Higher spin supermultiplets in three dimensions: (2,0) AdS supersymmetry

Within the framework of (2,0) anti-de Sitter (AdS) supersymmetry in three dimensions, we propose a multiplet of higher-spin currents. Making use of this supercurrent, we construct two off-shell gauge formulations for a massless multiplet of half-integer superspin $(s+\frac 12)$, for every integer $s>0$. In the $s=1$ case, one formulation describes the linearised action for (2,0) anti-de Sitter supergravity, while the other gives the type III minimal supergravity action in (2,0) AdS superspace, with both linearised supergravity actions originally derived in arXiv:1109.0496. The constructed massless actions are then used to formulate topologically massive higher-spin supermultiplets in (2,0) AdS superspace. Our results admit a natural extension to the case of $S^3$.


Introduction
In four dimensions (4D), there is an interesting correspondence between N = 1 antide Sitter (AdS) supergravity 1 [1] and massless higher-spin supermultiplets in AdS 4 [3]. Specifically, two off-shell formulations are known for pure N = 1 AdS supergravity, the minimal [4,5,6] (see, e.g., [7,8] for pedagogical reviews) and the non-minimal [9] theories. In AdS 4 there exist two series of massless off-shell gauge supermultiplets of half-integer superspin s + 1 2 , with s = 1, 2, . . . [3]. 2 The correspondence consists of the fact that, for the lowest superspin value corresponding to s = 1, one series yields the linearised action for minimal AdS supergravity, while the other leads to linearised non-minimal AdS supergravity. It has recently been pointed out [10] that a similar correspondence might 1 Townsend's work on N = 1 AdS supergravity [1] appeared shortly after Freedman and Das constructed N = 2 AdS supergravity [2]. The motivations for [1] and [2] were rather different. 2 Such a supermultiplet describes two ordinary massless spin-(s + 1 2 ) and spin-(s + 1) fields on-shell. occur in the case of 3D N = 2 supersymmetry, which is a natural cousin of the 4D N = 1 one.
Unlike four dimensions, where pure N = 1 AdS supergravity is unique on-shell, the feature specific to three dimensions is the existence of two distinct N = 2 AdS supergravity theories [11], which are known as the (1,1) and (2,0) AdS supergravity theories, originally constructed as Chern-Simons theories. Two off-shell formulations for (1,1) AdS supergravity have been developed, the minimal [12,13,14,15,16,17,18] and the nonminimal [17,18] theories, and one for (2,0) AdS supergravity [19,16,17,18]. Since there are three off-shell N = 2 AdS supergravity theories, one might expect the existence of three series of massless higher-spin gauge supermultiplets. In a recent paper [10], we have presented two series of massless higher-spin actions which are associated with the minimal and the non-minimal (1,1) AdS supergravity theories, respectively, generalising similar constructions in the super-Poincaré case [20]. The present paper is devoted to constructing higher-spin gauge multiplets with (2,0) AdS supersymmetry.
It is worth pointing out that the massless 3D constructions of [10,20], were largely modelled on the 4D results of [3,21]. With respect to 3D (2,0) AdS supersymmetry, unfortunately there is no 4D intuition to guide us, and new ideas are required in order to construct higher-spin gauge supermultiplets. In this paper our approach will be to utilise an observation that has often been used in the past to formulate off-shell supergravity multiplets [22,23,24,25,26,27]. The idea is to make use of a higher-spin extension of the supercurrent (also known as the multiplet of currents), the concept introduced by Ferrara and Zumino in the case of 4D N = 1 Poincaré supersymmetry [28] and extended to 4D N = 2 Poincaré supersymmetry by Sohnius [29]. Specifically, for a simple supersymmetric model in (2,0) AdS superspace we identify a multiplet of conserved higher-spin currents. In general, the multiplet of currents is always off-shell. Using the constructed higher-spin supercurrent, we may identify a corresponding supermultiplet of higher-spin fields. The procedure to follow is concisely described by Bergshoeff et al. [22]: "One first assigns a field to each component of the current multiplet, and forms a generalized inner product of field and current components." Our multiplet of currents is described by the conservation equations (1.1a) Here D α andD α are the covariant spinor derivatives of (2,0) AdS superspace [17], J α(2s) := J α 1 ...α 2s = J (α 1 ...α 2s ) =J α(2s) denotes the higher-spin supercurrent, and T α(2s−2) the corre-sponding trace supermultiplet constrained to be covariantly linear 3 In general, the trace supermultiplet is complex, In the s = 1 case, the above conservation equation coincides with that for the (2,0) AdS supercurrent [17].
Our work may have various generalisations and applications. For instance, the massless higher-spin actions constructed in section 4.1 are expected to possess nonlinear completions, say, in the spirit of the bosonic Chern-Simons constructions of [30,31,32]. Our results admit a natural extension to the case of S 3 , which may lead to higher-spin applications of the localisation techniques, see, e.g., [33,34] for reviews. The adequate superspace setting to formulate N = 2 supersymmetric theories on S 3 has been developed [35]. This paper is organised as follows. Section 2 provides a brief review of (2,0) AdS superspace. In section 3 we consider simple models for a chiral scalar supermultiplet and demonstrate how the higher-spin supercurrent (1.1) emerges. In section 4 we develop two off-shell formulations for a massless multiplet of half-integer superspin (s + 1 2 ) in (2,0) AdS superspace, with s a positive integer. Our results and their implications and possible extensions are discussed in section 5. In the appendix we collect important (2,0) AdS identities.

(2,0) AdS superspace
In this section we give a summary of the most important results concerning (2,0) AdS superspace, see [17] for the details.
The covariant derivatives of (2,0) AdS superspace have the form Here E A and Ω A denote the inverse supervielbein and the Lorentz connection, respectively, The Lorentz generators with two vector indices (M ab = −M ba ), with one vector index (M a ) and with two spinor indices (M αβ = M βα ) are defined in the appendix. The U(1) R generator J in (2.1) is defined to act on the covariant derivatives as follows: The covariant derivatives satisfy the following algebra [17]: Here the parameter S is related to the AdS scalar curvature as R = −24S 2 .
In accordance with the general formalism of [8], the isometries of (2,0) AdS superspace are generated by those real supervector fields ζ A E A which obey the superspace Killing equation [17] ζ and τ and l bc are some local U(1) R and Lorentz parameters, respectively. Every solution of (2.5) is called a Killing supervector field of (2,0) AdS superspace. As demonstrated in [17], eq. (2.5) implies that the parameters ζ α , τ and l αβ are uniquely expressed in terms of the vector ζ αβ , which obeys the equation It follows that ζ a is a Killing vector field, One may also prove the following relations The Killing supervector fields of (2,0) AdS superspace generate the supergroup OSp(2|2; R)× Sp(2, R), the isometry group of (2,0) AdS superspace. Rigid supersymmetric field theories on (2,0) AdS superspace are invariant under the isometry transformations. The isometry transformation associated with the Killing supervector field ζ A E A acts on a tensor superfield U (with its indices suppressed) by the rule Associated with a real scalar superfield L is the following supersymmetric invariant where E denotes the chiral integration measure.

Higher-spin supercurrents for chiral matter
In this section we study higher-spin supercurrents in simple models for a chiral scalar supermultiplet in (2, 0) AdS superspace.

Massless models
We first consider a massless model. Its action is invariant under the isometry transformations of (2,0) AdS superspace for any U(1) R charge w of the chiral superfield, The action is superconformal provided w = 1 2 . As in [10], it is useful to introduce auxiliary real variables ζ α ∈ R 2 . Given a tensor superfield U α(m) , we associate with it the following field which is a homogeneous polynomial of degree m in the variables ζ α . We introduce operators that increase the degree of homogeneity in the variable ζ α , We also introduce two nilpotent operators that decrease the degree of homogeneity in the variable ζ α , specifically Let us first consider the superconformal case, w = 1 2 . The analysis given in [10] implies that the theory possesses a real supercurrent J (2s) =J (2s) , for any positive integer s, which obeys the conservation equation This supercurrent proves to have the same form as in the (1,1) AdS case considered in [10]. Specifically, the higher-spin supercurrent 4 is given by Making use of the massless equations of motion, D 2 Φ = 0, one may check that this supermultiplet does obey the conservation equation (3.6).
Now we turn to the non-superconformal case, w = 1 2 . Direct calculations give where we have denoted The trace multiplet T (2s−2) is covariantly linear, as a consequence of the equations of motion and identities (A.2c). It is seen that T (2s−2) has non-zero real and imaginary parts, except for the s = 1 case which is characterised by Y = 0. For s = 1 the above results agree with [17]. The technical details of the derivation of (3.8) are collected in the appendix.
The above results can be used to derive higher-spin supercurrents in a non-minimal scalar supermultiplet model described by the action with Γ being a complex linear superfield. 5 The non-minimal theory (3.9) proves to be dual to (3.1) provided the U(1) R weight of Γ is opposite to that of Φ, Replacing Φ →Γ andΦ → Γ in (3.8) gives the higher-spin supercurrents in the nonminimal theory (3.9), which is similar to the 4D case [37,38].

Massive model
Let us add a mass term to the functional (3.1) and consider the following action with m a complex mass parameter. In the m = 0 case, the U(1) R weight of Φ is uniquely fixed to be w = 1, in order for the action to be R-invariant.
Making use of the massive equations of motion we obtain Φ 5 Unlike eq. (1.1b), the above condition on Γ is the only constraint obeyed by Γ.
The above consideration demonstrates that in the massive case higher-spin supercur-rentsĴ (2s) exist only for the odd values of s, eq. (3.15a). This conclusion is analogous to the earlier results in four dimensions [39,40,38]. As was demonstrated [38] in the 4D case, the even values of s are also allowed provided there are several massive chiral superfields in the theory. The analysis of [38] may be extended to the 3D (2,0) AdS case.

Massless higher-spin gauge theories
The explicit structure of the higher-spin supercurrent defined by eqs. (3.8a) and (3.8c) allows us to develop two off-shell formulations for a massless multiplet of half-integer superspin (s + 1 2 ), for every integer s > 0. We will call them type II and type III models in order to comply with the terminology introduced in [17] for the minimal formulations of N = 2 supergravity.
Associated with L α(2s−2) is the real field strength which is invariant under the gauge transformations (4.4), δ ξ L α(2s−2) = 0. It is not difficult to see that L α(2s−2) is a covariantly linear superfield, From (4.2b) we can read off the gauge transformation of the field strength Modulo an overall normalisation factor, there is a unique quadratic action which is invariant under the gauge transformations (4.2). It is given by By construction, the action is also invariant under (4.4).
Setting s = 1 in (4.8) gives the linearised action for (2,0) AdS supergravity, which was originally derived in section 10.1 of [17]. 6 It should be remarked that the second last term in (4.8) is not defined in the s = 1 case. However, this term contains an overall numerical factor (s − 1) and therefore it does not contribute for s = 1.

Type III series
Our second model for the massless superspin-(s + 1 2 ) multiplet is realised in terms of dynamical variables that are completely similar to (4.1), Here H α(2s) and V α(2s−2) are unconstrained real tensor superfields. The only difference from the type II case consists in a different gauge transformation law for the compensator V α(2s−2) . We postulate the following gauge transformation laws: where the gauge parameter λ α(2s−1) is unconstrained complex. The compensator V α(2s−2) is required to have its own gauge freedom of the form with the gauge parameter ξ α(2s−2) being covariantly chiral, but otherwise arbitrary.
A unique gauge-invariant action is given by This action involves the real linear field strength which is invariant under (4.11). It varies under the transformation (4.10) as (4.14) Setting s = 1 in (4.12) gives the type III minimal supergravity action in (2,0) AdS superspace, which was originally derived in section 10.2 of [17]. 7

Discussion
In this paper we did not carry out a systematic analysis (similar to that given by Dumitrescu and Seiberg [42] for ordinary supercurrents in Minkowski space) of the higherspin supercurrent (1.1). The explicit form of the multiplet of currents was deduced from the consideration of simple dynamical systems in (2,0) AdS superspace. However, the formal consistency of (1.1) follows from the structure of the massless higher-spin gauge theories constructed in section 4. For instance, within the framework of the type II formulation let us couple the prepotentials H α(2s) and L α(2s−2) to external sources source to be invariant under the gauge transformations (4.4) tells us that the real supermultiplet Z α(2s−2) is covariantly linear, If we also require S (s+ 1 2 ) source to be invariant under the gauge transformations (4.2), we obtain the conservation equationD Additionally, taking the type III formulation into account leads to the general conservation equationD where the real trace supermultiplets Y α(2s−2) and Z α(2s−2) are covariantly linear.
There is one special feature of the supergravity case, s = 1, for which the supercurrent conservation equation takes the form [17] with the real trace supermultiplets Y and Z being covariantly linear. Building on the thorough analysis of [42], it was pointed out in [17] that there exists a well-defined improvement transformation that results with Y = 0. For all the supersymmetric field theories in (2,0) AdS superspace considered in [17], the supercurrent is characterised by the condition Y = 0. Actually, this condition is easy to explain. The point is that every 3D N = 2 supersymmetric field theory with U(1) R-symmetry may be coupled to the (2,0) AdS supergravity, which implies Y = 0 upon freezing the supergravity multiplet to its maximally supersymmetric (2,0) AdS background. 8 However, in the higher-spin case it no longer seems possible to improve the trace supermultiplet Y α(2s−2) to vanish, as our analysis in section 3 indicates.
The massless models (4.8) and (4.12) describe no propagating degrees of freedom. However, in conjunction with the superconformal higher-spin actions in conformally flat backgrounds proposed in [10] they can be used to construct topologically massive higherspin supersymmetric theories. Specifically, let us consider the following gauge-invariant models: with κ and m dimensionless and massive parameters, respectively. Here is the superconformal higher-spin action [10], with W α(2s) (H) =W α(2s) (H) being the higher-spin super-Cotton tensor. It is the unique descendant of H α(2s) with the following properties: (i) W α(2s) is invariant under the gauge transformations (4.2a); (ii) W α(2s) obeys the conservation equations We believe that the higher-derivative actions (5.7a) and (5.7b) describe the on-shell massive superspin-(s + 1 2 ) multiplets formulated in [43]. 9 For a positive integer n > 0, a massive on-shell multiplet of superspin (n + 1)/2 is described by a real symmetric rank-n spinor T α(n) subject to the constraints [43] It may be shown that T α 1 ···αn = D a D a + (n + 2)iSD γD γ − n(n + 2)S 2 T α 1 ···αn , (5.11) where the second term on the right can be rewritten as follows: At the component level, the equations (5.10) may be shown to describe the on-shell massive fields in AdS 3 introduced in [44,45].
It is of interest to carry out N = 2 → N = 1 AdS superspace reduction of the massless models (4.8) and (4.12). Following [48], we can introduce a real basis for the spinor covariant derivatives which is obtained by replacing the complex operators D α and D α with ∇ I α , where I = 1, 2, defined by Defining ∇ a = D a , the new (2,0) AdS covariant derivatives satisfy the algebra The graded commutation relations for the operators ∇ a and ∇ 1 α have the following properties: (i) they do not involve ∇ 2 α ; and (ii) they are identical to those defining N = 1 AdS superspace, AdS 3|2 , see [48] for the details. These properties mean that AdS 3|2 is naturally embedded in (2,0) AdS superspace as a subspace. The 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. Every supersymmetric field theory in (2,0) AdS superspace may be reduced to AdS 3|2 . Carrying out the N = 2 → N = 1 AdS superspace reduction of the massless models (4.8) and (4.12) will give a new understanding of the difference between these models. It will also uncover whether one of the massless models (4.8) and (4.12) contain any new N = 1 supersymmetric higher spin actions compared with those derived in [49,50].

A (2,0) AdS identities
The Lorentz generators with two vector indices (M ab = −M ba ), one vector index (M a ) and two spinor indices (M αβ = M βα ) are related to each other by the rules: M a = 1 2 ε abc M bc and M αβ = (γ a ) αβ M a . These generators act on a vector V c and a spinor Ψ γ as follows: The covariant derivatives of (2,0) AdS superspace hold various identities, which can be easily derived from the covariant derivatives algebra (2.4). We have made use of the following identities: where D 2 = D α D α , andD 2 =D αD α . These relations imply the identity which guarantees the reality of the actions considered in the main body of the paper.
In deriving eq. (3.8), one may find the following identities useful. We start with the obvious relations We also state some other properties which we often use throughout our calculations