(2,0) Ads Supersymmetry in N=1 Ads Superspace

PHYSICAL REVIEW D 100, 045010 (2019)

Field theories with (2,0) AdS supersymmetry in N = 1 AdS superspace

† Jessica Hutomo* and Sergei M. Kuzenko Department of Physics M013, The University of Western Australia 35 Stirling Highway, Crawley W.A. 6009, Australia

(Received 11 June 2019; published 12 August 2019)

In three dimensions, it is known that field theories possessing extended ðp; qÞ anti-de Sitter (AdS) supersymmetry with N ¼ p þ q ≥ 3 can be realized in (2,0) AdS superspace. Here we present a formalism to reduce every field theory with (2,0) AdS supersymmetry to N ¼ 1 AdS superspace. As nontrivial examples, we consider supersymmetric nonlinear sigma models formulated in terms of N ¼ 2 chiral and linear supermultiplets. The ð2; 0Þ → ð1; 0Þ 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 1 sˆ, integer ðsˆ ¼ sÞ or half-integer ðsˆ ¼ s þ 2Þ, with s ¼ 1; 2; …, we obtain two off-shell formulations for a massless N ¼ 1 superspin-sˆ multiplet in AdS3. 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 N ¼ 1 supersymmetric higher-spin models thus derived are new. The constructed massless N ¼ 1 supersymmetric higher-spin actions in AdS3 are used to formulate (i) higher- spin supercurrent multiplets in N ¼ 1 AdS superspace, and (ii) new topologically massive higher-spin off- shell supermultiplets. Examples of N ¼ 1 higher-spin supercurrents are given for models of a complex scalar supermultiplet. We also present two new off-shell formulations for a massive N ¼ 1 gravitino supermultiplet in AdS3.

DOI: 10.1103/PhysRevD.100.045010

I. INTRODUCTION may be interpreted as a maximally symmetric solution of 2 In three spacetime dimensions, the AdS group is a ðp; qÞ AdS supergravity. Within the off-shell formulation N product of two simple groups, for -extended conformal supergravity which was first sketched in [4] and then fully developed in [5],AdSð3jp;qÞ originates as a maximally symmetric supergeometry with SOð2; 2Þ ≅ ðSLð2; RÞ ×SLð2; RÞÞ=Z2; ð1:1Þ covariantly constant torsion and curvature generated by a symmetric torsion SIJ ¼ SJI, with the structure-group N 2; R indices I, J taking values from 1 to . It turns out that and so are its supersymmetric extensions OSpðpj Þ × IJ 1 S is nonsingular and can be brought to the form OSpðqj2; RÞ. This implies that N -extended AdS super- N − gravity exists in several incarnations [2], which are known zfflfflfflfflfflfflffl}|fflfflfflfflfflfflffl{p zfflfflfflfflfflffl}|fflfflfflfflfflffl{q¼ p as the ðp; qÞ AdS supergravity theories, where the integers SIJ ¼ Sdiagðþ1; …; þ1; −1; …; −1Þ; ð1:3Þ p ≥ q ≥ 0 are such that N ¼ p þ q. The so-called ðp; qÞ AdS superspace [3] for some positive parameter S of unit dimension. For p ¼ N ≥ 4 and q ¼ 0, there exist more general AdS superspaces [3] than the conformally flat ones defined by (1.2). OSpðpj2; RÞ ×OSpðqj2; RÞ AdS 3 ¼ ð1:2Þ In the extended N ¼ p þ q ≥ 3 case, general ðp; qÞ ð jp;qÞ SL 2; R ×SO p ×SO q ð Þ ð Þ ð Þ supersymmetric field theories in AdS3 can be realized in 3 (2,0) AdS superspace, AdSð3j2;0Þ [3,6]. Such realizations *[email protected] are often useful for applications, for instance, in order to † [email protected] study the target space geometry of supersymmetric 1More general AdS supergroups exist for N ≥ 4.

2 Published by the American Physical Society under the terms of In the case of N ¼ 1 AdS supersymmetry, both notations the Creative Commons Attribution 4.0 International license. (1,0) and N ¼ 1 are used in the literature. We will often use the Further distribution of this work must maintain attribution to notation AdS3j2 for N ¼ 1 AdS superspace. 3 the author(s) and the published article’s title, journal citation, General aspects of (2,0) supersymmetric field theory in AdS3 and DOI. Funded by SCOAP3. were studied in [7].

2470-0010=2019=100(4)=045010(29) 045010-1 Published by the American Physical Society JESSICA HUTOMO and SERGEI M. KUZENKO PHYS. REV. D 100, 045010 (2019) nonlinear σ-models in AdS3 [6]. It is worth elaborating on A. Geometry of (2,0) AdS superspace: the σ-model story in some more detail. For the N ¼ 3 and Complex basis N 4 ¼ choices, manifestly ðp; qÞ supersymmetric formu- We begin by briefly reviewing the key results concerning lations have been constructed [3] for the most general (2,0) AdS superspace; see [6,7] for the details. There are σ nonlinear -models in AdS3 (these formulations make use two ways to describe the geometry of (2,0) AdS super- of the curved superspace techniques developed in [5]). This space, which correspond to making use of either a real or a manifestly supersymmetric setting is very powerful since it complex basis for the spinor covariant derivatives. We first σ allows one to generate arbitrary nonlinear -models with consider the formulation in the complex basis. ðp; qÞ AdS supersymmetry. However, it also has a draw- The covariant derivatives of (2,0) AdS superspace are back that the hyperkähler geometry of the σ-model target space is hidden. In order to uncover this geometry, the ¯ α DA ¼ðD ; Dα; D Þ¼EA þ ΩA þ iΦAJ; formulation of the nonlinear σ-model in (2,0) AdS super- a 4 ∂ space becomes truly indispensable [6]. M EA ¼ EA ; ð2:1Þ This work is somewhat similar in spirit to [6,13]; ∂zM however, our goals are quite different. Specifically, we M m μ ¯ develop a formalism to reduce every field theory with (2,0) where z ¼ðx ; θ ; θμÞ are local superspace coordinates, 1 AdS supersymmetry to N ¼ 1 AdS superspace. This and J is the generator of the R-symmetry group, Uð ÞR. The formalism is then applied to carry out the ð2; 0Þ → ð1; 0Þ generator J is defined to act on the covariant derivatives as AdS reduction of the off-shell massless higher-spin super- follows: multiplets in AdSð3j2;0Þ constructed in [1]. There are at least ¯ α ¯ α two motivations for pursuing such an application. First, ½J; Dα¼Dα; ½J; D ¼−D ; ½J; Da¼0: ð2:2Þ certain theoretical arguments imply that there exist more general off-shell massless higher-spin N ¼ 1 supermultip- The Lorentz connection, ΩA, can be written in several lets in AdS3 than those described in [14]. Second, N ¼ 1 equivalent forms, which are supermultiplets of conserved higher-spin currents have never been constructed in AdS3 (except for the super- 1 1 Ω Ω bc −Ω b Ω βγ conformal multiplets of conserved currents in Minkowski A ¼ 2 A Mbc ¼ A Mb ¼ 2 A Mβγ: ð2:3Þ superspace [15] which can readily be lifted to AdS3). Both issues will be addressed below. In particular, we will derive The relations between the Lorentz generators with two N 1 new off-shell higher-spin ¼ supermultiplets in AdS3, vector indices (Mab ¼ −Mba), one vector index (Ma) which will be used to construct new topologically massive and two spinor indices (Mαβ ¼ Mβα) are given in the higher-spin supermultiplets. Appendix A. The table of contents reflects the structure of the paper. The covariant derivatives of (2,0) AdS superspace obey Our notation and conventions follow [5]. the following graded commutation relations:

¯ ¯ II. ð2;0Þ → ð1;0Þ AdS SUPERSPACE REDUCTION fDα; Dβg¼0; fDα; Dβg¼0; ð2:4aÞ

The aim of this section is to elaborate on the details of the ¯ procedure for reducing the field theories in (2,0) AdS fDα; Dβg¼−2iðDαβ − 2SMαβÞ − 4iεαβSJ; ð2:4bÞ superspace to N ¼ 1 AdS superspace. Explicit examples of such a reduction are given by considering supersymmetric γ ¯ γ ¯ ½Da; Dβ¼ðγaÞβ SDγ; ½Da; Dβ¼ðγaÞβ SDγ; ð2:4cÞ nonlinear σ-models. 2 ½Da; Db¼−4S Mab: ð2:4dÞ

4Analogous results exist in four dimensions. The most general Here the parameter S is related to the AdS scalar curvature N ¼ 2 supersymmetric σ-model in AdS4 was constructed [8,9] as R ¼ −24S2. N 1 using a formulation in terms of ¼ covariantly chiral super- There exists a universal formalism to determine iso- fields, as an extension of the earlier analysis in the super-Poincar´e case [10,11]. One of the main virtues of the N ¼ 1 formulation metries of curved superspace backgrounds in diverse [8,9] is its geometric character; however the second supersym- dimensions [16,17]. This formalism was used in [7] to metry is hidden. General off-shell N ¼ 2 supersymmetric σ- compute the isometries of (2,0) AdS superspace (as well as models in AdS4 were actually formulated a few years earlier [12] supersymmetric backgrounds in off-shell N ¼ 2 super- in N ¼ 2 AdS superspace. The latter approach makes N ¼ 2 σ gravity theories [18]). The isometries of (2,0) AdS super- supersymmetry manifest, but the hyperkähler geometry of the - A model target space is hidden. The two σ-model formulations are space are generated by the Killing supervector fields ζ EA, related via the N ¼ 2 → N ¼ 1 AdS superspace reduction [13]. which are defined to solve the master equation

045010-2 FIELD THEORIES WITH (2,0) ADS SUPERSYMMETRY IN … PHYS. REV. D 100, 045010 (2019) 1 M ζ bc τ D 0 In a similar way we introduce real coordinates, z ¼ þ l Mbc þ i J; A ¼ ; ð2:5aÞ m θμ 2 ðx ; I Þ, to parametrize (2,0) AdS superspace. Defining ∇a ¼ Da, the algebra of (2,0) AdS covariant derivatives where (2.4) turns into5

B b β ¯ ¯ β b b ∇I ∇J 2 δIJ∇ − 4 δIJS 4ε εIJS ζ ¼ ζ DB ¼ ζ Db þ ζ Dβ þ ζβD ; ζ ¼ ζ ; ð2:5bÞ f α; βg¼ i αβ i Mαβ þ αβ J; ð2:12aÞ

τ bc 1 J γ J 2 and and l are some real Uð ÞR and Lorentz superfield ½∇a; ∇β¼SðγaÞβ ∇γ ; ½∇a; ∇b¼−4S Mab: ð2:12bÞ parameters, respectively. It follows from Eq. (2.5) that the parameters ζα, τ and lαβ are uniquely expressed in terms of 1 The action of the Uð ÞR generator on the spinor covariant the vector parameter ζαβ as follows: derivatives is given by

i β i α I J ζ D¯ ζ τ D ζ 2 D ζ − Sζ ½J; ∇α¼−iεIJ∇α: ð2:13Þ α ¼ 6 βα; ¼ 2 α;lαβ ¼ ð ðα βÞ αβÞ: ð2:6Þ As may be seen from (2.12), the graded commutation 1 relations for the operators ∇a and ∇α have the following The vector parameter ζαβ satisfies the equation properties: (1) These (anti)commutation relations do not 2 D ζ 0 ∇α ðα βγÞ ¼ : ð2:7Þ involve ,

1 1 This implies the standard Killing equation, f∇α; ∇βg¼2i∇αβ − 4iSMαβ; ð2:14aÞ

Daζb þ Dbζa ¼ 0: ð2:8Þ 1 γ 1 2 ½∇a; ∇β¼SðγaÞβ ∇γ ; ½∇a; ∇b¼−4S Mab: One may also prove the following relations: ð2:14bÞ

¯ i ¯ β ¯ (2) Relations (2.14) are identical to the algebra of the Dατ ¼ D lαβ ¼ 4Sζα; Dαζβ ¼ 0; D αlβγ ¼ 0: 3 ð Þ covariant derivatives of AdS3j2, see (2.4). 3 2 ð2:9Þ We thus see that AdS j is naturally embedded in (2,0) AdS superspace as a subspace. The real Grassmann variables of θμ θμ θμ The Killing supervector fields prove to generate the super- (2,0) AdS superspace, I ¼ð 1; 2Þ, may be chosen in 2 2; R 2 R 3 2 group OSpð j Þ ×Spð ; Þ, the isometry group of (2,0) such a way that AdS j corresponds to the surface defined AdS superspace. Rigid supersymmetric field theories in μ by θ2 ¼ 0. We also note that no Uð1Þ curvature is present (2,0) AdS superspace are required to be invariant under the R in the algebra of N ¼ 1 AdS covariant derivatives. These isometry transformations. An infinitesimal isometry trans- properties make possible a consistent ð2; 0Þ → ð1; 0Þ AdS formation acts on a tensor superfield U (with suppressed superspace reduction. indices) by the rule Now we will recast the fundamental properties of the 1 (2,0) AdS Killing supervector fields in the real representa- δ ζ bc τ ζU ¼ þ 2 l Mbc þ i J U: ð2:10Þ tion (2.11). The isometries of (2,0) AdS superspace are described in terms of those first-order operators

ζ ≔ ζB∇ ζb∇ ζβ∇J 1 2 B ¼ b þ J β;J¼ ; ; ð2:15aÞ B. Geometry of (2,0) AdS superspace: Real basis which solve the equation Instead of dealing with the complex basis for the (2,0) AdS spinor covariant derivatives, Eq. (2.1), it is more 1 ζ bc τ ∇ 0 convenient to switch to a real basis in order to carry out þ 2 l Mbc þ i J; A ¼ ; ð2:15bÞ reduction to N ¼ 1 AdS superspace AdS3j2. Following [3], such a basis is introduced by replacing the complex opera- for some real parameters τ and lab ¼ −lba. Equation (2.15b) ¯ I 1 2 tors Dα and Dα with ∇α ¼ð∇α; ∇αÞ defined as follows: is equivalent to 1 1 1 2 ¯ 1 2 5 εIJ ε D ffiffiffi ∇α − ∇α D − ffiffiffi ∇α ∇α The antisymmetric tensors and IJ are normalized as α ¼ p ð i Þ; α ¼ p ð þ i Þ: ð2:11Þ 1 2 2 2 ε ¼ ε1 2 ¼ 1.

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1 ∂ 1 I J IJ IJ IJ I M bc ∇αζβ ¼ −εαβε τ þ Sδ ζαβ þ δ lαβ; ð2:16aÞ ∇A ¼ð∇ ; ∇αÞ¼EA þ ΩA M þ iΦAJ; 2 a ∂zM 2 bc

I βI ð2:22Þ ∇αζb ¼ 2iζ ðγbÞαβ; ð2:16bÞ

I IJ its bar-projection is defined as ∇ατ ¼ −4iSε ζαJ; ð2:16cÞ ∂ 1 I I M bc ∇αlβγ ¼ 8iSεα βζ ; ð2:16dÞ ∇Aj¼EA j þ ΩA jM þ iΦAjJ: ð2:23Þ ð γÞ ∂zM 2 bc and We use the freedom to perform general coordinate, local Lorentz and U 1 transformations to choose the following ∇ ζ − ð ÞR a b ¼ lab ¼ lba; ð2:17aÞ gauge condition: β β ∇ ζ −Sζα γ 1 a I ¼ I ð aÞα; ð2:17bÞ ∇aj¼∇a; ∇αj¼∇α; ð2:24Þ ∇ τ 0; 2:17c a ¼ ð Þ where ∇ bc 4S2 δbζc − δcζb al ¼ ð a a Þ: ð2:17dÞ ∂ 1 M bc ∇ ¼ð∇ ; ∇αÞ¼E þ ω M ð2:25Þ A a A ∂zM 2 A bc Some nontrivial implications of the above equations which will be important for our subsequent consideration are 3 2 denotes the set of covariant derivatives for AdS j , which ∇I ζ 0 ∇I 0 obey the following graded commutation relations: ðα βγÞ ¼ ; ðαlβγÞ ¼ ; ð2:18aÞ ∇ ∇ 2 ∇ − 4 S γ JÞ f α; βg¼ i αβ i Mαβ; ð2:26aÞ ∇I ζJ 2SδIJζ ∇ ðIζγ 0 ðα βÞ ¼ αβ; ¼ ; ð2:18bÞ γ 2 ½∇a; ∇β¼SðγaÞβ ∇γ; ½∇a; ∇b¼−4S Mab: ð2:26bÞ Iα i I αβ i I αβ i IJ α ζ ¼ ∇βζ ¼ ∇βl ¼ − ε ∇ τ; ð2:18cÞ 6 12S 4S J 1 In such a coordinate system, the operator ∇αj contains no 1 θ γI J partial derivative with respect to 2. As a consequence, τ ¼ − εIJ∇ ζγ : ð2:18dÞ 1 1 4 ∇α ∇α ∇ ∇ ð 1 k UÞj ¼ α1 αk Uj, for any positive integer k, where U is a tensor superfield on (2,0) AdS superspace. Let Equation (2.17) implies that ζ is a Killing vector field, a us study how the N ¼ 1 descendants of U defined by 2 2 ∇ ζ ∇ ζ 0 Uα …α ≔ ð∇α ∇α UÞj transform under the (2,0) AdS a b þ b a ¼ ; ð2:19Þ 1 k 1 k isometries, with k a non-negative integer. while (2.17b) is a Killing spinor equation. The real We introduce the N ¼ 1 projection of the (2,0) AdS parameter τ is constrained by Killing supervector field (2.15)

∇2 2τ ∇1 2τ 8 Sτ ∇ τ 0 b β β 2 b b ð Þ ¼ð Þ ¼ i ; a ¼ : ð2:20Þ ζj¼ξ ∇b þ ξ ∇β þ ϵ ∇βj; ξ ≔ ζ j; β β β β ξ ≔ ζ1j; ϵ ≔ ζ2j: ð2:27Þ C. Reduction from (2,0) to N = 1 AdS superspace We also introduce the N ¼ 1 projections of the Lorentz Given a tensor superfield U x; θ on (2,0) AdS ð IÞ and Uð1Þ parameters in (2.15): superspace, its N ¼ 1 projection (or bar-projection) is R defined by λbc ≔ lbcj; ϵ ≔ τj: ð2:28Þ

Uj ≔ Uðx; θ Þjθ 0 ð2:21Þ I 2¼ It follows from (2.15) that the N ¼ 1 parameters ξB ¼ ðξb; ξβÞ and λbc obey the equation in a special coordinate system to be specified below. By definition, Uj depends on the real coordinates zM ¼ μ 1 xm; θμ , with θμ ≔ θ , which will be used to parametrize ξ λbc ∇ 0 ξ ξB∇ ξb∇ ξβ∇ ð Þ 1 þ 2 Mbc; A ¼ ; ¼ B ¼ b þ β; N ¼ 1 AdS superspace AdS3j2. For the (2,0) AdS covariant derivative ð2:29Þ

045010-4 FIELD THEORIES WITH (2,0) ADS SUPERSYMMETRY IN … PHYS. REV. D 100, 045010 (2019) which tells us that ξB is a Killing supervector field of N ¼ δ ϵβ ∇2∇2 ∇2 ϵ ∇2 ∇2 ϵUα1…α ¼ ð β α1 αk UÞj þ i ðJ α1 αk UÞj 1 AdS superspace [3]. This equation is equivalent to k Xk β 2 2 1 2 ¼ ϵ Uβα …α − ϵ ∇α ∇α ∇α ∇α ∇ ξ 0 ∇ ξβα −6 ξα 1 k 1 l−1 l lþ1 ðα βγÞ ¼ ; β ¼ i ; ð2:30aÞ l¼1 2 ∇α UÞj þ iqϵUα …α ; ð2:35Þ 1 k 1 k ∇ ξ λ Sξ α β ¼ αβ þ αβ; ð2:30bÞ 1 2 where q is the Uð ÞR charge of U defined by JU ¼ qU.In ∇1 the second term on the right, we have to push αl to the far ∇ λ 0 ∇ λβα −12 Sξα 2 ðα βγÞ ¼ ; β ¼ i : ð2:30cÞ left through the (l − 1) factors of ∇ ’s by making use of the 1 2 relation f∇α; ∇βg¼4εαβSJ and taking into account the These relations automatically follow from the (2,0) AdS relation Killing equations, Eqs. (2.16a)–(2.16d), upon N ¼ 1 ξa ξα λab 1 2 2 2 2 projection. Thus ð ; ; Þ parametrize the infinitesimal ∇α ∇α ∇α ∇α ∇α U ∇α Uα …α α …α : ð l 1 l−1 lþ1 k Þj ¼ l 1 l−1 lþ1 k isometries of AdS3j2 [3] (see also [14]). The remaining parameters ϵα and ϵ generate the second ð2:36Þ 1 supersymmetry and Uð ÞR transformations, respectively. 1 Using the Killing equations (2.18), it can be shown that As the next step, the Uð ÞR generator J should be pushed they satisfy the following properties: to the right until it hits U producing on the way insertions of ∇1. Then the procedure should be repeated. As a result, δ the variation ϵUα1…α is expressed in terms of the super- i 1 α k ϵα ¼ ∇αϵ; ϵ ¼ − ∇ ϵα; ð2:31aÞ fields Uα …α ;Uα …α ; Uα ;U. 4S 2 1 kþ1 1 k 1 So far we have been completely general and discussed infinitely many descendants Uα …α of U. However only a ∇2 8S ϵ 0 ∇ ϵ 0 1 k ði þ Þ ¼ ; a ¼ : ð2:31bÞ few of them are functionally independent. Indeed, Eq. (2.12a) tells us that These imply that the only independent components of ϵ are 2 2 ϵjθ 0 and ∇αϵjθ 0. They correspond to the Uð1Þ and ¼ ¼ R f∇α; ∇βg¼2i∇αβ − 4iSMαβ; ð2:37Þ second supersymmetry transformations, respectively. Given a matter tensor superfield U, its (2,0) AdS trans- 2 and thus every Uα1…αk for k> can be expressed in terms formation law ≤ 2 of U, Uα and Uα1α2 . Therefore, it suffices to consider k . Let us give two examples of matter superfields on (2,0) 1 bc δζU ¼ ζ þ l M þ iτJ U ð2:32Þ AdS superspace. We first consider a covariantly chiral 2 bc ϕ D¯ ϕ 0 1 scalar superfield ; α ¼ , with an arbitrary Uð ÞR charge q defined by Jϕ ¼ qϕ. It transforms under the turns into (2,0) AdS isometries as

δ ϕ ζ τ ϕ δζUj¼δξUjþδϵUj; ð2:33aÞ ζ ¼ð þ iq Þ : ð2:38Þ

When expressed in the real basis (2.11), the chirality 1 δ ξb∇ ξβ∇ λbc constraint on ϕ means ξUj¼ b þ β þ 2 Mbc Uj; ð2:33bÞ 2 1 ∇αϕ ¼ i∇αϕ: ð2:39Þ β 2 δϵUj¼ðϵ ð∇βUÞj þ iϵJUjÞ: ð2:33cÞ As a result, there is only one independent N ¼ 1 superfield upon reduction, It follows from (2.15) and (2.33) that every N ¼ 1 2 2 ≔ ∇α ∇α descendant Uα1…αk ð 1 k UÞj is a tensor superfield φ ≔ ϕj: ð2:40Þ on AdS3j2, We then get the following relations: 1 b β bc δξUα …α ¼ ξ ∇ þ ξ ∇β þ λ M Uα …α : ð2:34Þ 2 1 k b 2 bc 1 k ∇αϕj¼i∇αφ; ð2:41aÞ

∇2 2ϕ −∇2φ − 8 Sφ For the ϵ-transformation we get ð Þ j¼ iq : ð2:41bÞ

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−1 M The ϵ-transformation (2.35) is given by with E ¼ BerðEA Þ; or (ii) by integrating a covariantly chiral scalar L over the chiral subspace of the (2,0) AdS β c δϵφ ¼ iϵ ∇βφ þ iqϵφ: ð2:42Þ superspace, Z Z 1 Our second example is a real linear superfield 3 2 3 2 ¯ α ¯ ¯ 2 θEL − D L D L 0 L ¼ L; D L ¼ 0. The real linearity constraint relates the Sc ¼ d xd c ¼ 4 d xe cjθ¼0; c ¼ ; N ¼ 1 descendants of L as follows: ð2:48Þ ð∇2Þ2L ¼ð∇1Þ2L; ð2:43aÞ with E being the chiral density. The superfield Lagrangians L L 1β 2 and c are neutral and charged, respectively with respect ∇ ∇βL ¼ 0: ð2:43bÞ 1 to the group Uð ÞR: Thus, L is equivalent to two independent, real N ¼ 1 L 0 L −2L J ¼ ;Jc ¼ c: ð2:49Þ superfields: 2 The two types of supersymmetric actions are related to each ≔ L ≔ ∇ L X j;Wα i α j: ð2:44Þ other by the rule Z Z Here X is unconstrained, while Wα obeys the constraint 1 (2.43b) 3 2θ 2θ¯ L 3 2θEL L ≔− D¯ 2L d xd d E ¼ d xd c; c 4 : ∇α 0 Wα ¼ ; ð2:45Þ ð2:50Þ which means that Wα is the field strength of an N ¼ 1 Instead of reducing the above actions to components, in vector multiplet. Since L is neutral under the R-symmetry this paper we need their reduction to N ¼ 1 AdS supergroup Uð1Þ , JL ¼ 0, the second SUSY and Uð1Þ trans- R R space. We remind the reader that the supersymmetric action formation laws of the N ¼ 1 descendants of L are as 3j2 L follows: in AdS makes use of a real scalar Lagrangian . The superspace and component forms of the action are β 2 β Z Z δϵX ¼ δϵLj¼ϵ ð∇βLÞj ¼ −iϵ Wβ; ð2:46aÞ 1 3j2 3 ∇2 8S S ¼ d zEL ¼ 4 d xeði þ ÞLjθ¼0: ð2:51Þ 2 β 2 2 2 δϵWα ¼ ið∇αδϵLÞj ¼ iϵ ð∇β∇αLÞj − ϵ½J;∇αLj For the action (2.47) we get −ϵβ∇ − i ϵ ∇2 − ϵ∇ ¼ αβX 2 α X i αX: ð2:46bÞ Z Z i 3 2θ 2θ¯ L − 3j2 ∇2 2L D. The (2,0) AdS supersymmetric actions in AdS3j2 S ¼ d xd d E ¼ 4 d zEð Þ j; ð2:52Þ Every rigid supersymmetric field theory in (2,0) AdS −1 M superspace may be reduced to N ¼ 1 AdS superspace. with E ¼ BerðEA Þ. The chiral action (2.48) reduces to Here we provide the key technical details of the reduction. AdS3j2 as follows: In accordance with [5–7,18], there are two ways of Z Z constructing supersymmetric actions in (2,0) AdS super- 3 2 3 2 S ¼ d xd θEL ¼ 2i d j zEL j: ð2:53Þ space: (i) either by integrating a real scalar L over the full c c c (2,0) AdS superspace,6 Z Z Making use of the (2,0) AdS transformation law 1 δL ζL δL ζ − 2 τ L 3 2 2 ¯ 3 2 ¯ 2 ¼ , c ¼ð i Þ c, and the Killing equa- S ¼ d xd θd θEL ¼ d xeD D Ljθ 0 16 ¼ tion (2.15b), it can be checked explicitly that the N 1 Z ¼ 1 action defined by the right-hand side of (2.52),or(2.53) are 3 D¯ 2D2L ¼ 16 d xe jθ¼0 invariant under the (2,0) AdS isometry transformations. Z 1 3 DαD¯ 2D SD¯ αD L ¼ d xe 16 α þ i α jθ¼0 E. Supersymmetric nonlinear sigma models Z 1 To illustrate the ð2; 0Þ → ð1; 0Þ AdS superspace reduc- 3 D¯ D2D¯ α SDαD¯ L ¼ d xe 16 α þ i α jθ¼0; ð2:47Þ tion described above, here we discuss two interesting examples. Our first example is a general nonlinear σ-model with 6The component inverse vierbein is defined as usual, (2,0) AdS supersymmetry [6,7]. It is described by the m m −1 m ea ðxÞ¼Ea jθ¼0, with e ¼ detðea Þ. action

045010-6 FIELD THEORIES WITH (2,0) ADS SUPERSYMMETRY IN … PHYS. REV. D 100, 045010 (2019) Z Z ¯ generated by a real scalar parameter ϵ subject to the 3 2θ 2θ¯ ϕi ϕ¯ j 3 2θE ϕi S ¼ d xd d EKð ; Þþ d xd Wð Þþc:c ; constraints (2.31), which are

D¯ ϕi 0 i α i i α ¼ ; ð2:54Þ δϵφ ¼ iϵ ∇αφ þ ϵJ ðφÞ: ð2:64Þ ¯ σ where Kðϕi; ϕ¯ jÞ is the Kähler potential of a Kähler The family of supersymmetric -models (2.54) includes ϕi 1 a special subclass which is specified by the two conditions: manifold and Wð Þ is a superpotential. The Uð ÞR gen- ¯ ϕ’ ϕi 0 erator is realized on the dynamical superfields ϕi and ϕ¯ i as (i) all s are neutral, J ¼ ; and (ii) no superpotential is present, WðϕÞ¼0. In this case no restriction on the Kähler ¯ ¯ ¯ potential is imposed by Eq. (2.56), and the action (2.54) is iJ ¼ JiðϕÞ∂ þ JiðϕÞ∂¯; ð2:55Þ i i invariant under arbitrary Kähler transformations where JiðϕÞ is a holomorphic Killing vector field such that K → K þ Λ þ Λ¯ ; ð2:65Þ i Ji ϕ ∂ − D ϕ ϕ¯ D¯ D ð Þ iK ¼ 2 ð ; Þ; ¼ ; ð2:56Þ with ΛðϕiÞ a holomorphic function. The corresponding action in N ¼ 1 AdS superspace is obtained from (2.62) by D ϕ ϕ¯ for some Killing potential ð ; Þ. The superpotential has setting Dðφ; φ¯ Þ¼0 and WðφÞ¼0, and thus the action is to obey the condition manifestly Kähler invariant. Let us also consider a supersymmetric nonlinear σ-model Ji ϕ ∂ −2 ð Þ iW ¼ iW ð2:57Þ formulated in terms of several Abelian vector multiplets with action [7] in order for the action (2.54) to be invariant under the (2,0) Z AdS isometry transformations S ¼ −2 d3xd2θd2θ¯EFðLiÞ; D¯ 2Li ¼ 0; L¯ i ¼ Li; δϕi ¼ðζ þ iτJÞϕi: ð2:58Þ ð2:66Þ

In the real representation (2.11), the chirality condition where FðxiÞ is a real analytic function of several variables, i on ϕ turns into which is defined modulo linear inhomogeneous shifts

2 1 ∇ ϕi ∇ ϕi i α ¼ i α : ð2:59Þ FðxÞ → FðxÞþbix þ c; ð2:67Þ

N 1 ϕi i It follows that upon ¼ reduction, leads to just one with real parameters bi and c. The real linear scalar L is the superfield, field strength of a vector multiplet. Upon reduction to N ¼ 1 AdS superspace, Li generates two different N ¼ 1 φi ≔ ϕij: ð2:60Þ superfields:

i i i 2 i In particular, we have the following relations: X ≔ L j;Wα ≔ i∇αL j: ð2:68Þ

i 2 i i Here the real scalar X is unconstrained, while the real ∇αϕ j¼i∇αφ ; ð2:61aÞ i spinor Wα obeys the constraint ∇2 2ϕi −∇2φi − 8SJi φ ð Þ j¼ ð Þ: ð2:61bÞ α i ∇ Wα ¼ 0; ð2:69Þ Using the reduction rules (2.52) and (2.53), we obtain i Z which means that Wα is the field strength of an N ¼ 1 N 1 3j2 α i j¯ vector multiplet. Reducing the action (2.66) to ¼ AdS S ¼ d zEf−iK ¯ðφ; φ¯ Þ∇ φ ∇αφ¯ þ SDðφ; φ¯ Þ ij superspace gives þð2iWðφÞþc:c:Þg; ð2:62Þ Z i 3j2 α i j αi j S ¼ − d zEgijðXÞf∇ X ∇αX þ W Wαg; ð2:70Þ where we have made use of the standard notation 2

∂pþqKðφ; φ¯ Þ where we have introduced the target-space metric ¯ ¯ ≔ Ki1i j1j ¯ ¯ : ð2:63Þ p q ∂φi1 ∂φip ∂φ¯ j1 ∂φ¯ jq 2 ∂ FðXÞ g ðXÞ¼ : ð2:71Þ ij ∂Xi∂Xj The action (2.62) is manifestly N ¼ 1 supersymmetric. One may explicitly check that it is also invariant under the The vector multiplets in (2.70) can be dualized into scalar second supersymmetry and R-symmetry transformations ones, which gives

045010-7 JESSICA HUTOMO and SERGEI M. KUZENKO PHYS. REV. D 100, 045010 (2019) Z λ i 3j2 α i j ij α where the gauge parameter αð2s−1Þ is unconstrained S ¼ − d zEfg ðXÞ∇ X ∇αX þg ðXÞ∇ Y ∇αY g; dual 2 ij i j complex. Equation (3.2a) implies that the complex gauge parameter gα 2 is a covariantly longitudinal linear ð2:72Þ ð sÞ superfield, with gijðXÞ being the inverse metric. Riemannian metrics of ¯ ¯ gα 2 ≔ D α λα …α ; D α gα …α 0: : the type (2.71) appeared in the literature 20 years ago in the ð sÞ ð 1 2 2sÞ ð 1 2 2sþ1Þ ¼ ð3 3Þ context of N ¼ 4 supersymmetric quantum mechanics [19] and N ¼ 4 superconformal mechanics [20]. H The gauge transformation of αð2sÞ, Eq. (3.2a), corresponds to the superconformal gauge prepotential [21,22]. The III. MASSLESS HIGHER-SPIN MODELS: L prepotential αð2s−2Þ is a compensating multiplet. In addi- TYPE II SERIES L tion to (3.2b), the compensator αð2s−2Þ also possesses its There exist two off-shell formulations for a massless own gauge freedom of the form 1 multiplet of half-integer superspin ðs þ 2Þ in (2,0) AdS superspace [1], with s ¼ 2; 3; …, which are called the ¯ ¯ δξLα 2 −2 ¼ ξα 2 −2 þ ξα 2 −2 ; Dβξα 2 −2 ¼ 0; ð3:4Þ type II and type III series7 by analogy with the terminology ð s Þ ð s Þ ð s Þ ð s Þ used in [7] for the linearized off-shell formulations for ξ N ¼ 2 supergravity (s ¼ 1). In this section we describe the with the gauge parameter αð2s−2Þ being covariantly chiral. L ð2;0Þ → ð1;0Þ AdS superspace reduction of the type II Associated with αð2s−2Þ is the real field strength theory. The reduction of the type III theory will be given L DβD¯ L L L¯ in Sec. IV. αð2s−2Þ ¼ i β αð2s−2Þ; αð2s−2Þ ¼ αð2s−2Þ; ð3:5Þ

A. The type II theory which is a covariantly linear superfield, We fix an integer s>1. In accordance with [1], the 1 D2L 0 ⇔ D¯ 2L 0 massless type II multiplet of superspin ðs þ 2Þ is described αð2s−2Þ ¼ αð2s−2Þ ¼ : ð3:6Þ in terms of two unconstrained real tensor superfields VðIIÞ H L It is inert under the gauge transformation (3.4), 1 ¼f αð2sÞ; αð2s−2Þg; ð3:1Þ δ L 0 λ ðsþ2Þ ξ αð2s−2Þ ¼ . From (3.2b) we can read off the -gauge H H L L transformation of the field strength: where αð2sÞ ¼ ðα1…α2sÞ and αð2s−2Þ ¼ ðα1…α2s−2Þ are symmetric in their spinor indices. 1 The dynamical superfields are defined modulo gauge δ L DβD¯ 2λ − D¯ βD2λ¯ λ αð2s−2Þ ¼ ð βαð2s−2Þ βαð2s−2ÞÞ; transformations of the form 4 − s DβD¯ γ ¯ δ H D¯ λ − D λ¯ ≡ ¯ ¼ 2 1 ðgβγαð2s−2Þ þ gβγαð2s−2ÞÞ λ αð2sÞ ¼ ðα1 α2…α2sÞ ðα1 α2…α2sÞ gαð2sÞ þ gαð2sÞ; s þ 2is βγ ð3:2aÞ − D g¯βγα 2 −2 : ð3:7Þ 2s þ 1 ð s Þ i ¯ β β ¯ δλLα 2 −2 ¼ − ðD λβα 2 −2 þ D λβα 2 −2 Þ; ð3:2bÞ ð s Þ 2 ð s Þ ð s Þ The type II theory is described by the action

Z 1 s 1 s ðIIÞ H L − 3 2θ 2θ¯ Hαð2sÞDβD¯ 2D H − D D¯ Hβγαð2s−2Þ Dδ D¯ ρ H S 1 ½ αð2sÞ; αð2s−2Þ¼ d xd d E β αð2sÞ ð½ β; γ Þ½ ; δραð2s−2Þ ðsþ2Þ 2 8 8 s D Hβγαð2s−2Þ DδρH 2 SHαð2sÞDβD¯ H þ 2 ð βγ Þ δραð2s−2Þ þ is β αð2sÞ 2s − 1 − Lαð2s−2Þ Dβ D¯ γ H 2Lαð2s−2ÞL 2 ð ½ ; βγαð2s−2Þ þ αð2s−2ÞÞ − 1 2 − 1 ðs Þð s Þ βαð2s−3Þ ¯ 2 γ αð2s−2Þ − ðDβL D D Lγα 2 −3 þ c:c:Þ − 4ð2s − 1ÞSL Lα 2 −2 : ð3:8Þ 4s ð s Þ ð s Þ

7Type I series will be referred to as the longitudinal formulation for the gauge massless half-integer superspin multiplets in (1,1) AdS superspace [21] and Minkowski superspace [22]. The type I series and its dual are naturally related to the off-shell formulations for massless higher-spin N ¼ 1 supermultiplets in four dimensions [23–25]. The type II and type III series have no four-dimensional counterpart.

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It is invariant under the gauge transformations (3.2) representation (2.11), we observe that the chirality con- and (3.4). dition (3.4) reads βαð2s−3Þ ¯ 2 γ The structure DβL D D Lγα 2s−3 in (3.8) is not ð Þ ∇2ξ ∇1ξ defined for s ¼ 1. However it comes with the factor (s − 1) β αð2s−2Þ ¼ i β αð2s−2Þ: ð3:15Þ and therefore drops out from (3.8) for s ¼ 1. The action (3.8) for s ¼ 1 coincides with the linearized action for (2,0) AdS The gauge transformation (3.4) allows us to impose a gauge supergravity, which was originally derived in Sec. 10. 1 condition of [7]. L 0 αð2s−2Þj¼ : ð3:16Þ

B. Reduction of the gauge prepotentials to AdS3j2 Thus, upon reduction to N ¼ 1 superspace, we have the following real superfields Let us turn to reducing the gauge prepotentials (3.1) to 8 N ¼ 1 AdS superspace. Our first task is to work out such Ψ ≔ ∇2L H β;αð2s−2Þ i β αð2s−2Þj; ð3:17aÞ a reduction for the superconformal gauge multiplet αð2sÞ. In the real representation (2.11), the longitudinal linear ≔ i ∇2 2L constraint (3.3) takes the form Lαð2s−2Þ 4 ð Þ αð2s−2Þj: ð3:17bÞ

2 1 ∇ gα …α i∇ gα …α : 3:9 Here Ψβ;α 2 −2 is a reducible superfield which belongs to ðα1 2 2sþ1Þ ¼ ðα1 2 2sþ1Þ ð Þ ð s Þ the representation 2 ⊗ ð2s − 1Þ of SLð2; RÞ, Ψβ;α …α ¼ θ 1 2s−2 It follows that gαð2sÞ has two independent 2-components, Ψ β;ðα1…α2s−2Þ. The condition (3.16) is preserved by the which are residual gauge freedom generated by a real unconstrained N 1 η ∇2β ¼ superfield αð2s−2Þ defined by gαð2sÞj; gαð2s−1Þβj: ð3:10Þ i The gauge transformation of Hα 2 , Eq. (3.2), allows us to ξ − η η¯ η ð sÞ αð2s−2Þj¼ 2 αð2s−2Þ; αð2s−2Þ ¼ αð2s−2Þ: ð3:18Þ choose two gauge conditions η H 0 ∇2βH 0 We may now determine how the -transformation acts on αð2sÞj¼ ; αð2s−1Þβj¼ : ð3:11Þ the superfields (3.17a) and (3.17b). We obtain In this gauge we stay with the following unconstrained real δηΨβ;α 2 −2 ¼ i∇βηα 2 −2 ; ð3:19aÞ N ¼ 1 superfields: ð s Þ ð s Þ 2 δηLα 2 −2 ¼ 0; ð3:19bÞ Hα 2 1 ≔ i∇ Hα …α ; 3:12a ð s Þ ð sþ Þ ðα1 2 2sþ1Þj ð Þ

i 2 2 where we have used the chirality constraint (3.15) and the Hα 2 ≔ ∇ Hα 2 : : ð sÞ 4 ð Þ ð sÞj ð3 12bÞ expression (3.18) for the residual gauge transformation. Next, we analyze the λ-gauge transformation and reduce There exists a residual gauge freedom which preserves the 3j2 the N ¼ 2 field strength Lα 2 −2 to AdS . In the real basis gauge conditions (3.11). It is described by unconstrained ð s Þ for the covariant derivatives, the real linearity constraint real N ¼ 1 superfields ζα 2 and ζα 2 −1 defined by ð sÞ ð s Þ (3.6) is equivalent to two constraints: i ¯ gα 2s j¼− ζα 2s ; ζα 2s ¼ ζα 2s ; ð3:13aÞ ∇2 2L ∇1 2L ð Þ 2 ð Þ ð Þ ð Þ ð Þ αð2s−2Þ ¼ð Þ αð2s−2Þ; ð3:20aÞ 2 1 2β s þ ¯ 1β 2 ∇ gα 2 −1 βj¼ ζα 2 −1 ; ζα 2 −1 ¼ ζα 2 −1 : ∇ ∇ Lα 2 −2 ¼ 0: ð3:20bÞ ð s Þ 2s ð s Þ ð s Þ ð s Þ β ð s Þ ð3:13bÞ These constraints imply that the resulting N ¼ 1 compo- L The gauge transformation laws of the superfields (3.12) are nents of αð2s−2Þ are given by given by L ∇2L αð2s−2Þj; i β αð2s−2Þj; ð3:21Þ δ ∇ ζ Hαð2sþ1Þ ¼ i ðα1 α2…α2 1Þ; ð3:14aÞ sþ of which the former is unconstrained and the latter is a δHα 2 ∇ α ζα …α : 3:14b ð sÞ ¼ ð 1 2 2sÞ ð Þ constrained N ¼ 1 superfield that proves to be a gauge- L invariant field strength, as we shall see below. The relation Our next step is to reduce the compensator αð2s−2Þ between Lα 2 −2 and the prepotential Lα 2 −2 is given by to N ¼ 1 AdS superspace. Making use of the ð s Þ ð s Þ (3.5), which can be expressed as

8In the super-Poincar´e case, the N ¼ 2 → N ¼ 1 reduction of L − i ∇1 2 ∇2 2 L H αð2s−2Þ ¼ fð Þ þð Þ g αð2s−2Þ: ð3:22Þ αð2sÞ has been carried out in [26]. 2

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We now compute the bar-projection of (3.22) in the gauge Applying the N ¼ 1 reduction rule (2.52) to the type II (3.16) and make use of the definition (3.17b) to obtain action (3.8), we find that it becomes a sum of two actions,

ðIIÞ Lα 2 −2 −2Lα 2 −2 : 3:23 H L k ð s Þj¼ ð s Þ ð Þ S 1 ½ αð2sÞ; αð2s−2Þ¼S 1 ½Hαð2sþ1Þ;Lαð2s−2Þ ðsþ2Þ ðsþ2Þ Making use of (3.22) and (3.17), the bar-projection of ⊥ Ψ þ SðsÞ½Hαð2sÞ; β;αð2s−2Þ: ð3:28Þ ∇2L N 1 i β αð2s−2Þ leads to the ¼ field strength Explicit expressions for these N ¼ 1 actions will be given W ≔ ∇2L − ∇γ∇ − 4 Sδγ Ψ in the next subsection. β;αð2s−2Þ i β αð2s−2Þj¼ ið β i βÞ γ;αð2s−2Þ: ð3:24Þ C. Massless higher-spin N =1 AdS W W W¯ supermultiplets in 3 Here β;αð2s−2Þ is a real superfield, β;αð2s−2Þ ¼ β;αð2s−2Þ, and is a descendant of the real unconstrained prepotential The gauge transformations (3.14a), (3.14b), (3.27a) and (3.27c) tell us that in fact we are dealing with two different Ψβ;α 2 −2 defined modulo gauge transformation (3.19). The ð s Þ N 1 supersymmetric higher-spin gauge theories. field strength proves to be gauge invariant under (3.19), and ¼ Given a positive integer n>0, we say that a super- it satisfies the condition symmetric gauge theory describes a multiplet of superspin ∇βW 0 n=2 if it is formulated in terms of a superconformal gauge β;αð2s−2Þ ¼ ; ð3:25Þ prepotential HαðnÞ and possibly a compensating multiplet. as a consequence of (3.20b) and the identity (A7b). Let us The gauge freedom of the real tensor superfield Hα is L ðnÞ express the gauge transformation of αð2s−2Þ, Eq. (3.7) in δ n −1 bn=2c∇ ζ terms of the real basis for the covariant derivatives. This ζHαðnÞ ¼ i ð Þ ðα1 α2…αnÞ; ð3:29Þ leads to ζ with the gauge parameter αðn−1Þ being real but otherwise is 1β 2γ unconstrained. δLα 2 −2 ¼ f∇ ∇ ðgβγα 2 −2 þ g¯βγα 2 −2 Þ ð s Þ 2s þ 1 ð s Þ ð s Þ ∇βγ − ¯ 1. Longitudinal formulation for massless þ ðgβγαð2s−2Þ gβγαð2s−2ÞÞg; ð3:26Þ - 1 superspin ðs+ 2Þ multiplet ∇2δL In a similar way, one should also rewrite β αð2s−2Þ in the One of the two N ¼ 1 theories provides an off-shell real basis. This allows us to derive the gauge transforma- formulation for the massless superspin-ðs þ 1Þ multiplet. It W 2 tions for Lαð2s−2Þ and β;αð2s−2Þ is formulated in terms of the real unconstrained gauge superfields δ − s ∇βγζ Lαð2s−2Þ ¼ 2 2 1 βγαð2s−2Þ; ð3:27aÞ ð s þ Þ Vk 1 ¼fHαð2sþ1Þ;Lαð2s−2Þg; ð3:30Þ ðsþ2Þ δW ∇γ∇ − 4 Sδγ ζ β;αð2s−2Þ ¼ ið β i βÞ γαð2s−2Þ: ð3:27bÞ which are defined modulo gauge transformations We can then read off the transformation law for the δHα 2 1 ¼ i∇ α ζα …α ; ð3:31aÞ Ψ ð sþ Þ ð 1 2 2sþ1Þ prepotential β;αð2s−2Þ δ − s ∇βγζ δΨ −ζ ∇ η Lαð2s−2Þ ¼ 2 2 1 βγαð2s−2Þ; ð3:31bÞ β;αð2s−2Þ ¼ βαð2s−2Þ þ i β αð2s−2Þ; ð3:27cÞ ð s þ Þ ζ where we have also taken into account the η-gauge where the parameter αð2sÞ is unconstrained real. The freedom (3.19). gauge-invariant action is

Z 1 s i i k − 3j2 − αð2sþ1ÞQ − ∇ βαð2sÞ∇2∇γ S 1 ½Hαð2sþ1Þ;Lαð2s−2Þ¼ d zE H Hαð2sþ1Þ βH Hγαð2sÞ ðsþ2Þ 2 2 8 is ∇ βγαð2s−1Þ∇ρδ 2 − 1 αð2s−2Þ∇βγ∇δ þ 4 βγH Hρδαð2s−1Þ þð s ÞL Hβγδαð2s−2Þ αð2s−2Þ 2 i βαð2s−3Þ γ þ 2ð2s − 1Þ L ði∇ − 4SÞLα 2 −2 − ðs − 1Þ∇βL ∇ Lγα 2 −3 ð s Þ s ð s Þ 1 S ∇ βαð2sÞ∇γ 2 1 αð2sþ1Þ ∇2 − 4 S þ s βH Hγαð2sÞ þ 2 ð s þ ÞH ð i ÞHαð2sþ1Þ ; ð3:32Þ

045010-10 FIELD THEORIES WITH (2,0) ADS SUPERSYMMETRY IN … PHYS. REV. D 100, 045010 (2019) where Q is the quadratic Casimir operator of the 3D N ¼ 1 is the linearized action for N ¼ 1 AdS supergravity. In the AdS supergroup (A9). The action (3.32) coincides with the flat-superspace limit, the action is equivalent to the one off-shell N ¼ 1 supersymmetric action for massless half- given in [27]. integer superspin in AdS in the form given in [14]. This 3j2 supersymmetric gauge theory in AdS was described in 2. Transverse formulation for massless [14]. Its flat-superspace limit was presented earlier in [26]. superspin-s multiplet In what follows, we will refer to the above theory as the 1 The other N ¼ 1 theory provides a formulation for the longitudinal formulation for the massless superspin-ðs þ 2Þ multiplet. massless superspin-s multiplet. It is described by the ∇ βαð2s−3Þ∇γ unconstrained real superfields The structure βL Lγαð2s−3Þ in (3.32) is not defined for s ¼ 1. However it comes with the factor (s − 1) 1 V⊥ Ψ and drops out from (3.32) for s ¼ . The resulting action ðsÞ ¼fHαð2sÞ; β;αð2s−2Þg; ð3:34Þ

k S 3 ½Hαð3Þ;L which are defined modulo gauge transformations of the ð2Þ Z form 1 i i − 3j2 − αð3ÞQ − ∇ βαð2Þ∇2∇γ ¼ 2 d zE 2 H Hαð3Þ 8 βH Hγαð2Þ δ ∇ ζ Hαð2sÞ ¼ ðα1 α2…α2sÞ; ð3:35aÞ i ∇ βγα∇ρδ ∇βγ∇δ 2 ∇2 − 4S þ 4 βγH Hρδα þ L Hβγδ þ Lði ÞL δΨ −ζ ∇ η 3 β;αð2s−2Þ ¼ βαð2s−2Þ þ i β αð2s−2Þ; ð3:35bÞ S ∇ βαð2Þ∇γ αð3Þ ∇2 − 4 S þ βH Hγαð2Þ þ 2 H ð i ÞHαð3Þ ζ η where the gauge parameters αð2s−1Þ and αð2s−2Þ are ð3:33Þ unconstrained real. The gauge-invariant action is given by

Z 1 s 1 ⊥ 3j2 αð2sÞ 2 S Hα 2 ; Ψβ;α 2 −2 − d zE H i∇ 8sS Hα 2 ðsÞ½ ð sÞ ð s Þ¼ 2 2 ð þ Þ ð sÞ

βαð2s−1Þ γ β;αð2s−2Þ γ − is∇βH ∇ Hγα 2 −1 − ð2s − 1ÞW ∇ Hγβα 2 −2 ð s Þ ð s Þ − 1 i β;αð2s−2Þ s βαð2s−3Þ γ; − ð2s − 1Þ W Wβ;α 2 −2 þ Wβ; W γα 2 −3 2 ð s Þ s ð s Þ − 2 2 − 1 SΨβ;αð2s−2ÞW ið s Þ β;αð2s−2Þ ; ð3:36aÞ

W where β;αð2s−2Þ denotes the field strength W − ∇γ∇ − 4 Sδγ Ψ ∇βW 0 β;αð2s−2Þ ¼ ið β i βÞ γ;αð2s−2Þ; β;αð2s−2Þ ¼ : ð3:36bÞ The action (3.36) defines a new N ¼ 1 supersymmetric higher-spin theory which did not appear in [14,21,26] even in the super-Poincar´e case. W βαð2s−3ÞWγ; 1 − 1 The structure β; γαð2s−3Þ in (3.36) is not defined for s ¼ . However it comes with the factor (s ) and drops out from (3.36) for s ¼ 1. The resulting gauge-invariant action Z 1 1 ⊥ 3j2 αð2Þ 2 βα γ S Hα 2 ; Ψβ − d zE H i∇ 8S Hα 2 − i∇βH ∇ Hγα ð1Þ½ ð Þ ¼ 2 2 ð þ Þ ð Þ − Wβ∇γ − i WβW − 2 SΨβW Hγβ 2 β i β ð3:37Þ provides an off-shell realization for a massless gravitino multiplet in AdS3. In the flat-superspace limit, this model reduces to the one described in [26]. 1 Ψ In the s> case, the gauge freedom of the prepotential β;αð2s−2Þ (3.35) allows us to impose a gauge condition

2Xs−2 Ψ 0 ⇔ Ψ ε φ ðα1;α2…α2s−1Þ ¼ β;αð2s−2Þ ¼ βαk α1…αˆk…α2s−2 ; ð3:38Þ k¼1

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φ for some field αð2s−3Þ. Since we gauge away the symmetric 1 Ψ ζ δ V D¯ βλ − Dβλ¯ part of β;αð2s−2Þ, the two gauge parameters αð2s−1Þ and λ αð2s−2Þ ¼ 2 ð βαð2s−2Þ βαð2s−2ÞÞ; ð4:2bÞ η s αð2s−2Þ are related. The theory is now realized in terms of the following dynamical variables: λ where the gauge parameter αð2s−1Þ is unconstrained complex, and the longitudinal linear parameter gα 2 is defined φ ð sÞ fHαð2sÞ; αð2s−3Þg; ð3:39Þ H as in (3.3). As in the type II case, αð2sÞ is the superconformal gauge multiplet, while Vα 2 −2 is a compensat- with the gauge freedom ð s Þ ing multiplet. The only difference from the type II case

δHα 2 ¼ −∇ α α ηα …α ; ð3:40aÞ occurs in the gauge transformation law for the compensa- ð sÞ ð 1 2 3 2sÞ V tor αð2s−2Þ. δφ ∇βη The compensator Vα 2 −2 also possesses its own gauge αð2s−3Þ ¼ i βαð2s−3Þ: ð3:40bÞ ð s Þ freedom of the form It follows that in the flat-superspace limit, S ¼ 0, and in the δ V ξ ξ¯ D¯ ξ 0 gauge (3.38), the action (3.36) coincides with Eq. (B.25) of ξ αð2s−2Þ ¼ αð2s−2Þ þ αð2s−2Þ; β αð2s−2Þ ¼ ; ð4:3Þ [21]. The component structure of this model will be discussed in Appendix B1. ξ with the gauge parameter αð2s−2Þ being covariantly chiral, but otherwise arbitrary. IV. MASSLESS HIGHER-SPIN MODELS: Associated with Vα 2 −2 is the real field strength TYPE III SERIES ð s Þ N 1 V DβD¯ V V V¯ In this section we carry out the ¼ AdS superspace αð2s−2Þ ¼ i β αð2s−2Þ; αð2s−2Þ ¼ αð2s−2Þ; ð4:4Þ reduction of the type III theory [1] following the procedure employed in Sec. III. δ V 0 which is inert under (4.3), ξ αð2s−2Þ ¼ . It is not difficult to see that V α 2 −2 is covariantly linear, A. The type III theory ð s Þ 1 We fix a positive integer s> . In accordance with [1], D2V 0 ⇔ D¯ 2V 0 1 αð2s−2Þ ¼ αð2s−2Þ ¼ : ð4:5Þ the massless type III multiplet of superspin ðs þ 2Þ is described in terms of two unconstrained real tensor super- λ fields It varies under the -gauge transformation as

VðIIIÞ H V i β ¯ 2 ¯ β 2 ¯ 1 ¼f αð2sÞ; αð2s−2Þg; ð4:1Þ δλV α 2 −2 ¼ ðD D λβα 2 −2 þ D D λβα 2 −2 Þ: ðsþ2Þ ð s Þ 4s ð s Þ ð s Þ H i β ¯ γ which are symmetric in their spinor indices, αð2sÞ ¼ ¼ − D D ðgβγα 2 −2 − g¯βγα 2 −2 Þ 2s þ 1 ð s Þ ð s Þ H α …α and Vα 2 −2 ¼ V α …α . ð 1 2sÞ ð s Þ ð 1 2s−2Þ 2 The dynamical superfields are defined modulo gauge − Dβγ ¯ gβγαð2s−2Þ: ð4:6Þ transformations of the form 2s þ 1

δ H D¯ λ − D λ¯ ¯ Modulo normalization, there exists a unique action being λ αð2sÞ ¼ ðα1 α2…α2 Þ ðα1 α2…α2 Þ ¼ gαð2sÞ þ gαð2sÞ; s s invariant under the gauge transformations (4.2) and (4.3).It ð4:2aÞ is given by

Z 1 s 1 ðIIIÞ H V − 3 2θ 2θ¯ Hαð2sÞDβD¯ 2D H S 1 ½ αð2sÞ; αð2s−2Þ¼ d xd d E β αð2sÞ ðsþ2Þ 2 8 1 − D D¯ Hβγαð2s−2Þ Dδ D¯ ρ H 16 ð½ β; γ Þ½ ; δραð2s−2Þ 1 D Hβγαð2s−2Þ DδρH SHαð2sÞDβD¯ H þ 4 ð βγ Þ δραð2s−2Þ þ i β αð2sÞ 2s − 1 1 − V αð2s−2ÞDβγH V αð2s−2ÞV 2 βγαð2s−2Þ þ 2 αð2s−2Þ 1 − 1 2 − 1 D Vβαð2s−3ÞD¯ 2DγV 2 2 − 1 SVαð2s−2ÞV þ 8 ðs Þð s Þð β γαð2s−3Þ þ c:c:Þþ sð s Þ αð2s−2Þ : ð4:7Þ

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βαð2s−3Þ ¯ 2 γ 2 Although the structure DβV D D Vγα 2 −3 in (4.7) V ∇ V ð s Þ αð2s−2Þj; i β αð2s−2Þj: ð4:13Þ is not defined for s ¼ 1, it comes with the factor (s − 1) and drops out from (4.7) for the s ¼ 1 case. In this case the The relation between the field strength V α 2s−2 and the action coincides with the type III supergravity action in ð Þ prepotential Vα 2 −2 is given by (4.4), which can be (2,0) AdS superspace, which was originally derived in ð s Þ expressed as Sec. 10. 2 of [7]. V − i ∇1 2 ∇2 2 V B. Reduction of the gauge prepotentials to AdS3j2 αð2s−2Þ ¼ 2 fð Þ þð Þ g αð2s−2Þ: ð4:14Þ The reduction of the superconformal gauge multiplet We now compute the bar-projection of (4.14) in the gauge H 3j2 αð2sÞ to AdS has been carried out in the previous (4.8) and make use of the definition (4.9b) to obtain H section. We saw that in the gauge (3.11), αð2sÞ is described V −2 by the two unconstrained real superfields Hαð2sþ1Þ and αð2s−2Þj¼ Vαð2s−2Þ: ð4:15Þ Hαð2sÞ defined according to (3.12), with their gauge trans- 2 formation laws given by Eqs. (3.14a) and (3.14b), respec- The bar-projection of i∇ V α 2 −2 leads to the N ¼ 1 field- V β ð s Þ tively. Now it remains to reduce the prepotential αð2s−2Þ to strength N ¼ 1 AdS superspace, following the same approach as outlined in the type II series. The gauge transformation 2 Ωβ;α 2 −2 ≔ i∇ V α 2 −2 j (4.3) allows us to choose a gauge condition ð s Þ β ð s Þ − ∇γ∇ − 4 Sδ γ ϒ ¼ ið β i β Þ γ;αð2s−2Þ; ð4:16Þ V 0 αð2s−2Þj¼ : ð4:8Þ Ω Ω¯ which is a real superfield, β;αð2s−2Þ ¼ β;αð2s−2Þ, and is a The compensator Vα 2 −2 is then equivalent to the follow- ϒ ð s Þ descendant of the real unconstrained prepotential β;αð2s−2Þ ing real N ¼ 1 superfields, which we define as follows: defined modulo gauge transformation (4.11). One may check that the field strength is invariant under (4.11) and ϒ ≔ ∇2V β;αð2s−2Þ i β αð2s−2Þj; ð4:9aÞ obeys the condition

i ∇βΩ 0 ≔ ∇2 2V β;αð2s−2Þ ¼ : ð4:17Þ Vαð2s−2Þ 4 ð Þ αð2s−2Þj: ð4:9bÞ V Let us express the gauge transformation of αð2s−2Þ, The residual gauge freedom, which preserves the gauge Eq. (4.6) in terms of the real basis for the covariant condition (4.8) is described by a real unconstrained N ¼ 1 η derivatives. This leads to superfield αð2s−2Þ defined by 1 δV − ∇1β∇2γ − ¯ αð2s−2Þ ¼ f ðgβγαð2s−2Þ gβγαð2s−2ÞÞ ξ − i η η¯ η 2s þ 1 αð2s−2Þj¼ αð2s−2Þ; αð2s−2Þ ¼ αð2s−2Þ: ð4:10Þ 2 ∇βγ ¯ þ ðgβγαð2s−2Þ þ gβγαð2s−2ÞÞg; ð4:18Þ As a result, we may determine how (4.9a) and (4.9b) vary ∇2δV under η-transformation One should also express its corollary β αð2s−2Þ in the real basis for the covariant derivatives. We determine the gauge δ ϒ ∇ η Ω η β;αð2s−2Þ ¼ i β αð2s−2Þ; ð4:11aÞ transformations law for Vαð2s−2Þ and β;αð2s−2Þ to be 1 δηVα 2 −2 ¼ 0: ð4:11bÞ β ð s Þ δVα 2 −2 ¼ ∇ ζβα 2 −2 ; ð4:19aÞ ð s Þ 2s ð s Þ Next, we analyze the λ-gauge transformation and reduce 3j2 1 γ δ δ γ the field strength V α 2 −2 to AdS . In the real basis for the δΩ ∇ ∇ ∇ − 4 S∇ δ ζ ð s Þ β;αð2s−2Þ ¼ 2 1 ð β i β Þ δγαð2s−2Þ: covariant derivatives, the real linearity constraint (4.5) turns s þ into ð4:19bÞ

∇2 2V ∇1 2V ð Þ αð2s−2Þ ¼ð Þ αð2s−2Þ; ð4:12aÞ From (4.19b) we read off the transformation law for the ϒ prepotential β;αð2s−2Þ: ∇1β∇2V 0 β αð2s−2Þ ¼ : ð4:12bÞ δϒ i ∇γζ 2 1 ∇ η β;αð2s−2Þ ¼ ð γβαð2s−2Þ þð s þ Þ β αð2s−2ÞÞ; V N 1 2s þ 1 This tells us that αð2s−2Þ is equivalent to two real ¼ superfields ð4:20Þ

045010-13 JESSICA HUTOMO and SERGEI M. KUZENKO PHYS. REV. D 100, 045010 (2019) where we have also taken into account the η-gauge free- 1. Longitudinal formulation for massless dom (4.11). superspin-s multiplet N 1 Performing ¼ reduction to the original type III One of the two N ¼ 1 theories provides an off-shell N 1 action (4.7), we arrive at two decoupled ¼ actions realisation for massless superspin-s multiplet described in terms of the real unconstrained superfields

ðIIIÞ H V ⊥ ϒ k S 1 ½ αð2sÞ; αð2s−2Þ¼S 1 ½Hαð2sþ1Þ; β;αð2s−2Þ V ¼fHα 2 ;Vα 2 −2 g; ð4:22Þ ðsþ2Þ ðsþ2Þ ðsÞ ð sÞ ð s Þ k þ SðsÞ½Hαð2sÞ;Vαð2s−2Þ: ð4:21Þ which are defined modulo gauge transformations of the form

δ ∇ ζ We will present the exact form of these actions in the next Hαð2sÞ ¼ ðα1 α2…α2sÞ; ð4:23aÞ subsection. 1 β δVα 2 −2 ¼ ∇ ζβα 2 −2 ; ð4:23bÞ ð s Þ 2s ð s Þ C. Massless higher-spin N = 1 supermultiplets in AdS3 ζ Upon reduction to N ¼ 1 superspace, the type III theory where the gauge parameter αð2s−1Þ is unconstrained real. leads to two N ¼ 1 supersymmetric gauge theories. The gauge-invariant action is given by

Z 1 s 1 k 3j2 αð2sÞ 2 S ½Hα 2 ;Vα 2 −2 ¼ − d zE H ði∇ þ 4SÞHα 2 ðsÞ ð sÞ ð s Þ 2 2 ð sÞ i − ∇ βαð2s−1Þ∇γ − 2 − 1 αð2s−2Þ∇βγ 2 βH Hγαð2s−1Þ ð s ÞV Hβγαð2s−2Þ 1 2 − 1 αð2s−2Þ ∇2 8 S − 1 ∇ βαð2s−3Þ∇γ þð s Þ 2 V ði þ s ÞVαð2s−2Þ þðs Þ βV Vγαð2s−3Þ : ð4:24Þ

Modulo an overall normalization factor, (4.24) coincides with the off-shell N ¼ 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]. ∇ βαð2s−3Þ∇γ 1 − 1 Although the structure βV Vγαð2s−3Þ in (4.24) is not defined for s ¼ , it comes with the factor (s ) and thus drops out from (4.24) for s ¼ 1. The resulting gauge-invariant action

Z 1 1 k 3j2 αð2Þ 2 i βα γ S ½Hα 2 ;V¼− d zE H ði∇ þ 4SÞHα 2 − ∇βH ∇ Hγα ð1Þ ð Þ 2 2 ð Þ 2 1 − ∇βγ ∇2 8S V Hβγ þ 2 Vði þ ÞV ð4:25Þ

describes an off-shell massless gravitino multiplet in AdS3. which are defined modulo gauge transformations of the In the flat-superspace limit, it reduces to the gravitino form multiplet model described in [28] (see also [26]).

δHα 2 1 ¼ i∇ α ζα …α ; ð4:27aÞ 2. Transverse formulation for massless ð sþ Þ ð 1 2 2sþ1Þ - 1 superspin ðs+ 2Þ multiplet The other theory provides an off-shell formulation for δϒ i ∇γζ 2 1 ∇ η 1 β;αð2s−2Þ ¼ 2 1 ð γβαð2s−2Þ þð s þ Þ β αð2s−2ÞÞ: massless superspin-ðs þ 2Þ multiplet. It is described by the s þ unconstrained superfields ð4:27bÞ V⊥ ϒ 1 ¼fHαð2sþ1Þ; β;αð2s−2Þg; ð4:26Þ ðsþ2Þ The gauge-invariant action is

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Z 1 s i i ⊥ ϒ − 3j2 − αð2sþ1ÞQ − ∇ βαð2sÞ∇2∇γ S 1 ½Hαð2sþ1Þ; β;αð2s−2Þ¼ d zE H Hαð2sþ1Þ βH Hγαð2sÞ ðsþ2Þ 2 2 8 i i ∇ βγαð2s−1Þ∇ρδ − 2 − 1 Ωβ;αð2s−2Þ∇γδ þ 8 βγH Hρδαð2s−1Þ 4 ð s Þ Hγδβαð2s−2Þ i − 2 − 1 Ωβ;αð2s−2ÞΩ − 2 − 1 Ω βαð2s−3ÞΩγ; 8 ð s Þð β;αð2s−2Þ ðs Þ β; γαð2s−3ÞÞ 1 S αð2sþ1Þ ∇2 − 4 S ∇ βαð2sÞ∇γ þ H ð i ÞHαð2sþ1Þ þ 2 βH Hγαð2sÞ 2 − 1 Sϒβ;αð2s−2ÞΩ þ isð s Þ β;αð2s−2Þ ; ð4:28aÞ

Ω where β;αð2s−2Þ denotes the real field strength Ω − ∇γ∇ − 4 Sδ γ ϒ ∇βΩ 0 β;αð2s−2Þ ¼ ið β i β Þ γ;αð2s−2Þ; β;αð2s−2Þ ¼ : ð4:28bÞ

This action defines a new N ¼ 1 supersymmetric higher-spin theory which did not appear in [14,21,26]. Ω βαð2s−3ÞΩγ; 1 − 1 The structure β; γαð2s−3Þ in (4.28a) is not defined for s ¼ . However it comes with the factor (s ) and hence drops out from (4.28a) for s ¼ 1. The resulting gauge-invariant action

Z 1 i i ⊥ ϒ − 3j2 − αð3ÞQ − ∇ βαð2Þ∇2∇γ S 3 ½Hαð3Þ; β¼ d zE H Hαð3Þ βH Hγαð2Þ ð2Þ 2 2 8

i ∇ βγα∇ρδ − i Ωβ∇γδ þ 8 βγH Hρδα 4 Hγδβ 1 S αð3Þ ∇2 − 4 S ∇ βαð2Þ∇γ þ H ð i ÞHαð3Þ þ 2 βH Hγαð2Þ − i ΩβΩ SϒβΩ 8 β þ i β ð4:29Þ

Z provides an off-shell formulation for a linearized super- 3j2 L Σ nþ1Wβ;αðnÞΣ Sfirst-order ¼ d zEf ð β;αðnÞÞþi β;αðnÞg; gravity multiplet in AdS3. In the flat-superspace limit, it reduces to the linearised supergravity model proposed ð5:2Þ in [26]. Σ where β;αðnÞ is unconstrained and the Lagrange V. ANALYSIS OF THE RESULTS multiplier is Let s>0 be a positive integer. For each superspin value, 1 W nþ1 ∇γ∇ − 4 Sδγ Ψ ∇βW 0 integer (s) or half-integer ðs þ 2Þ, we have constructed two β;αðnÞ ¼ i ð β i βÞ γ;αðnÞ; β;αðnÞ ¼ ; off-shell formulations which have been called longitudinal ð5:3Þ and transverse. Now we have to explain this terminology. Consider a field theory in AdS3j2 that is described in for some unconstrained prepotential Ψγ;α . Varying (5.2) terms of a real tensor superfield VαðnÞ. We assume the ðnÞ Ψ action to have the form with respect to γ;αðnÞ gives Z β nþ1 k 3j2 L nþ1∇ ∇ ∇γΣβ;α − 4iSΣγ;α ¼ 0 ⇒ Σβ;α ¼ i ∇βVα ; S ½VαðnÞ¼ d zE ði βVαðnÞÞ: ð5:1Þ ðnÞ ðnÞ ðnÞ ðnÞ ð5:4Þ ∇ It is natural to call βVαðnÞ a longitudinal superfield, by analogy with a longitudinal vector field. This theory and then Sfirst-order reduces to the original action (5.1).On possesses a dual formulation that is obtained by introducing the other hand, we may start from Sfirst-order and integrate Σ a first-order action β;αðnÞ out. This will lead to a dual action of the form

045010-15 JESSICA HUTOMO and SERGEI M. KUZENKO PHYS. REV. D 100, 045010 (2019) Z 9 ⊥ Ψ 3j2 L W with a gauge one-form. However, such a duality trans- S ½ γ;αðnÞ¼ d zE dualð β;αðnÞÞ: ð5:5Þ formation cannot be lifted to AdS3.

This is a gauge theory since the action is invariant under VI. NONCONFORMAL HIGHER-SPIN gauge transformations SUPERCURRENTS

δΨ nþ1∇ η In the previous sections, we have shown that there exist γ;αðnÞ ¼ i γ αðnÞ: ð5:6Þ two different off-shell formulations for the massless higher- spin N ¼ 1 supermultiplets. Massless half-integer super- The gauge-invariant field strength Wβ;α can be called a ðnÞ spin theory can be realised in terms of the dynamical transverse superfield, due to the constraint (5.3) it obeys. variables (3.30) and (4.26), while the models (3.34) and It is natural to call the dual formulations (5.1) and (5.5) (4.22) define massless multiplet of integer superspin s, with as longitudinal and transverse, respectively. s>1. These models lead to different N ¼ 1 higher-spin Now, let us consider the transverse and longitudinal supercurrent multiplets. Our aim in this section is to formulations for the massless superspin-s models, which describe the general structure of N ¼ 1 supercurrent are given by Eqs. (3.26) and (4.24), respectively. These multiplets in AdS. actions depend parametrically on S, the curvature of AdS ⊥ Ψ superspace. We denote by SðsÞ½Hαð2sÞ; β;αð2s−2ÞFS and A. N = 1 supercurrents: Half-integer superspin case k S 0 S ½Hαð2sÞ;Vαð2s−2Þ these actions in the limit ¼ , ðsÞ FS Our half-integer supermultiplet in the longitudinal for- which corresponds to a flat superspace. The dynamical ⊥ k mulation (3.30) can be coupled to external sources systems S Hα 2 ; Ψβ;α 2 −2 and S Hα 2 ;Vα 2 −2 ðsÞ½ ð sÞ ð s ÞFS ðsÞ½ ð sÞ ð s ÞFS Z 1 prove to be related to each other by the Legendre trans- ðsþ2Þ 3j2 αð2sþ1Þ 4 αð2s−2Þ ⊥ Ψ Ssource ¼ d zEfiH Jαð2sþ1Þ þ L Sαð2s−2Þg: formation described above. Thus SðsÞ½Hαð2sÞ; β;αð2s−2ÞFS k ð6:1Þ and SðsÞ½Hαð2sÞ;Vαð2s−2ÞFS are dual formulations of the same theory. This duality does not survive if S is nonvanishing. The condition that the above action is invariant under The same feature characterizes the longitudinal and the gauge transformations (3.31) gives the conservation transverse formulations for the massless superspin- equation 1 ðs þ 2Þ multiplet, which are described by the actions β 2s (3.32) and (4.28), respectively. The flat-superspace coun- ∇ Jβα 2 ¼ − ∇ α α Sα α : ð6:2Þ ð sÞ 2 1 ð 1 2 3 2sÞ terparts of these higher-spin models, which we denote by ð s þ Þ k ⊥ ϒ S 1 ½Hαð2sþ1Þ;Lαð2s−2Þ and S 1 ½Hαð2sþ1Þ; β;αð2s−2Þ , ðsþ2Þ FS ðsþ2Þ FS For the transverse theory (4.26) described by the ϒ are dual to each other. However, this duality does not survive prepotentials fHαð2sþ1Þ; β;αð2s−2Þg, we construct an action if we turn on a nonvanishing AdS curvature. functional of the form The above discussion can be illustrated by considering Z 1 the model for linearized gravity in AdS3. It is described by ðsþ2Þ 3j2 αð2sþ1Þ S ¼ d zEfiH Jα 2 1 the action source ð sþ Þ β;αð2s−2Þ Z þ 2isϒ Uβ;α 2 −2 g: ð6:3Þ 1 ð s Þ 3 hαð4Þ□h −∇ hβð2Þαð2Þ∇γð2Þh Sgravity ¼ d xe αð4Þ βð2Þ αð2Þγð2Þ 8 Requiring that the action is invariant under the gauge 1 1 transformations (4.27) leads to ∇αð2Þy∇βð2Þh − ∇αð2Þy∇ y þ2 αð2Þβð2Þ 4 αð2Þ β 2s β 2 α 4 2 2 ∇ Jβα 2 ∇ α Uα α ; ∇ Uβ;α 2 −2 0: 8S h ð Þh 6S y ð sÞ ¼ ð 1 2 2sÞ ð s Þ ¼ þ αð4Þ þ ; ð5:7Þ 2s þ 1 ð6:4Þ which is invariant under gauge transformations From the above consideration, it follows that the most 2 general conservation equation in the half-integer superspin αð2Þ δξhα 4 ¼ ∇ α α ζα α ; δξy ¼ ∇ ζα 2 : ð5:8Þ ð Þ ð 1 2 3 4Þ 3 ð Þ case takes the form

S 0 9 h In the flat-space limit, ¼ , the model possesses a dual There is another dual realization in which αð4Þ turns into a h formulation in which the scalar compensator y is replaced gauge one-form b;αð4Þ with an additional gauge freedom.

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β 2s models for a chiral scalar superfield. Unlike in 4D N ¼ 1 ∇ Jβα 2 ¼ ð∇ α Uα α − ∇ α α Sα α Þ; ð sÞ 2s þ 1 ð 1 2 2sÞ ð 1 2 3 2sÞ supergravity where every supersymmetric matter theory ð6:5aÞ can be coupled to only one of the off-shell supergravity formulations (either old-minimal or new-minimal), here in β the (2,0) AdS case our trace multiplets require both type II ∇ Uβ;α 2 −2 ¼ 0: ð6:5bÞ ð s Þ and type III compensators to couple to. The general conservation equation (6.11) naturally gives N 1 B. N = 1 supercurrents: Integer superspin case rise to the ¼ higher-spin supercurrent multiplets discussed in the previous subsection. One may show that In complete analogy with the half-integer superspin case, in the real basis, (6.11) turns into we couple the prepotentials (4.22) in terms of which the integer superspin-s is described, to external sources 1β 1 2 ∇ Jβα 2 −1 ∇ Y α α − ∇ Zα α ; 6:12a Z ð s Þ ¼ ðα1 2 2s−1Þ ðα1 2 2s−1Þ ð Þ ðsÞ 3j2 αð2sÞ 2 αð2s−2Þ Ssource ¼ d zEfH Jαð2sÞ þ sV Rαð2s−2Þg: ∇2βJ ∇1 Z ∇2 Y βαð2s−1Þ ¼ α α2α2 −1Þ þ α α2α2 −1Þ; ð6:12bÞ ð6:6Þ ð 1 s ð 1 s For such an action to be invariant under the gauge freedom The real linearity constraints on the trace supermultiplets (4.23), the sources must be conserved are equivalent to

∇β ∇ 2 2 1 2 1β 2 Jβα 2s−1 ¼ α1 Rα2 α2 −1 : ð6:7Þ ∇ Y ∇ Y ∇ ∇ Y 0 ð Þ ð s Þ ð Þ αð2s−2Þ ¼ð Þ αð2s−2Þ; β αð2s−2Þ ¼ ; Next, we turn to the transverse formulation (3.34) ð6:13aÞ Ψ characterized by the prepotentials fHαð2sÞ; β;αð2s−2Þg and construct an action functional 2 2 1 2 1β 2 ð∇ Þ Zα 2 −2 ¼ð∇ Þ Zα 2 −2 ; ∇ ∇ Zα 2 −2 ¼ 0: Z ð s Þ ð s Þ β ð s Þ ðsÞ 3j2 αð2sÞ Ψβ;αð2s−2Þ ð6:13bÞ Ssource ¼ d zEfH Jαð2sÞ þ i Tβ;αð2s−2Þg: J ð6:8Þ It follows from (6.12) and (6.13) that αð2sÞ contains two independent real N ¼ 1 supermultiplets: Demanding that the action be invariant under the gauge transformations (3.35), we derive the following conditions: ≔ J Jαð2sÞ αð2sÞj; ð6:14aÞ β β ∇ Jβα 2 −1 ¼ iTα 2 −1 ; ∇ Tβ;α 2 −2 ¼ 0: ð6:9Þ ð s Þ ð s Þ ð s Þ 2 Jα 2 1 ≔ i∇ Jα α ; 6:14b ð sþ Þ ðα1 2 2sþ1Þj ð Þ From the above consideration, the most general conservation equation for the multiplet of currents in the integer N 1 Y while the independent real ¼ components of αð2s−2Þ superspin case is given by Z and αð2s−2Þ are defined by β ∇ Jβα 2 −1 ¼ ∇ α Rα α þ iTα 2 −1 ; ð6:10aÞ ð s Þ ð 1 2 2s−1Þ ð s Þ ≔ Y ≔ ∇2Y Rαð2s−2Þ αð2s−2Þj;Uβ;αð2s−2Þ i β αð2s−2Þj; ∇β 0 Tβ;αð2s−2Þ ¼ : ð6:10bÞ ð6:15aÞ

N N ≔ Z ≔ ∇2Z C. From = 2 supercurrents to = 1 supercurrents Sαð2s−2Þ αð2s−2Þj;Tβ;αð2s−2Þ i β αð2s−2Þj: In our recent paper [1], we constructed the general ð6:15bÞ conservation equation for the N ¼ 2 higher-spin supercurrent multiplets in (2,0) AdS superspace, which takes the Making use of (6.13), one may readily show that form ∇β 0 D¯ βJ D¯ Y Z Uβ;αð2s−2Þ ¼ ; ð6:16aÞ βαð2s−1Þ ¼ ðα1 ð α2…α2s−1Þ þ i α2…α2s−1ÞÞ: ð6:11Þ J ∇β 0 Here αð2sÞ denotes the higher-spin supercurrent, while the Tβ;αð2s−2Þ ¼ : ð6:16bÞ Y Z trace supermultiplets αð2s−2Þ and αð2s−2Þ are covariantly linear. The explicit form of this multiplet of currents was On the other hand, Eq. (6.12) implies that the N ¼ 1 presented by considering simple N ¼ 2 supersymmetric superfields obey the following conditions:

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2 ζα ∈ R2 ∇β s ∇ − ∇ auxiliary real variables and associate with any Jβαð2sÞ ¼ ð ðα1 Uα2α2sÞ ðα1α2 Sα3α2sÞÞ; 2s þ 1 tensor superfield UαðmÞ the following index-free field: 6:17a ð Þ α α ζ ≔ ζ 1 …ζ m UðmÞð Þ Uα1…αm ; ð7:6Þ ∇β ∇ Jβαð2s−1Þ ¼ ðα1 Rα2α2 −1Þ þ iTαð2s−1Þ: ð6:17bÞ s which is a homogeneous polynomial of degree m in ζα. α Indeed, the right-hand side of Eq. (6.17a) coincides Furthermore, we make use of the bosonic variables ζ and α with (6.5a). Therefore, Eqs. (6.16a) and (6.17a) define the corresponding partial derivatives ∂=∂ζ to convert the the N ¼ 1 higher-spin current multiplets associated with spinor and vector covariant derivatives into index-free the massless half-integer superspin formulations (3.30) and operators. In the case of (2,0) AdS superspace, we (4.26). In a similar way, it can be observed that Eqs. (6.16b) introduce operators which increase the degree of homo- α and (6.17b) correspond to the N ¼ 1 higher-spin super- geneity in ζ : currents for the two integer superspin models (3.34) D ≔ ζαD D¯ ≔ ζαD¯ D ≔ ζαζβD and (4.22). ð1Þ α; ð1Þ α; ð2Þ i αβ: ð7:7Þ VII. EXAMPLES OF N = 1 HIGHER-SPIN SUPERCURRENTS We also introduce two operators that decrease the degree of α In this section we give an explicit realization of the homogeneity in ζ : N ¼ 1 higher-spin supercurrent introduced earlier. ∂ ∂ Consider a massless chiral scalar multiplet in (2,0) AdS α ¯ ¯ α D −1 ≔ D α ; D −1 ≔ D α : ð7:8Þ superspace with action [1] ð Þ ∂ζ ð Þ ∂ζ Z 3 2 2 ¯ ¯ ¯ The operators associated with the real spinor covariant S ¼ d xd θd θEΦΦ; DαΦ ¼ 0: ð7:1Þ I derivatives, ∇α, may be defined in a similar way:

∇I ≔ ζα∇I ∇ ≔ ζαζβ∇ The chiral superfield is charged under the R-symmetry 1 α; ð2Þ i αβ; ð7:9Þ 1 ð Þ group Uð ÞR, ∂ Φ − Φ ∇I ≔ ∇Iα : 7:10 J ¼ r ;r¼ const: ð7:2Þ ð−1Þ ∂ζα ð Þ

This action is a special case of the supersymmetric non- Analogous operators are introduced in the case of N ¼ 1 linear sigma model studied in Sec. II D with a vanishing AdS superspace. They are superpotential, WðΦÞ¼0. Making use of (2.62), the reduction of the action (7.1) to N ¼ 1 AdS superspace α α β ∇ 1 ≔ ζ ∇α; ∇ 2 ≔ iζ ζ ∇αβ; ð7:11Þ is given by ð Þ ð Þ Z α ∂ 3j2 α ∇ −1 ≔ ∇ α : ð7:12Þ S ¼ d zEf−i∇ φ¯ ∇αφ þ 4rSφφ¯ g; ð7:3Þ ð Þ ∂ζ

φ ≔ Φ It was shown in [1] that by using the massless equations where we have denoted j. This action is manifestly D2Φ 0 N 2 N ¼ 1 supersymmetric. It also possesses hidden second of motion, ¼ , the ¼ higher-spin supercurrent 1 multiplet associated with the theory (7.1) is described by supersymmetry and Uð ÞR invariance. These transformations are the conservation equation

α α D −1 J 2 D 1 T 2 −2 : 7:13a δϵφ ¼ iϵ ∇αφ − iϵrφ; δϵφ¯ ¼ iϵ ∇αφ¯ þ iϵrφ¯ ; ð7:4Þ ð Þ ð sÞ ¼ ð Þ ð s Þ ð Þ

α J J¯ where ϵ is given in terms or ϵ according to (2.31), and the Here the real supercurrent ð2sÞ ¼ ð2sÞ is given by real parameter ϵ is constrained by (2.31b). It can be seen φ φ¯ Xs 1 2s that and obey the equations of motion k k ¯ ¯ s−k−1 J 2 ¼ ð−1Þ D D 1 ΦD D 1 Φ ð sÞ 2 2 1 ð2Þ ð Þ ð2Þ ð Þ k 0 k þ ði∇2 þ 4rSÞφ ¼ 0; ði∇2 þ 4rSÞφ¯ ¼ 0: ð7:5Þ ¼ 2s þ Dk Φ¯ Ds−kΦ ; ð7:13bÞ When studying higher-spin supercurrents in the (2,0) 2k ð2Þ ð2Þ and (1,0) AdS superspaces, it is advantageous to make use T of a condensed notation employed in [1]. We introduce while the trace multiplet ð2s−2Þ has the form

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T 2 S 1 − 2 2 1 1 T 1 2 ð2s−2Þ ¼ i ð rÞð s þ Þðs þ Þ As is seen from (7.13c), ð2s−2Þ vanishes for r ¼ = ,in −1 which case Φ is an N ¼ 2 superconformal multiplet. Xs 1 2s k T × ð−1Þ The complex trace multiplet ð2s−2Þ may be split into its 2s − 2k 1 2 1 k¼0 þ k þ real and imaginary parts: Dk Φ¯ Ds−k−1Φ × ð2Þ ð2Þ : ð7:13cÞ T Y − Z T ð2s−2Þ ¼ ð2s−2Þ i ð2s−2Þ; ð7:14aÞ One may check that ð2s−2Þ is covariantly linear,

D¯ 2T 0 D2T 0 ð2s−2Þ ¼ ; ð2s−2Þ ¼ : ð7:13dÞ with

Xs−1 2 − 1 2s k s þ k k ¯ s−k−1 Y 2 −2 ¼ 2iSð1 − 2rÞð2s þ 1Þðs þ 1Þ ð−1Þ D ΦD Φ; ð7:14bÞ ð s Þ 2k 3 2s − 2k 1 2 1 ð2Þ ð2Þ k¼0 ð þ Þð þ Þ k þ Xs−1 1 2s k k ¯ s−k−1 Z 2 −2 ¼ −2Sð1 − 2rÞð2s þ 1Þðs þ 1Þðs þ 2Þ ð−1Þ D ΦD Φ: ð7:14cÞ ð s Þ 2k 3 2s − 2k 1 2 1 ð2Þ ð2Þ k¼0 ð þ Þð þ Þ k þ J N 1 In accordance with (6.14), the supercurrent ð2sÞ reduces to two different multiplets upon projection to ¼ superspace: Xs 2s 2s kþ1 k s−k−1 k s−k J 2 ≔ J 2 j¼ ð−1Þ ∇ ∇ 1 φ¯ ∇ ∇ 1 φ − ∇ φ¯ ∇ φ ; ð7:15aÞ ð sÞ ð sÞ 2 1 ð2Þ ð Þ ð2Þ ð Þ 2 ð2Þ ð2Þ k¼0 k þ k

2 1 ¯ J 2 1 ≔ i∇ J 2 j¼− pffiffiffi ðD 1 þ D 1 ÞJ 2 j; ð sþ Þ ð1Þ ð sÞ 2 ð Þ ð Þ ð sÞ Xs 1 2s kþ1 k s−k s−1 k s−k ¼ð2s þ 1Þ ð−1Þ ∇ φ¯ ∇ ∇ 1 φ þð−1Þ ∇ φ∇ ∇ 1 φ¯ ; ð7:15bÞ 2s − 2k 1 2 ð2Þ ð2Þ ð Þ ð2Þ ð2Þ ð Þ k¼0 þ k 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):

2s β ∇ −1 J 2 1 ¼ ð∇ 1 U 2 −1 þ i∇ 2 S 2 −2 Þ; ∇ Uβ; 2 −2 ¼ 0; ð7:16aÞ ð Þ ð sþ Þ 2s þ 1 ð Þ ð s Þ ð Þ ð s Þ ð s Þ with

≔ Z Sð2s−2Þ ð2s−2Þj Xs−1 1 2s ¼ −2Sð1 − 2rÞð2s þ 1Þðs þ 1Þðs þ 2Þ ð−1Þk ∇k φ¯ ∇s−k−1φ; ð7:16bÞ 2k 3 2s − 2k 1 2 1 ð2Þ ð2Þ k¼0 ð þ Þð þ Þ k þ

1 ¯ Uβ; 2 −2 ≔−pffiffiffi ðDβ þ DβÞY 2 −2 j; ð s Þ 2 ð s Þ Xs−1 2k − s þ 1 2s ¼ −2iSð1 − 2rÞð2s þ 1Þðs þ 1Þ ð−1Þk ð2k þ 3Þð2s − 2k þ 1Þ 2k þ 1 k¼0 ∇k φ¯ ∇s−k−1∇ φ −1 sþ1∇k φ∇s−k−1∇ φ¯ × ð2Þ ð2Þ β þð Þ ð2Þ ð2Þ β 2 S − − 1 ζ ∇k φ¯ ∇s−k−2∇ φ −1 sþ1∇k φ∇s−k−2∇ φ¯ þ i ðs k Þ βð ð2Þ ð2Þ ð1Þ þð Þ ð2Þ ð2Þ ð1Þ Þ : ð7:16cÞ

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It may also be verified that the N ¼ 1 supercurrent multiplet for integer superspin obeys the conditions (6.10) on-shell:

∇ ∇ ∇β 0 ð−1ÞJð2sÞ ¼ ð1ÞRð2s−2Þ þ iTð2s−1Þ; Tβ;ð2s−2Þ ¼ ð7:17aÞ with

≔ Y Rð2s−2Þ ð2s−2Þj Xs−1 2k − s þ 1 2s ¼ 2iSð1 − 2rÞð2s þ 1Þðs þ 1Þ ð−1Þk ∇k φ¯ ∇s−k−1φ; ð7:17bÞ 2k 3 2s − 2k 1 2 1 ð2Þ ð2Þ k¼0 ð þ Þð þ Þ k þ

1 ¯ Tβ; 2 −2 ≔−pffiffiffi ðDβ þ DβÞY 2 −2 j; ð s Þ 2 ð s Þ Xs−1 1 2s ¼ 2Sð1 − 2rÞð2s þ 1Þðs þ 1Þðs þ 2Þ ð−1Þk ð2k þ 3Þð2s − 2k þ 1Þ 2k þ 1 n k¼0 k s−k−1 s k s−k−1 × ∇ 2 φ¯ ∇ 2 ∇βφ þð−1Þ ∇ 2 φ∇ 2 ∇βφ¯ ð Þ ð Þ ð Þ ð Þ o 2 S − − 1 ζ ∇k φ¯ ∇s−k−2∇ φ −1 s∇k φ∇s−k−2∇ φ¯ þ i ðs k Þ βð ð2Þ ð2Þ ð1Þ þð Þ ð2Þ ð2Þ ð1Þ Þ : ð7:17cÞ

The above technique can also be used to construct provided the explicit examples of these supercurrents in N ¼ 1 higher-spin supercurrents for the Abelian vector simple models for a chiral scalar superfield. multiplets model described by the action (2.66). We will not There are several interesting applications of the results elaborate on such a construction in the present work. obtained in this paper. In particular, the massless higher- In four dimensions, various aspects of the higher-spin spin N ¼ 1 supermultiplets in AdS3, which were derived supercurrent multiplets were studied in [29–33] in the in Secs. III and IV, can be used to construct off-shell N ¼ 1 super-Poincar´e case and in [34] for N ¼ 1 AdS topologically massive supermultiplets in AdS3 by extend- supersymmetry. In particular, the general nonconformal ing the approaches advocated in [14,22,26]. Such a massive higher-spin supercurrent multiplets for N ¼ 1 supersym- supermultiplet is described by a gauge-invariant action metric field theories in Minkowski space were proposed in being the sum of massless and superconformal higher-spin [31,32], and their AdS counterparts were formulated in actions, following the philosophy of topologically massive [34]. Explicit realizations of the higher-spin supercurrents theories [28,35–37]. N 1 n were derived in [34] for various ¼ supersymmetric Given a positive integer n, the conformal superspin- 2 theories in AdS4, including a model of N massive chiral action [14,26,38] is Z scalar superfields with an arbitrary mass matrix. n 2 i ðn= Þ − 3j2 αðnÞ SSCS ½HαðnÞ¼ 2 1 d zEH WαðnÞðHÞ; ð8:1Þ VIII. APPLICATIONS AND OPEN PROBLEMS 2bn= cþ

Let us briefly summarize the main results obtained in where WαðnÞðHÞ denotes the higher-spin super-Cotton this paper. In Sec. II we developed a formalism to reduce tensor. The latter is a unique descendant of HαðnÞ with every field theory with (2,0) AdS supersymmetry to N ¼ 1 the properties AdS superspace. In Secs. III and IV we applied this δ 0 reduction procedure to the off-shell massless higher- WαðnÞð ζHÞ¼ ; ð8:2aÞ spin supermultiplets in AdSð3j2;0Þ constructed in [1].For 1 ∇β 0 each superspin value, integer (s) or half-integer ðs þ 2Þ, the Wβαðn−1Þ ¼ ; ð8:2bÞ reduction produced two off-shell gauge formulations, δ longitudinal and transverse, for massless N ¼ 1 super- where ζHαðnÞ is the gauge transformation (3.29). These multiplets in AdS3. The transverse formulations for mass- properties imply the gauge invariance of (8.1). In a flat N 1 less higher-spin ¼ supermultiplets in AdS3 are new superspace, WαðnÞ has the form [38] gauge theories. In Sec. V, we proved that for each superspin n value the longitudinal and transverse theories are dually i β β S ¼ 0 ⇒ Wα …α ¼ − ∇ 1 ∇α …∇ n ∇α Hβ …β : equivalent only in the flat superspace limit. In Sec. VI we 1 n 2 1 n 1 n formulated, for the first time, the nonconformal higher-spin supercurrent in N ¼ 1 AdS superspace. In Sec. VII we ð8:3Þ

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The construction of WαðnÞ in arbitrary conformally flat The theory defined by (8.6a) was introduced in [14], while backgrounds is described in [39]. its flat-superspace counterpart appeared earlier in [26]. The Given a positive integer s, there are two off-shell gauge- other model, Eq. (8.6b), is a new formulation for a massive 1 invariant formulations for atopologically massive superspin-s superspin-ðs þ 2Þ multiplet in AdS3. multiplet in AdS3. The corresponding actions are In the Minkowski superspace limit, the dynamical systems (8.6a) and (8.6b) are equivalent, since they are k μ ðsÞ SðsÞ½Hαð2sÞ;Vαð2s−2Þj ¼SSCS½Hαð2sÞ related to each other by the superfield Legendre transformation described in Sec. V. It is also an interesting open μ2s−1 k þ SðsÞ½Hαð2sÞ;Vαð2s−2Þ; problem to understand whether the models (8.6) and (8.6b) ð8:4aÞ in AdS3 generate equivalent dynamics. We now present two off-shell formulations for the massive ⊥ ðsÞ N ¼ 1 gravitino supermultiplet in AdS3 and analyze the S ½Hα 2s ; Ψβ;α 2s−2 jμ¼S ½Hα 2s ðsÞ ð Þ ð Þ SCS ð Þ corresponding equations of motion.11 The massive extension μ2s−1 ⊥ Ψ þ SðsÞ½Hαð2sÞ; β;αð2s−2Þ: of the longitudinal theory (4.25) is described by the action Z ð8:4bÞ 1 jj 3j2 i αβ 2 i αβ γ S 1 μ ¼ − d zE H ∇ Hαβ − ∇βH ∇ Hγα The dynamical system (8.4a) was introduced in [14],while ð Þ; 2 2 2 its flat-superspace counterpart appeared earlier in [26].The − ∇αβ i ∇2 μ 2S αβ other theory, Eq. (8.4b), is a new formulation for massive V Hαβ þ 2 V V þð þ ÞH Hαβ superspin-s multiplet in AdS3. 2 In the Minkowski superspace limit, the dynamical − 2ðμ − 2SÞV ; ð8:7Þ systems (8.4a) and (8.4b) are equivalent, since they are related to each other by the superfield Legendre trans- with μ a real mass parameter. The massive gravitino action is formation described in Sec. V. On the mass shell, dynamics thus constructed from the massless one by adding masslike can be recast in terms of the gauge-invariant field strength terms. In the limit μ → 0, the action reduces to (4.25). Wα 2 which obeys the equations [26] ð sÞ The equations of motion for the dynamical superfields Hαβ V β i 2 and are D Wβα α ¼ 0; − D Wα 2 ¼ mσWα 2 ; σ ¼1 1 2s−1 2 ð sÞ ð sÞ γ 2 2∇ αHβ γ − i∇ Hαβ − 2∇αβV − 4μHαβ ¼ 0; ð8:8aÞ ð8:5Þ ð Þ

αβ 2 where the mass m and helicity parameter σ are determined ∇ Hαβ ¼ði∇ þ 8S − 4μÞV: ð8:8bÞ by μ.10 It is an interesting open problem to understand αβ 2 whether the AdS models (8.4a) and (8.4b) lead to equiv- Multiplying (8.8a) by ∇ and noting that ½∇αβ; ∇ ¼0 alent dynamics, modulo a redefinition of the mass param- yields eter μ. There are two off-shell gauge-invariant formulations for − ∇2∇αβ 4□ − 4μ∇αβ 0 1 i Hαβ þ V Hαβ ¼ : ð8:9Þ a topologically massive superspin-ðs þ 2Þ multiplet in AdS3. The corresponding actions are Substituting (8.8b) into (8.9) leads to k μ S 1 ½Hαð2sþ1Þ;Lαð2s−2Þj 0 ðsþ2Þ V ¼ : ð8:10Þ 1 ðsþ2Þ 2s−1 k S Hα 2 1 μ S Hα 2 1 ;Lα 2 −2 ; ¼ SCS ½ ð sþ Þþ ðsÞ½ ð sþ Þ ð s Þ Now that V ¼ 0 on-shell, Eq. (8.8b) turns into 8:6a ð Þ αβ ∇ Hαβ ¼ 0; ð8:11Þ ⊥ ϒ μ S 1 ½Hαð2sþ1Þ; β;αð2s−2Þj ðsþ2Þ while (8.8) can equivalently be written as 1 ðsþ2Þ μ2s−1 ⊥ ϒ ¼ S ½Hαð2sþ1Þþ S 1 ½Hαð2sþ1Þ; β;αð2s−2Þ: SCS ðsþ2Þ γ −i∇ ∇αHβγ − ð2μ þ 4SÞHαβ ¼ 0: ð8:12Þ ð8:6bÞ

11The construction of the models (8.7) and (8.16) is similar to 10Equations (8.5) describe the irreducible massive multiplet of those used to derive the off-shell formulations for massive 1 superhelicity κ ¼ðs þ 4Þσ [40], with the N ¼ 1 superhelicity superspin-1 and superspin-3=2 multiplets in four dimensions operator being defined according to [41]. [42–52].

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Making use of the identity (A7b), it immediately follows There exist alternative off-shell gauge-invariant formu- from (8.12) that lations for massive higher-spin supermultiplets in AdS3 proposed in [14] for N ¼ 1 AdS supersymmetry and in [1] α ∇ Hαβ ¼ 0; ð8:13Þ for (2,0) AdS supersymmetry. In the N ¼ 1 case the corresponding action is Z and then (8.12) is equivalent to n ðn=2Þ i 3j2 αðnÞ S ½Hα ¼− d zEH i massive ðnÞ 2bn=2cþ1μ − ∇2 μ 2S 2 Hαβ ¼ð þ ÞHαβ: ð8:14Þ μ i ∇2 × þ 2 WαðnÞðHÞ; ð8:19Þ Therefore, we have demonstrated that the model (8.7) leads to the following conditions on the mass shell: with μ ≠ 0 a real parameter. This action may be viewed as a deformation of the superconformal model (8.1).Itis V ¼ 0; ð8:15aÞ invariant under the gauge transformation (3.29) as a consequence of the condition (8.2b) and the identity (A7c). α αβ ∇ Hαβ ¼ 0 ⇒ ∇ Hαβ ¼ 0; ð8:15bÞ In the flat superspace limit, the action (8.19) leads to the equation of motion i 2 − ∇ Hαβ ¼ðμ þ 2SÞHαβ: ð8:15cÞ 2 − i 2 μ 2 D WαðnÞ ¼ WαðnÞ: ð8:20Þ Such conditions are required to describe an irreducible Since Wα is transverse, the equation of motion implies on-shell massive gravitino multiplet in 3D N ¼ 1 AdS ðnÞ that Wα describes a massive higher-spin supermultiplet, superspace [40]. ðnÞ compare with (8.5). The (2,0) supersymmetric extension of In the transverse formulation (3.37), the action for a the model (8.19) is presented in [1]. massive gravitino multiplet is given by It should be pointed out that there also exists an on-shell Z 1 construction of gauge-invariant Lagrangian formulations ⊥ 3j2 i αβ 2 αβ γ 2;1 S − d zE H ∇ Hαβ − i∇βH ∇ Hγα for massive higher-spin supermultiplets in R and AdS3, ð1Þ;μ ¼ 2 2 which were developed in [53,54]. It is obtained by − αβ∇ W − i WαW μ 4S αβ combining the massive bosonic and fermionic higher-spin H α β 2 α þð þ ÞH Hαβ actions [55,56], and therefore this construction is intrinsi- cally on-shell. The formulations given in [53–56] are based − μ 2S ΨαW 2μΨαΨ ið þ Þð α þ αÞ : ð8:16Þ on the gauge-invariant approaches to the dynamics of massive higher-spin fields, which were advocated by In the limit μ → 0, the action reduces to (3.37). One may Zinoviev [57] and Metsaev [58]. It is an interesting open check that the equations of motion for this model imply that problem to understand whether there exists an off-shell uplift of these models. N 2 Ψα ¼ 0; ð8:17aÞ All off-shell higher-spin ¼ supermultiplets in AdS3, both with (2,0) and (1,1) AdS supersymmetry [1,21], are α αβ ∇ Hαβ ¼ 0 ⇒ ∇ Hαβ ¼ 0; ð8:17bÞ reducible gauge theories (in the terminology of the Batalin-Vilkovisky quantization [59]), similar to the massless higher-spin supermultiplets in AdS4 [25]. The − i ∇2 μ 4S 2 Hαβ ¼ð þ ÞHαβ: ð8:17cÞ Lagrangian quantization of such theories is nontrivial. In the four-dimensional case, the quantization of the theories The actions (8.7) and (8.16) can be made into gauge- proposed in [25] was carried out in [60]. All off-shell invariant ones using the Stueckelberg construction. higher-spin N ¼ 1 supermultiplets in AdS3, which we In the Minkowski superspace limit, the massive models have constructed in this paper, are irreducible gauge (8.7) and (8.16) lead to the identical equations of motion theories that can be quantized using the Faddeev-Popov described in terms of Hαβ: procedure [61] as in the nonsupersymmetric case, see e.g., [62,63]. This opens the possibility to develop heat kernel 3 2 α i 2 techniques for higher-spin theories in AdS j ,asan D Hαβ ¼ 0; − D Hαβ ¼ μHαβ: ð8:18Þ 2 extension of the four-dimensional results [16,64,65]. In the AdS case, Eqs. (8.15) and (8.17) lead to equivalent ACKNOWLEDGMENTS dynamics modulo a redefinition of μ. It is an interesting open problem to understand whether there exists a duality We are very grateful to Dmitri Sorokin for fruitful transformation relating these models. discussions and to Darren Grasso for comments on the

045010-22 FIELD THEORIES WITH (2,0) ADS SUPERSYMMETRY IN … PHYS. REV. D 100, 045010 (2019) manuscript. We thank the referee for the important sug- β ∇ ∇α∇β ¼ 4iS∇α; ðA7bÞ gestion to improve the presentation in Sec. VII. The work of J. H. is supported by an Australian Government 2 2 β ∇ ∇α ¼ −∇α∇ þ 4iS∇α ¼ 2i∇αβ∇ þ 2iS∇α Research Training Program (RTP) scholarship. The work of β S. M. K. is supported in part by the Australian Research − 4iS∇ Mαβ; ðA7cÞ Council, Project No. DP160103633. 1 2 2 2 αβ 2 αβ − ∇ ∇ ¼ □ − 2iS∇ þ 2S∇ Mαβ − 2S M Mαβ; APPENDIX A: NOTATION, CONVENTIONS 4 AND N = 1 ADS IDENTITIES ðA7dÞ

2 α a 1 αβ We summarize our notation and conventions which where ∇ ¼ ∇ ∇α and □ ¼ ∇ ∇ ¼ − ∇ ∇αβ.An η −1 1 1 a 2 follow [5]. The Minkowski metric is ab ¼ diagð ; ; Þ. important corollary of (A7a) and (A7c) is The spinor indices are raised and lowered by the rule ∇ ∇ ∇2 0 ⇒ ∇ ∇2 0 α αβ β ½ α β; ¼ ½ αβ; ¼ : ðA8Þ ψ ¼ ε ψ β; ψ α ¼ εαβψ : ðA1Þ The left-hand side of (A7d) can be expressed in terms 2 R ε Here the antisymmetric SLð ; Þ invariant tensors αβ ¼ of the quadratic Casimir operator of the 3D N ¼ 1 AdS αβ βα 12 −εβα and ε ¼ −ε are normalized as ε12 ¼ −1; ε ¼ 1. supergroup [14]: β We make use of real Dirac γ-matrices, γ ≔ ððγ Þα Þ a a 1 defined by Q − ∇2∇2 S∇2 Q ∇ 0 ¼ 4 þ i ; ½ ; A¼ : ðA9Þ β βγ ðγ Þ ≔ ε ðγ Þ ¼ð−iσ2; σ3; σ1Þ: ðA2Þ a α a αγ We also note the following commutation relation:

They obey the algebra 1 2 1 2 1 2 2 2β 2 2 ½ð∇ Þ ð∇ Þ − 4iSð∇ Þ ; ∇α¼16S∇αβ∇ − 16S ∇α γ γ η 1 ε γc 2 2β 2 1 a b ¼ ab þ abc ; ðA3Þ − 32S ∇ Mαβ − 32iS ∇αJ: where the Levi-Civita tensor is normalized as ε012 ¼ ðA10Þ −ε012 ¼ 1. Some useful relations involving γ-matrices are Given an arbitrary superfield F and its complex con- a ρσ ρ σ σ ρ jugate F¯ , the following relation holds: ðγ ÞαβðγaÞ ¼ −ðδαδβ þ δαδβÞ; ðA4aÞ

ϵðFÞ ¯ ε γb γc ε γ ε γ ∇αF ¼ −ð−1Þ ∇αF; ðA11Þ abcð Þαβð Þγδ ¼ γðαð aÞβÞδ þ δðαð aÞβÞγ; ðA4bÞ where ϵðFÞ denotes the Grassmann parity of F. tr½γaγbγcγd¼2ηabηcd − 2ηacηdb þ 2ηadηbc: ðA4cÞ

Given a three-vector A , it can equivalently be described as a APPENDIX B: COMPONENT STRUCTURE OF a symmetric rank-2 spinor Aαβ ¼ Aβα, N = 1 HIGHER-SPIN ACTIONS 1 a αβ In this Appendix we will discuss the component struc- Aαβ ≔ ðγ ÞαβA ;A¼ − ðγ Þ Aαβ: ðA5Þ a a 2 a ture of the two new off-shell N ¼ 1 supersymmetric higher-spin theories: the transverse massless superspin-s The relationship between the Lorentz generators with multiplet (3.36), and the transverse massless superspin- two vector indices (Mab ¼ −Mba), one vector index (Ma) 1 ðs þ 2Þ multiplet (4.28a). For simplicity we will carry out and two spinor indices (Mαβ ¼ Mβα) is as follows: Ma ¼ 1 bc a our analysis in flat Minkowski superspace. In accordance ε M and Mαβ γ M . These generators act on a 2 abc ¼ð Þαβ a with (2.51), the component form of an N ¼ 1 super- Ψ vector Vc and a spinor γ by the rules symmetric action is computed by the rule Z Z 2η Ψ ε Ψ MabVc ¼ c½aVb;Mαβ γ ¼ γðα βÞ: ðA6Þ 3j2 i 3 2 ¯ S ¼ d zL ¼ 4 d xD Ljθ¼0;L¼ L: ðB1Þ We collect some useful identities for N ¼ 1 AdS covariant derivatives, which we denote by ∇A ¼ð∇a;∇αÞ. Making use of the (anti)commutation relation (2.12a) and 1. Massless superspin-s action (2.12b), we obtain the following identities: Let us first work out the component structure of 1 2 the massless integer superspin model (3.36).Intheflat- ∇α∇β ¼ εαβ∇ þ i∇αβ − 2iSMαβ; ðA7aÞ 2 superspace limit, the transverse action (3.36) takes the form

045010-23 JESSICA HUTOMO and SERGEI M. KUZENKO PHYS. REV. D 100, 045010 (2019) Z 1 s ⊥ 3j2 i αð2sÞ 2 S Hα 2 ; Ψβ;α 2 −2 − d z H D Hα 2 ðsÞ½ ð sÞ ð s Þ¼ 2 2 ð sÞ

βαð2s−1Þ γ βαð2s−2Þ γ − isDβH D Hγα 2 −1 − ð2s − 1ÞW D Hγβα 2 −2 ð s Þ ð s Þ − 1 i β;αð2s−2Þ s βαð2s−3Þ γ; − ð2s − 1Þ W Wβ;α 2 −2 þ Wβ; W γα 2 −3 : ðB2Þ 2 ð s Þ s ð s Þ

Ψ 0 As described in (3.38), it is possible to choose a gauge condition ðα1;α2α2s−1Þ ¼ , such that the above action turns into Z 1 s ⊥ 3j2 i αð2sÞ 2 S Hα 2 ; Ψβ;α 2 −2 − d z H D Hα 2 ðsÞ½ ð sÞ ð s Þ¼ 2 2 ð sÞ − βαð2s−1Þ γ − 2 − 1 φαð2s−3Þ∂βγ δ isDβH D Hγαð2s−1Þ ðs Þ D Hβγδαð2s−3Þ 2 − 1 − 2 2 − 3 i αð2s−3Þ iðs Þðs Þð s Þ δλαð2s−5Þ βγ − ðs − 1Þφ □φα 2 −3 − ∂δλφ ∂ φβγα 2 −5 s ð s Þ sð2s − 1Þ ð s Þ − 1 2 − 3 iðs Þð s Þ βαð2s−4Þ 2 γ þ Dβφ D D φγα 2 −4 : ðB3Þ 2sð2s − 1Þ ð s Þ

≔− It is invariant under the following gauge transformations: hαð2sÞ Hαð2sÞj; ðB9Þ δ −∂ η Hαð2sÞ ¼ ðα1α2 α3…α2sÞ; ðB4aÞ s hα 2 1 ≔ i D α Hα α j; ð sþ Þ 2s þ 1 ð 1 2 2sþ1Þ δφ βη αð2s−3Þ ¼ iD βαð2s−3Þ; ðB4bÞ ≔ β yαð2s−1Þ iD Hβα1α2s−1 j; ðB10Þ η where the gauge parameter αð2s−2Þ is a real unconstrained superfield. ≔ i 2 Fαð2sÞ D Hαð2sÞj: ðB11Þ The gauge freedom (B4) can be used to impose a Wess- 4 Zumino gauge Applying the reduction rule (B1) to the N ¼ 1 action φ 0 φ 0 (B3), we find that it splits into bosonic and fermionic parts: αð2s−3Þj¼ ;Dðα1 α2α2s−2Þj¼ : ðB5Þ ⊥ Ψ In order to preserve these gauge conditions, the residual SðsÞ½Hαð2sÞ; β;αð2s−2Þ¼Sbos þ Sferm: ðB12Þ gauge freedom has to be constrained by The bosonic action is given by β 2 β D ηβα 2 −3 0;Dηα 2 −2 2i∂ α ηα α β : ð s Þj¼ ð s Þj¼ ð 1 2 2s−2Þ j Z 1 s ðB6Þ − 3 2 1 − αð2sÞ Sbos ¼ 2 d x ð sÞF Fαð2sÞ These imply that there remain two independent, real 1 2 αð2s−1Þβ∂γ − − 1 αð2sÞ□ η þ sF βhαð2s−1Þγ ðs Þh hαð2sÞ components of αð2s−2Þ: 2 2s − 1 2s − 3 − ð Þð Þ αð2s−4Þ□ ξα 2 −2 ≔ ηα 2 −2 j; λα 2 −1 ≔ iD α ηα α j: ðB7Þ y yαð2s−4Þ ð s Þ ð s Þ ð s Þ ð 1 2 2s−1Þ 2sðs − 1Þ 2s − 1 2s − 3 In the gauge (B5), the independent component fields of − ð Þð Þ αð2s−4Þ∂βγ∂δλ φ 4 − 1 y hβγδλαð2s−4Þ αð2s−3Þ can be chosen as ðs Þ − 2 2 − 1 2 − 3 2 − 5 − ðs Þð s Þð s Þð s Þ 2s − 2 β 2 yα 2 −4 ≔− D φβα α j; 16sðs − 1Þ ð s Þ 2s − 1 1 2s−4 i ∂ δλαð2s−6Þ∂βγ ≔ 2φ × δλy yβγαð2s−6Þ : ðB13Þ yαð2s−3Þ 2 D αð2s−3Þj: ðB8Þ

We define the component fields of Hαð2sÞ as Integrating out the auxiliary field Fαð2sÞ leads to

045010-24 FIELD THEORIES WITH (2,0) ADS SUPERSYMMETRY IN … PHYS. REV. D 100, 045010 (2019)

Z 1 s 2 − 1 s 3 αð2sÞ s δλαð2s−2Þ βγ S ¼ − d x h □hα 2 − ∂δλh ∂ hβγα 2 −2 bos 2 2s − 2 ð sÞ 2 ð s Þ 2 − 3 s αð2s−4Þ βγ δλ αð2s−4Þ − sy ∂ ∂ hβγδλα 2 −4 þ 2y □yα 2 −4 2s ð s Þ ð s Þ − 2 2 − 5 ðs Þð s Þ δλαð2s−6Þ βγ þ ∂δλy ∂ yβγα 2 −6 : ðB14Þ 4ðs − 1Þ ð s Þ

This action is invariant under the gauge transformations The fermionic sector of the component action is described by the real dynamical fields hαð2sþ1Þ, yαð2s−1Þ, δξhα 2 ¼ ∂ α α ξα α ; ðB15Þ ð sÞ ð 1 2 3 2sÞ yαð2s−3Þ, defined modulo gauge transformations of the form 2s − 2 βγ δλhα 2 1 ¼ ∂ α α λα α ; ðB17Þ δξyα 2 −4 ¼ ∂ ξβγα α : ðB16Þ ð sþ Þ ð 1 2 3 2sþ1Þ ð s Þ 2s − 1 1 2s−4 1 β The gauge transformations for the fields hα 2 and yα 2 −4 δλyα 2 −1 ¼ ∂ α λα α β; ðB18Þ ð sÞ ð s Þ ð s Þ 2s 1 ð 1 2 2s−1Þ can be easily read off from the gauge transformations of the þ φ superfields Hαð2sÞ and αð2s−3Þ, respectively. Modulo an δ ∂βγλ λyαð2s−3Þ ¼ βγα1α2 −3 : ðB19Þ overall normalization factor, (B14) corresponds to the s ð2sÞ massless Fronsdal spin-s action SF described in [14]. The gauge-invariant action is

Z 1 s i − 3 αð2sÞβ∂ γ 2 2 − 1 αð2s−1Þ∂βγ 4 2 − 1 αð2s−2Þβ∂ γ Sferm ¼ 2 2 d x h β hαð2sÞγ þ ð s Þy hβγαð2s−1Þ þ ð s Þy β yαð2s−2Þγ 2 − 1 2 − 3 αð2s−3Þ βγ ðs Þð s Þ αð2s−4Þβ γ þ ð2s þ 1Þðs − 1Þy ∂ yβγα 2 −3 − y ∂β yα 2 −4 γ : ðB20Þ s ð s Þ sð2s − 1Þ ð s Þ

1 ð2sþ1Þ It may be shown that Sferm coincides with the Fang-Fronsdal spin-ðs þ 2Þ action, SFF [14]. We have thus proved that at the component level and upon elimination of the auxiliary field, the transverse theory (B3) is equivalent to a sum of two massless models: the bosonic Fronsdal spin-s model and the fermionic Fang-Fronsdal 1 spin-ðs þ 2Þ model.

superspin- 1 2. Massless ðs + 2Þ action We will now elaborate on the component structure of the massless half-integer superspin model in the transverse ϒ formulation (4.28). The theory is described in terms of the real unconstrained prepotentials Hαð2sþ1Þ and β;αð2s−2Þ.In Minkowski superspace, the action (4.28) simplifies into Z 1 s i i ⊥ ϒ − 3j2 − αð2sþ1Þ□ − βαð2sÞ 2 γ S 1 ½Hð2sþ1Þ; β;αð2s−2Þ¼ d z H Hαð2sþ1Þ DβH D D Hγαð2sÞ ðsþ2Þ 2 2 8 i i ∂ βγαð2s−1Þ∂ρδ − 2 − 1 Ωβ;αð2s−2Þ∂γδ þ 8 βγH Hρδαð2s−1Þ 4 ð s Þ Hβγδαð2s−2Þ i − 2 − 1 Ωβ;αð2s−2ÞΩ − 2 − 1 Ω βαð2s−3ÞΩγ; 8 ð s Þð β;αð2s−2Þ ðs Þ β; γαð2s−3ÞÞ ; ðB21Þ with the following gauge symmetry

δHα 2 1 D α ζα …α ; ð sþ Þ ¼ i ð 1 2 2sþ1Þ ðB22aÞ

i γ δϒβ;α 2 −2 ¼ ðD ζγβα 2 −2 þð2s þ 1ÞDβηα 2 −2 Þ: ðB22bÞ ð s Þ 2s þ 1 ð s Þ ð s Þ

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Ω ϒ The action (B21) involves the real field strength β;αð2s−2Þ Let us now represent the prepotential β;αð2s−2Þ in terms of its irreducible components, Ω − γ ϒ βΩ 0 β;αð2s−2Þ ¼ iD Dβ γ;αð2s−2Þ;Dβ;αð2s−2Þ ¼ : 2Xs−2 ðB23Þ ϒ ε β;αð2s−2Þ ¼ Yβα1…α2s−2 þ βαk Zα1…αˆk…α2s−2 ; ðB27Þ k¼1 The gauge transformations (B22) allow us to impose a Wess-Zumino gauge on the prepotentials: where we have introduced the two irreducible components of ϒβ;α 2 −2 by the rule 0 β 0 ð s Þ Hαð2sþ1Þj¼ ;DHβα1α2s j¼ ; β ϒβ;α 2 −2 j¼0;Dϒβ;α 2 −2 j¼0: ðB24Þ ≔ ϒ ð s Þ ð s Þ Yβα1α2s−2 ðβ;α1α2s−2Þ;

1 β; The residual gauge symmetry preserving the gauge con- Zα …α ≔ ϒ βα …α : ðB28Þ 1 2s−3 2 − 1 1 2s−3 ditions (B24) is characterized by s 2 The next step is to determine the remaining independent ζ 0 2ζ − is ∂β ζ Dðα1 α2α2 1Þj¼ ;Dαð2sÞj¼ ðα1 α2α2 Þβj; component fields of Hα 2 1 and ϒβ;α 2 −2 in the Wess- sþ s þ 1 s ð sþ Þ ð s Þ Zumino gauge (B24). ðB25aÞ In the bosonic sector, we have the following set of fields: 1 η η − γζ Dβ αð2s−2Þj¼Dðβ αð2s−2ÞÞj¼ D γβαð2s−2Þj; hα 2 2 ≔−D α Hα α j; ðB29aÞ 2s þ 1 ð sþ Þ ð 1 2 2sþ2Þ ðB25bÞ ≔ yαð2sÞ Dðα1 Yα2α2sÞj; ðB29bÞ 2η − i ∂βγζ D αð2s−2Þj¼ βγαð2s−2Þj: ðB25cÞ 2s þ 1 1 ≔− 2 − 1 zαð2s−2Þ ð s ÞDðα1 Zα2α2 −2Þj; ðB29cÞ As a result, there are three independent, real gauge s s parameters at the component level, which we define as ≔−2 − 1 β zαð2s−4Þ ð s ÞD Zβαð2s−4Þj: ðB29dÞ s β ξα 2 ≔ ζα 2 j; λα 2 −1 ≔−i D ζβα 2 −1 j; ð sÞ ð sÞ ð s Þ 2s þ 1 ð s Þ Reduction of the action (B21) to components leads to the ρ ≔ η αð2s−2Þ αð2s−2Þj: ðB26Þ following bosonic action:

Z 1 s 1 3 − 3 − αð2sþ2Þ□ ∂ δλαð2sÞ∂βγ Sbos ¼ 2 d x 4 h hαð2sþ2Þ þ 16 δλh hβγαð2sÞ 1 1 δλαð2sÞ β αð2s−2Þ βγ δλ þ ð2s − 1Þ∂δλh ∂ α yα α β − ð2s − 1Þðs − 1Þz ∂ ∂ hβγδλα 2 −2 4 ð 1 2 2sÞ 4 ð s Þ 1 1 − 2 − 1 αð2sÞ□ − − 2 2 − 1 ∂ δλαð2s−2Þ∂βγ 4 ð s Þy yαð2sÞ 8 ðs Þð s Þ δλy yβγαð2s−2Þ 1 − − 1 2 − 1 αð2sÞ□ − − 1 2 2 − 1 2 − 3 ∂ δλαð2s−4Þ∂βγ ðs Þð s Þz zαð2sÞ 4 ðs Þðs þ Þð s Þð s Þ δλz zβγαð2s−4Þ − 1 2 − 1 ∂ βγαð2s−2Þ∂δ þðs Þð s Þ βγy ðα1 zα2α2s−2Þδ 2 − 3 s s 2 αð2s−4Þ − ð4s − 12s þ 11Þz □zα 2 −4 4 ðs − 1Þð2s − 1Þ ð s Þ 3 s δλαð2s−6Þ βγ þ ðs − 2Þð2s − 3Þð2s − 5Þ∂δλz ∂ zβγα 2 −6 8ðs − 1Þð2s − 1Þ ð s Þ 1 1 2 − 3 αð2s−4Þ∂βγ∂δλ þ 4 ðs þ Þð s Þz yβγδλαð2s−4Þ 1 βγαð2s−4Þ δ þ ðs − 2Þð2s þ 1Þð2s − 3Þ∂βγz ∂ α zα α δ ; ðB30Þ 2 ð 1 2 2s−4Þ

045010-26 FIELD THEORIES WITH (2,0) ADS SUPERSYMMETRY IN … PHYS. REV. D 100, 045010 (2019) which proves to be invariant under gauge transformations and their gauge transformation laws are given by of the form δλhα 2 1 ∂ α α λα α ; ð sþ Þ ¼ ð 1 2 3 2sþ2Þ ðB33aÞ δξhα 2 2 ∂ α α ξα α ; ð sþ Þ ¼ ð 1 2 3 2sþ2Þ ðB31aÞ 1 1 β β δλyα 2 −1 ∂ α λα α β; δ − ∂ ξ − ∂ ρ ð s Þ ¼ ð 1 2 2s−1Þ ðB33bÞ ξ;ρyαð2sÞ ¼ ðα1 α2α2 Þβ ðα1α2 α3α2 Þ; 2s 1 s þ 1 s s þ ðB31bÞ δ ∂βγλ λyαð2s−3Þ ¼ βγαð2s−3Þ: ðB33cÞ 1 βγ 1 β δξ ρzα 2 −2 ¼ ∂ ξβγα 2 −2 þ ∂ α ρα α β; ; ð s Þ 2 2 1 ð s Þ ð 1 2 2s−2Þ The above fermionic fields correspond to the dynamical sð s þ Þ s 1 variables of the Fang-Fronsdal spin-ðs þ 2Þ model. As ðB31cÞ follows from (B33a), (B33b) and (B33c), their gauge free- 1 βγ dom is equivalent to that of the massless spin-ðs þ 2Þ gauge δρzα 2 −4 ∂ ρβγα 2 −4 : B31d ð s Þ ¼ ð s Þ ð Þ field. Indeed, direct calculations of the component action give the standard massless gauge-invariant spin-ðs þ 1Þ Let us consider the fermionic sector. We find that the 2 ð2sþ1Þ independent fermionic fields are action SFF . The component structure of the obtained supermultiplets ≔ i 2 is a three-dimensional counterpart of so-called (reducible) hαð2sþ1Þ D Hαð2sþ1Þj; ðB32aÞ 4 higher-spin triplet systems. In AdSD an action for bosonic higher-spin triplets was constructed in [66] and for fer- ≔ i 2 mionic triplets in [67,68]. Our superfield construction yαð2s−1Þ D Yαð2s−1Þj; ðB32bÞ 8 provides a manifestly off-shell supersymmetric generali- zation of these systems. It might be of interest to extend it i 2 ≔ 2 − 1 to AdS4. yαð2s−3Þ 2 sð s ÞD Zαð2s−3Þj; ðB32cÞ

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