parts, It then follows from gauge transformations (5.10) and (5.14) that we are in fact dealing with two different gauge theories. The first model is formulated in terms of the real unconstrained gauge superfields which are defined modulo gauge transformations of the form where the gauge parameter ζ α(2s) is real unconstrained. The other gauge theory is constructed in terms of the superfields which possesses the gauge freedom where the parameter ζ α(2s−1) is real unconstrained. Applying the superspace reduction procedure to the action (5.3) yields two decoupled N = 1 supersymmetric theories, each formulated in terms of the gauge fields (5.16) and (5.18) respectively (5.20) Explicit expressions for the decoupled supersymmetric actions are given in the following subsection.

Massless higher-spin N = 1 supermultiplets
The first of the decoupled N = 1 supersymmetric actions, which is realised by the dynamical variables (5.16), takes the following form Upon inspection, it is apparent that the superfield Φ α(2s−1) is auxiliary. So by making use of its equation of motion, we can eliminate the auxiliary field Φ α(2s−1) from the action (5.21). It can be shown that the resulting action coincides with the off-shell N = 1 supersymmetric action for massless half-integer superspin in AdS 3 (3.6).
The other N = 1 theory is formulated in terms of the gauge superfields (5.18) The superfield Ω α(2s−1) is auxiliary, so upon elimination via its equation of motion we find that the resulting action coincides with the off-shell N = 1 action massless superspin-s multiplet in AdS 3 (3.14).

Second supersymmetry transformations
As discussed in subsection 4.4, by construction, the N = 1 reduced actions (5.21) and (5.23) are invariant under the second supersymmetry transformations (4.37b). For convenience, we recall the form of these transformations where U(x, θ I ) is a N = (1, 1) superfield, with indices being suppressed. The second supersymmetry tranformations act on the N = 1 fields (5.16) and (5.18) in the following fashion It is apparent from (5.26a) and (5.26b) that the second supersymmetry transformation (5.25) breaks the WZ gauge conditions (5.7), which we recall for convenience In order to resolve this, it is necessary to supplement the variation (5.25) with the ǫdependent gauge transformations: The modified second supersymmetry transformations now take the following form The sole purpose of introducing the ǫ-dependent gauge transformations (5.28) is to restore the original Wess-Zumino gauge (5.27). Fixing the form of the N = 1 components of g α(2s) (ǫ) as 6 Massless half-integer superspin: Longitudinal formulation In this section, we develop the superspace reduction procedure for the longitudinal formulation, following the prescription advocated in section 5.

Longitudinal formulation
For s ≥ 2, the longitudinal formulation for the massless superspin-(s + 1 2 ) multiplet is described in terms of the variables Here, the real superfield H α(2s) is identical to that of (5.1), and the complex superfield G α(2s−2) is longitudinal linear (4.4a). Modulo an overall normalisation factor, the longi-tudinal formulation is uniquely described by the action which is invariant under the gauge transformations where λ α(2s−1) is complex unconstrained and g α(2s) is longitudinal linear, as in (5.2a).
In the case where s = 1, the compensator G is covariantly chiral. If we introduce the field redefinition G = 3σ in action (6.2), along with choosing s = 1, then the model can be shown to coincide with the linearised action for minimal (1, 1) AdS supergravity [21]. In the case s = 1, the gauge transformations (6.3) yield It is an easy exercise to show that δ λ G is covariantly chiral.

Reduction of gauge prepotentials to AdS 3|2
The reduction of the superconformal gauge prepotential H α(2s) was addressed in section 5. We need only reduce the compensator G α(2s−2) to N = 1 AdS superspace. We start by converting the longitudinal constraint of G α(2s−2) (4.4a) to the real basis (4.18) Performing a Taylor expansion of G α(2s−2) (θ I ) about θ 2 , and using (6.6), we find the independent θ 2 -components of G α(2s−2) to be Utilising the gauge transformations (6.3b) and the real representation (4.18), we find We can use the residual gauge freedom (5.9) to compute the gauge transformations of the complex N = 1 superfields (6.7) It is useful to separate the complex superfields (6.7) into real and imaginary components It then becomes apparent from gauge transformations (5.10) and (6.9) that we are in fact dealing with two different gauge theories. The first gauge theory is formulated in terms of the dynamical variables which are defined modulo gauge transfomations of the form The other gauge model is described in terms of the superfields which possess the following gauge freedom After carrying out the reduction to N = 1 AdS superspace, the action (6.2) decouples into two N = 1 theories, which are formulated in terms of superfields (6.11) and (6.13) respectively In the following subsection, we provide the explicit forms of the decoupled N = 1 actions.

Massless higher-spin N = 1 supermultiplets
The first N = 1 gauge theory, described in terms of the dynamical variables (6.11), takes the form The action (6.16) is invariant under the gauge transformations (6.12). We can eliminate the auxiliary field Φ α(2s−3) from (6.16) by using the equation of motion The resulting action, up to an overall factor, then coincides with model (3.6).
The other N = 1 model which is constructed in terms of the dynamical variables (6.13) assumes the form which is invariant under the gauge transformations (6.14). It is evident that the superfield Ω α(2s−3) is auxiliary, so upon elimination via its equation of motion one obtains the action (3.14).

Second supersymmetry transformations
Let us note that the reduced actions (6.15) are invariant under second supersymmetry. More explicitly, the second supersymmetry transformation acts on the dynamical superfields (6.11) and (6.13) by the following rule It is evident from the variations (6.21a) and (6.21b) that second supersymmetry is not compatible with the imposed WZ gauge conditions (5.27). Recall that the longitudinal and transverse formulations are both constructed in terms of the real superfield H α(2s) , which is defined modulo gauge transformations (5.2a). We used this gauge freedom to impose (5.27) in both theories.
Using a similar approach as in subsection 5.4, the WZ gauge conditions (5.27) in the longitudinal formulation can be restored provided we accompany the second supersymmetry transformations with the following ǫ-dependent gauge transformations: The modified second supersymmetry transformations now read The transformations (6.24) act on the N = 1 superfields (6.11) and (6.13) as followŝ ..α 2s )βγ (6.25b)

Massless integer superspin: Longitudinal formulation
In [1], two dually equivalent off-shell formulations, called transverse and longitudinal, for the massless multiplets of integer superspin were developed in (1, 1) AdS superspace. In the following section, we reduce the longitudinal model to N = 1 AdS superspace.

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. The dynamical superfields U α(2s−2) and G α(2s) are 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.
Modulo an overall normalisation factor, the longitudinal formulation for the massless superspin-s multiplet is described by the action which is invariant under the gauge transformations (7.2).

Reduction of gauge prepotentials to AdS 3|2
We wish to reduce the gauge superfields (7.1) to N = 1 AdS superspace. We start by reducing the superfield U α(2s−2) . Converting the transverse linear constraint of γ α(2s−2) (4.5a) to the real basis (4.18) gives Taking a Taylor expansion of γ α(2s−2) (θ I ) about θ 2 , then using (7.4), we find the independent θ 2 -components of γ α(2s−2) to be The gauge transformation (7.2a) allows us to choose the gauge conditions It must be noted that the gauge condition (7.6) is less restrictive than those proposed in the analogous reduction procedure performed in Minkowski superspace, as given in appendix B of [1]. This was done in order to ensure that one of the decoupled N = 1 actions coincides with (3.17). It can be shown that if one chooses to impose the same gauge conditions as detailed in [1], i.e then one will obtain a N = 1 action which would instead coincide with (3.24).
Let us point out that there also appeared a new off-shell formulation for the massless integer superspin multiplet in (1,1) AdS superspace [1]. This formulation proves to be a generalised version of the longitudinal action (7.3), for the gauge-invariant action involves not only U α(2s−2) , Ψ α(2s−1) andΨ α(2s−1) , but also new compensating superfields. Furthermore, the prepotential Ψ α(2s−1) (associated to the longitudinal linear field strength G α(2s) ), enjoys a larger gauge symmetry, which is that of the superconformal complex superspin-s multiplet [1]. Upon reduction to N = 1 AdS superspace, we found that this new formulation decoupled into two N = 1 supersymmetric higher-spin models which coincide with the right-hand side of eq. (7.23).

Massless higher-spin N = 1 supermultiplets
The first N = 1 supersymmetric action which is described by the dynamical variables (7.19) takes the form It is apparent that the superfield Φ α(2s−1) is auxiliary, thus, upon elimination via its equation of motion, we arrive at the resulting action .

(7.26)
Here we have made use of the notation U α(2s−1) = U (α 1 ;α 2 ...α 2s−1 ) . It is a tedious but straightforward exercise to verify that the above N = 1 action coincides with the transverse formulation for massless superspin-s multiplet given by (3.17). To prove this, one first needs to introduce the N = 1 field strength W β; α(2s−2) associated with the prepotential U β;α(2s−2) , The action (7.26) can then be written more compactly as which is exactly (3.17) modulo an overall normalisation factor.
The other decoupled N = 1 action which is formulated in terms of the dynamical variables (7.21) assumes the form It is clear from the action (7.29) that the superfield Ω α(2s−1) is auxiliary. Integrating it out by using its equation of motion, we find that the resulting action, modulo a normalisation factor, coincides with (3.14).

Massless integer superspin: Transverse formulation
We now apply the same reduction procedure to the off-shell transverse formulation for the massless multiplets of integer superspin.

Reduction of gauge prepotentials to AdS 3|2
We begin by reducing the real superfield U α(2s−2) . Following the prescription endorsed in section 7, we have the freedom to impose the gauge condition The residual gauge symmetry which preserves this gauge is described by the real unconstrained N = 1 superfield As a result of choosing the gauge condition (8.4), we are left with the remaining independent N = 1 superfields with respect to U α(2s−2) Using (8.3a) enables the computation of the corresponding gauge transformations where we have introduced the following definition and gauge parameter redefinition ρ α(2s−1) =τ α(2s−1) + i∇ (α 1 ζ α 2 ...α 2s−1 ) .

Massless higher-spin N = 1 supermultiplets
The first of the decoupled N = 1 supersymmetric actions which is formulated in terms of the dynamical variables (8.18) takes the form which is invariant under the gauge transformations (8.19). It is apparent that the superfield Φ α(2s+1) is auxiliary, so upon elimination via its equation of motion, 24) one arrives at the action, which, up to an overall normalisation factor, coincides with (7.26).
The superfield Ω α(2s+1) is auxiliary, so integrating it out using its equation of motion, we obtain action (3.14) up to an overall normalisation factor.

Second supersymmetry transformations
We begin by computing the second supersymmetry transformations of the real N = 1 fields (8.18) and (8.20) It is clear from the variation (8.27a) that the second supersymmetry transformation (5.25) breaks the gauge condition (8.4). We should then supplement the second supersymmetry transformation with the ǫ-dependent gauge transformations: The modified second supersymmetry transformations have the form They act on the N = 1 gauge superfields (8.18) and (8.20) by the rulê

Discussion
This section provides a summary of the results obtained in sections 5−8, where we carried out the (1,1) → (1,0) AdS superspace reduction of the four massless higher-spin multiplets with (1,1) AdS supersymmetry.
It is useful to recall from section 3 that there exist four off-shell series of massless higher-spin N = 1 multiplets in AdS 3 (two series for each half-integer (ŝ = s + 1 2 ) and integer (ŝ = s) superspinŝ ≥ 1): As demonstrated in [3], the models S (s+ 1 2 ) [H α(2s+1) , L α(2s−2) ] and S ⊥ (s+ 1 in AdS 3|2 . The same feature characterises the two formulations for the massless integer superspin multiplet: S ⊥ (s) [H α(2s) , Ψ β;α(2s−2) ] is dual to S (s) [H α(2s) , V α(2s−2) ] only in flat superspace. In appendix B, we study in detail the component structure of these four massless N = 1 supersymmetric higher-spin gauge models in AdS 3 . Upon elimination of the corresponding auxiliary fields, the results are as follows: Sections 5 and 6 detail the (1,1) → (1,0) AdS superspace reduction of the transverse and longitudinal formulations for the massless half-integer superspin multiplets, respectively. In (1,1) AdS superspace, these off-shell formulations are dually equivalent. When reduced to N = 1 AdS superspace, we found that the actual difference between the two models lies in the structure of auxiliary superfields. Let us refer to eq. (5.20) which shows that the transverse formulation gives rise to two decoupled N = 1 supersymmetric theories: In the longitudinal case, from eq. (6.15) we have that We further showed that the superfields Φ α(2s−1) , Ω α(2s−1) , Φ α(2s− 3) , Ω α(2s−3) are all auxiliary. Once they are eliminated, each formulation then leads to the same N = 1 supersymmetric higher-spin actions: Transverse: Sections 7 and 8 concern the (1,1) → (1,0) AdS superspace reduction of the longitudinal and transverse formulations for the massless integer superspin multiplets, respectively. In (1,1) AdS superspace, they are dual to each other. As in the half-integer case, we demonstrated that upon reduction to N =1 AdS superspace, these formulations differ by the structure of the auxiliary superfields. As shown in eq. (7.23), the longitudinal model is equivalent to On the other hand, eq. (8.22) shows that the transverse model yields The superfields Φ α(2s−1) , Ω α(2s−1) , Φ α(2s+1) , Ω α(2s+1) are all auxiliary and thus can be integrated out from their corresponding actions. We showed that the resulting N = 1 actions are the same in both formulations: Longitudinal: S ⊥ (s) [H α(2s) , Ψ β; α(2s−2) ] (9.8b) We are now in a position to compare the above (1,1) → (1,0) AdS reduction results with the N = 1 supersymmetric higher-spin gauge theories obtained via (2,0) → (1,0) AdS reduction [3]. There exist two off-shell formulations for massless half-integer superspin multiplets with (2,0) AdS supersymmetry, which are not dual to each other. 8 They are known as type II and type III series [2]. Upon reduction to N = 1 superspace, type II series yields Type II: On the other hand, type III series leads to Type III: We point out that unlike the situation for the (1,1) AdS multiplets, the N = 1 models obtained from the (2, 0) → (1, 0) AdS reduction do not involve any auxiliary superfields. Additionally, reductions of the (1,1) AdS multiplets only produce three of the four series of off-shell N = 1 multiplets, with the exception of the transverse half-integer model, The dualities relating the four N = 1 supersymmetric models in the flat limit, |µ| → 0, deserve further comment. In flat superspace, one may take any of the above N = 2 superfield formulations and construct several dual models. This can be done by replacing its N = 1 sector(s) by its dual versions. For concreteness, let us consider the transverse formulation for the massless half-integer superspin, eq. (9.5a), in flat superspace. We denote the flat superspace action by S ⊥ ] FS , respectively. At this stage, it is unknown if there exists a N = 2 superfield formulation which leads to the two transverse models described by (iv). This will be an interesting open problem to investigate further.
The massless higher-spin N = 1 models (9.1) do not have any propagating degrees of freedom. However, they can be deformed in order to generate off-shell topologically massive higher-spin supersymmetric theories. There has been recent progress made in the construction of these off-shell theories with various supersymmetries. It is based on the approaches developed in [39] for N = 1 and in [26] for N = 2 Poincaré supersymmetries. They were later extended to AdS 3 for the following cases: N = 1 [3,37], N = (1, 1) [1] and N = (2, 0) [2]. The gauge-invariant actions for such massive multiplets are obtained by adding a superconformal and massless higher-spin action together, following the philosophy of topologically massive theories [43][44][45][46].
In accordance with [3,37], there exist four off-shell formulations for topologically massive higher-spin N = 1 multiplets in AdS 3 . For a positive integer s, there are two off-shell gauge-invariant models for a topologically massive superspin-(s + 1 2 ) multiplet in AdS 3 : Here κ is a dimensionless parameter and m is a real massive parameter. For a topologically massive superspin-s multiplet in AdS 3 , we have the following models: Another way of deriving the four massive N = 1 models, (9.12)-(9.13), is by performing either a (1,1) → (1,0) or (2,0) → (1,0) AdS reduction of the off-shell topologically massive higher-spin supersymmetric theories presented in [1] and [2], respectively. In what follows, we will recall the structure of off-shell massive higher-spin supermultiplets in (1,1) AdS superspace [1] and take a closer look at their reduction.
As pointed out in [1], it is expected that the topologically massive actions (9.16a) and (9.16b) describe the on-shell massive supermultiplets in (1,1) AdS superspace [40]. Given a positive integer n > 0, a massive on-shell multiplet of superspin (n + 1)/2 is realised in terms of a real symmetric rank-n spinor T α(n) constrained by It may be shown that One can also construct a massive model which leads directly to the equations (9.17), as an extension of the flat-space bosonic constructions of [51,52]. It is given by The action (9.19) is invariant under gauge transformations (9.15). To check gauge invariance, one makes use of the properties that W α(n) (H) is (i) gauge-invariant; and (ii) transverse linear,D β W βα 1 ...α n−1 = D β W βα 1 ...α n−1 = 0. The action (9.19) becomes superconformal in the m → ∞ limit.
The (2,0) AdS analogue of the model (9.19), which was described in [2], takes the form where D α ,D α denote the spinor covariant derivatives of (2,0) AdS superspace. One can then obtain the following set of constraints describing an on-shell massive superspin-(n + 1)/2 multiplet in (2,0) AdS superspace: We note that the models (9.19) and (9.24) are not dual to each other since they have different types of supersymmetry, namely (1,1) and (2,0) AdS supersymmetry, respectively.
It is worth pointing out that there also exists an on-shell construction of gaugeinvariant Lagrangian formulations for massive higher-spin N = 1 supermultiplets in R 2,1 and AdS 3 [53][54][55], extending previous works on the non-supersymmetric cases [56,57]. These frame-like formulations are based on the gauge-invariant approach (also known as the Stueckelberg approach) to the dynamics of massive higher-spin fields, which were proposed by Zinoviev [58,59] and Metsaev [60]. As demonstrated in [53][54][55], in three dimensions it is possible to rewrite the corresponding Lagrangians in terms of a set of gauge-invariant curvatures, resulting in a more elegant formulation. An interesting open problem is to understand if there exists an off-shell uplift of these models.

Acknowledgements:
We are grateful to the referee of this work for pointing out important references as well as for the suggestion to work out the component results described in appendix B. The work of JH and SMK is supported in part by the Australian Research Council, project No. DP200101944. The work of DH is supported by the Jean Rogerson Postgraduate Scholarship and an Australian Government Research Training Program Scholarship at The University of Western Australia.
They have the following properties: A three-vector x a can be equivalently realised as a symmetric second-rank spinor x αβ defined as The relationships between Lorentz generators with two vector indices (M ab = −M ba ), one vector index (M a ) and two spinor indices (M αβ = M βα ) are: These generators act on a vector V c and a spinor Ψ γ by the rules The (brackets) parentheses correspond to (anti-)symmetrisation of tensor or spinor indices, which encode a normalisation factor, for example V [a 1 a 2 ...an] := 1 n! π∈Sn sgn(π)V a π(1) ...a π(n) , V (α 1 ...αn) := 1 n! π∈Sn V α π(1) ...α π(n) , (A. 8) with S n being the symmetric group of n elements.
We provide a summary of essential identities for (1,1) AdS covariant derivatives, which we denote by ∇ I A = (∇ a , ∇ I α ), where I = 1, 2. Making use of the algebra in the real basis (4.19), we can derive the following results: B Component structure of N = 1 supersymmetric higher-spin actions in AdS 3 In this appendix, we study the component structure of the longitudinal and transverse formulations for both the half-integer and integer superspin multiplets which are reviewed in section 3. In accordance with (4.42), any N = 1 supersymmetric action in AdS 3 can be reduced to components by the rule In what follows, we will denote the torsion-free covariant derivative on AdS 3 by D a . It is related to the vector covariant derivative ∇ a in (2.1) by the simple rule D a := ∇ a | θ=0 , provided an appropriate Wess-Zumino gauge chosen.

B.1 Fronsdal-type actions in AdS 3
We begin by reviewing the AdS 3 counterparts of the (Fang-)Fronsdal actions in AdS 4 [61,62], mostly following the presentation in [37]. Here, the gauge parameter λ α(n−2) is real and unconstrained. Modulo an overall constant, the corresponding gauge-invariant action is Only the action S (n,+) FF was considered in [37]. Both actions S In the flat-space limit, the action (B.7) reduces to the model considered by Tyutin and Vasiliev [63] (see also [37] for a review). Ref. [57] also made use of the action (B.7) in the frame-like formulation, but only a particular choice of sign was considered. In order to relate the action (B.7) to (B.4), it is necessary to decompose the dynamical field h βγ;α(n−2) into its irreducible components. The irreducible fields contained in h βγ;α(n−2) can be defined as follows: The Fronsdal action in AdS 4 [61] is second-order in derivatives. It can be generalised to AdS 3 as follows. Given an integer n ≥ 4, we introduce two real dynamical variables which are defined modulo gauge transformations of the form with the gauge parameter λ α(n−2) being real unconstrained. Modulo an overall constant, the gauge-invariant action is given by where Q is the following Casimir operator of the AdS group SO(2, 2): The model (B.11) is the d = 3 counterpart of the Fronsdal action provided n is even, n = 2s, with s ≥ 2 an integer.

B.2 Dual formulation of the Fronsdal-type action in Minkowski space
Here we introduce a one-parameter family of dual formulations for the flat-space counterpart of the model (B.11). These results will be important for our analysis in the next subsection.
Let us consider the flat-space limit of the model (B.11) where ∂ αβ are the partial derivatives of three-dimensional Minkowski space. This action is invariant under the gauge transformations Both fields h α(n) and y α(n−4) appear in (B.13) with derivatives, and therefore a duality transformation may be performed upon each of them. Here we will only dualise y α(n−4) and keep h α(n) intact. 9 We may think of ∂ βγ y α(n−4) as a longitudinal tensor field, by analogy with a longitudinal vector field. Our dual formulation for (B.15) is obtained by introducing the following first-order action where we have introduced the Lagrangian The gauge transformation law (B.14a) allows us to interpret h α(n) as a conformal spinn 2 gauge field, while (B.14b) is compatible with the interpretation of y α(n−4) as a conformal compensator (following the modern supergravity terminology). Performing a duality transformation on y α(n−4) is equivalent to the introduction of an alternative conformal compensator. which involves the real coefficients A and B constrained by The field H βγ; α(n−4) in (B.16) is unconstrained and the Lagrange multiplier F βγ; α(n−4) is given by for some unconstrained prepotential Φ γδ; α(n−4) . Varying the first-order action (B.16) with respect to Φ γδ; α(n−4) gives which is equivalent to This equation allows us to express H βγ; α(n−4) in terms of h α(n) and F βγ; α(n−4) : We have obtained an one-parameter family of dual actions.
As described above, the one-parameter first-order action (B.16) is equivalent to the original model (B.15). The action (B.16) is invariant under the gauge λ-transformations (B.14a), which act on H βγ; α(n−4) and Φ βγ; α(n−4) as follows: The dual action (B.25) is also invariant under the gauge ρ-transformation, We emphasise that the parameters A and B are constrained by (B.18).

B.3 Transverse formulation of the superspin-(s + 1 2 ) multiplet
The transverse formulation of the massless superspin-(s + 1 2 ) multiplet is described by the action (3.10), which is invariant under the gauge transformations (3.9). The gauge freedom (3.9) can be used to impose the following Wess-Zumino gauge The residual gauge freedom preserving the gauge conditions (B.28) is given by This implies that there are three real independent gauge parameters at the component level, which we choose as The next task is to identify the remaining independent component fields of H α(2s+1) and Υ β;α(2s−2) in the Wess-Zumino gauge (B.28).
The However, it must be noted that this duality does not hold in the presence of a nonvanishing AdS curvature, since (B.37) cannot be written solely in terms of the field strength F βγ; α(2s−2) .

B.4 Longitudinal formulation of the superspin-(s + 1 2 ) multiplet
The longitudinal formulation of the massless superspin-(s+ 1 2 ) multiplet is described by the action (3.6), which is invariant under the gauge transformations (3.5). The component structure of this model was studied in [37] only in the flat superspace limit. Here we extend these results to AdS 3 .

(B.60)
We can express the field Φ βγ;α(2s−2) in terms of its irreducible components The study of the fermionic sector requires the decomposition of the reducible superfield Ψ β;α(2s−2) into irreducible parts. This procedure is completely analogous to that of the prepotential Υ β;α(2s−2) (B.31). In this case, we find that the remaining independent fermionic fields are given by

B.6 Longitudinal formulation of the superspin-s multiplet
The longitudinal formulation of the massless superspin-s multiplet is described by the action (3.14) and is invariant under gauge transformations (3.13). This formulation corresponds to the massless first-order model, whose component reduction in the flatsuperspace limit has been studied in [37].
The gauge freedom (3.13) can be used to impose the following Wess-Zumino gauge