Abstract
In three dimensions, it is known that field theories possessing extended $(p,q)$ anti-de Sitter (AdS) supersymmetry with ${\cal N}=p+q \geq 3$ can be realised in (2,0) AdS superspace. Here we present a formalism to reduce every field theory with (2,0) AdS supersymmetry to ${\cal N}=1$ AdS superspace. As nontrivial examples, we consider supersymmetric nonlinear sigma models formulated in terms of ${\cal N}=2$ chiral and linear supermultiplets. The $(2,0) \to (1,0)$ AdS reduction technique is then applied to the off-shell massless higher-spin supermultiplets in (2,0) AdS superspace constructed in [1]. As a result, for each superspin value $\hat s$, integer $(\hat s= s)$ or half-integer $(\hat s= s+1/2)$, with $s=1,2,\dots $, we obtain two off-shell formulations for a massless ${\cal N}=1$ superspin-$\hat s$ multiplet in AdS${}_3$. These models prove to be related to each other by a superfield Legendre transformation in the flat superspace limit, but the duality is not lifted to the AdS case. Two out of the four series of ${\cal N}=1$ supersymmetric higher-spin models thus derived are new. The constructed massless ${\cal N}=1$ supersymmetric higher-spin actions in AdS${}_3$ are used to formulate (i) higher-spin supercurrent multiplets in ${\cal N}=1$ AdS superspace; and (ii) new topologically massive higher-spin off-shell supermultiplets. Examples of ${\cal N}=1$ higher-spin supercurrents are given for models of a complex scalar supermultiplet. We also present two new off-shell formulations for a massive ${\cal N}=1$ gravitino supermultiplet in AdS${}_3$.
Highlights
In three spacetime dimensions, the anti-de Sitter (AdS) group is a product of two simple groups, SOð2; 2Þ ≅ ðSLð2; RÞ × SLð2; RÞÞ=Z2; ð1:1Þ and so are its supersymmetric extensions OSpðpj2; RÞ × OSpðqj2; RÞ.1 This implies that N -extended AdS supergravity exists in several incarnations [2], which are known as the ðp; qÞ AdS supergravity theories, where the integers p ≥ q ≥ 0 are such that N 1⁄4 p þ q
Let us turn to reducing the gauge prepotentials (3.1) to N 1⁄4 1 AdS superspace
We have shown that there exist two different off-shell formulations for the massless higherspin N 1⁄4 1 supermultiplets
Summary
The AdS group is a product of two simple groups, SOð2; 2Þ ≅ ðSLð2; RÞ × SLð2; RÞÞ=Z2; ð1:1Þ and so are its supersymmetric extensions OSpðpj; RÞ × OSpðqj; RÞ.1 This implies that N -extended AdS supergravity exists in several incarnations [2], which are known as the ðp; qÞ AdS supergravity theories, where the integers p ≥ q ≥ 0 are such that N 1⁄4 p þ q. Defining ∇a 1⁄4 Da, the algebra of (2,0) AdS covariant derivatives (2.4) turns into ζ 1⁄4 ζBDB 1⁄4 ζbDb þ ζβDβ þ ζβDβ; ζb 1⁄4 ζb; ð2:5bÞ f∇Iα; ∇Jβg 1⁄4 2iδIJ∇αβ − 4iδIJSMαβ þ 4εαβεIJSJ; ð2:12aÞ and τ and lbc are some real Uð1ÞR and Lorentz superfield parameters, respectively. ∇αÞ ωAbcMbc ð2:25Þ denotes the set of covariant derivatives for AdS3j2, which obey the following graded commutation relations: f∇α; ∇βg 1⁄4 2i∇αβ − 4iSMαβ; ð2:26aÞ In such a coordinate system, the operator ∇1αj contains no partial derivative with respect to θ2. We first consider a covariantly chiral scalar superfield φ; Dαφ 1⁄4 0, with an arbitrary Uð1ÞR charge q defined by Jφ 1⁄4 qφ It transforms under the (2,0) AdS isometries as δζUj 1⁄4 δξUj þ δεUj; δξUj 1⁄4. Making use of the (2,0) AdS transformation law δL 1⁄4 ζL, δLc 1⁄4 ðζ − 2iτÞLc, and the Killing equation (2.15b), it can be checked explicitly that the N 1⁄4 1 action defined by the right-hand side of (2.52), or (2.53) are invariant under the (2,0) AdS isometry transformations
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