On higher-spin N=2{\mathcal{N}=2} supercurrent multiplets

We elaborate on the structure of higher-spin $\mathcal{N}=2$ supercurrent multiplets in four dimensions. It is shown that associated with every conformal supercurrent $J_{\alpha(m) \dot{\alpha}(n)}$ (with $m,n$ non-negative integers) is a descendant $J^{ij}_{\alpha(m+1) \dot{\alpha}(n+1)}$ with the following properties: (a) it is a linear multiplet with respect to its $\mathsf{SU}(2)$ indices, that is $D_\beta^{(i} J^{ jk)}_{\alpha(m+1) \dot{\alpha}(n+1) }=0$ and $\bar D_{\dot \beta}^{(i} J^{jk)}_{ \alpha(m+1) \dot{\alpha}(n+1) }=0$; and (b) it is conserved, $\partial^{\beta \dot{\beta}} J^{ij}_{\beta \alpha(m) \dot{\beta} \dot{\alpha}(n)}=0$. Realisations of the conformal supercurrents $J_{\alpha(s) \dot{\alpha}(s)}$, with $s=0,1, \dots$, are naturally provided by a massless hypermultiplet and a vector multiplet. It turns out that such supercurrents and their linear descendants $J^{ij}_{\alpha(s+1) \dot{\alpha}(s+1)}$ do not occur in the harmonic-superspace framework recently described in arXiv:2212.14114. Making use of a massive hypermultiplet, we derive non-conformal higher-spin $\mathcal{N}=2$ supercurrent multiplets.Comment: 15 page

Self-duality for N\cal N-extended superconformal gauge multiplets

We develop a general formalism of duality rotations for $\cal N$-extended superconformal gauge multiplets in conformally flat backgrounds as an extension of the approach given in arXiv:2107.02001. Additionally, we construct $\mathsf{U}(1)$ duality-invariant models for the ${\mathcal N}=2$ superconformal gravitino multiplet recently described in arXiv:2305.16029. Each of them is automatically self-dual with respect to a superfield Legendre transformation. A method is proposed to generate such self-dual models, including a family of superconformal theories.Comment: 24 page

N=3\mathcal{N}=3 conformal superspace in four dimensions

We develop a superspace formulation for ${\cal N}=3$ conformal supergravity in four spacetime dimensions as a gauge theory of the superconformal group $\mathsf{SU}(2,2|3)$. Upon imposing certain covariant constraints, the algebra of conformally covariant derivatives $\nabla_A = (\nabla_a,\nabla_\alpha^i,\bar{\nabla}_i^{\dot \alpha})$ is shown to be determined in terms of a single primary chiral spinor superfield, the super-Weyl spinor $W_\alpha$ of dimension $+1/2$ and its conjugate. Associated with $W_\alpha$ is its primary descendant $B^i{}_j$ of dimension $+2$, the super-Bach tensor, which determines the equation of motion for conformal supergravity. As an application of this construction, we present two different but equivalent action principles for ${\cal N}=3$ conformal supergravity. We describe the model for linearised $\mathcal{N}=3$ conformal supergravity in an arbitrary conformally flat background and demonstrate that it possesses $\mathsf{U}(1)$ duality invariance. Additionally, upon degauging certain local symmetries, our superspace geometry is shown to reduce to the $\mathsf{U}(3)$ superspace constructed by Howe more than four decades ago. Further degauging proves to lead to a new superspace formalism, called $\mathsf{SU}(3)$ superspace, which can also be used to describe ${\mathcal N}=3$ conformal supergravity. Our conformal superspace setting opens up the possibility to formulate the dynamics of the off-shell ${\mathcal N}=3$ super Yang-Mills theory coupled to conformal supergravity.Comment: 32 pages; v2: comments, references, and an appendix adde

The N=2\mathcal{N}=2 superconformal gravitino multiplet

We propose a new gauge prepotential $\Upsilon_i$ describing the four-dimensional ${\mathcal N}=2$ superconformal gravitino multiplet. The former naturally arises via a superspace reduction of the ${\mathcal N}=3$ conformal supergravity multiplet. A locally superconformal chiral action for $\Upsilon_i$, which is gauge-invariant in arbitrary conformally-flat backgrounds, is derived. This construction readily yields a new superprojector, which maps an isospinor superfield $\Psi_i$ to a multiplet characterised by the properties of a conformal supercurrent associated with $\Upsilon_i$. Our main results are also specialised to ${\mathcal N}=2$ anti-de Sitter superspace.Comment: 9 pages; V2: typos correcte

Conformal interactions between matter and higher-spin (super)fields

In even spacetime dimensions, the interacting bosonic conformal higher-spin (CHS) theory can be realised as an induced action. The main ingredient in this definition is the model $\mathcal{S}[\varphi,h]$ describing a complex scalar field $\varphi$ coupled to an infinite set of background CHS fields $h$, with $\mathcal{S}[\varphi,h]$ possessing a non-abelian gauge symmetry. Two characteristic features of the perturbative constructions of $\mathcal{S}[\varphi , h]$ given in the literature are: (i) the background spacetime is flat; and (ii) conformal invariance is not manifest. In the present paper we provide a new derivation of this action in four dimensions such that (i) $\mathcal{S}[\varphi , h]$ is defined on an arbitrary conformally-flat background; and (ii) the background conformal symmetry is manifestly realised. Next, our results are extended to the $\mathcal{N}=1$ supersymmetric case. Specifically, we construct, for the first time, a model $\mathcal{S}[\Phi, H]$ for a conformal scalar/chiral multiplet $\Phi$ coupled to an infinite set of background higher-spin superfields $H$. Our action possesses a non-abelian gauge symmetry which naturally generalises the linearised gauge transformations of conformal half-integer superspin multiplets. The other fundamental features of this model are: (i) $\mathcal{S}[\Phi, H]$ is defined on an arbitrary conformally-flat superspace background; and (ii) the background $\mathcal{N}=1$ superconformal symmetry is manifest. Making use of $\mathcal{S}[\Phi, H]$, an interacting superconformal higher-spin theory can be defined as an induced action.Comment: 69 pages; V2: References and new comments added, typos corrected; V3: Published versio

Embedding formalism for N{\mathcal N}-extended AdS superspace in four dimensions

The supertwistor and bi-supertwistor formulations for ${\mathcal N}$-extended anti-de Sitter (AdS) superspace in four dimensions, ${\rm AdS}^{4|4\mathcal N}$, were derived two years ago in arXiv:2108.03907. In the present paper, we introduce a novel realisation of the ${\mathcal N}$-extended AdS supergroup $\mathsf{OSp}(\mathcal{N}|4;\mathbb{R})$ and apply it to develop a coset construction for ${\rm AdS}^{4|4\mathcal N}$ and the corresponding differential geometry. This realisation naturally leads to an atlas on ${\rm AdS}^{4|4\mathcal N}$ (that is a generalisation of the stereographic projection for a sphere) that consists of two charts with chiral transition functions for ${\mathcal N}>0$. A manifestly $\mathsf{OSp}(\mathcal{N}|4;\mathbb{R})$ invariant model for a superparticle in ${\rm AdS}^{4|4\mathcal N}$ is proposed. Additionally, by employing a conformal superspace approach, we describe the most general conformally flat $\mathcal N$-extended supergeometry. This construction is then specialised to the case of ${\rm AdS}^{4|4\mathcal N}$.Comment: 49 page

On higher-spin N N \mathcal{N} = 2 supercurrent multiplets

Abstract We elaborate on the structure of higher-spin N $$\mathcal{N}$$ = 2 supercurrent multiplets in four dimensions. It is shown that associated with every conformal supercurrent J α m α ⋅ n $${J}_{\alpha (m)\overset{\cdot }{\alpha }(n)}$$ (with m, n non-negative integers) is a descendant J α m + 1 α ⋅ n + 1 ij $${J}_{\alpha \left(m+1\right)\overset{\cdot }{\alpha}\left(n+1\right)}^{ij}$$ with the following properties: (a) it is a linear multiplet with respect to its SU(2) indices, that is D β ( i J α m + 1 α ⋅ n + 1 jk ) = 0 $${D}_{\beta}^{\Big(i}{J}_{\alpha \left(m+1\right)\overset{\cdot }{\alpha}\left(n+1\right)}^{jk\Big)}=0$$ and D ¯ β ̇ ( i J α m + 1 α ⋅ n + 1 jk ) = 0 $${\overline{D}}_{\dot{\beta}}^{\Big(i}{J}_{\alpha \left(m+1\right)\overset{\cdot }{\alpha}\left(n+1\right)}^{jk\Big)}=0$$ ; and (b) it is conserved, ∂ β β ⋅ J βα m β ⋅ α ⋅ n ij = 0 $${\partial}^{\beta \overset{\cdot }{\beta }}{J}_{\beta \alpha (m)\overset{\cdot }{\beta}\overset{\cdot }{\alpha }(n)}^{ij}=0$$ . Realisations of the conformal supercurrents J α s α ⋅ s $${J}_{\alpha (s)\overset{\cdot }{\alpha }(s)}$$ , with s = 0, 1, …, are naturally provided by a massless hypermultiplet and a vector multiplet. It turns out that such supercurrents and their linear descendants J α s + 1 α ⋅ s + 1 ij $${J}_{\alpha \left(s+1\right)\overset{\cdot }{\alpha}\left(s+1\right)}^{ij}$$ do not occur in the harmonic-superspace framework recently described by Buchbinder, Ivanov and Zaigraev. Making use of a massive hypermultiplet, we derive non-conformal higher-spin N $$\mathcal{N}$$ = 2 supercurrent multiplets. Additionally, we derive the higher symmetries of the kinetic operators for both a massive and massless hypermultiplet. Building on this analysis, we sketch the construction of higher-derivative gauge transformations for the off-shell arctic multiplet Υ(1), which are expected to be vital in the framework of consistent interactions between Υ(1) and superconformal higher-spin gauge multiplets