Anti-de Sitter flag superspace

Abstract

This work aims to develop a global formulation for ${\cal N}=2$ harmonic/projective anti-de Sitter (AdS) superspace $\text{AdS}^{4|8}\times S^2 \simeq \text{AdS}^{4|8}\times {\mathbb C}P^1$ that allows for a simple action of superconformal (and hence AdS isometry) transformations. First of all, we provide an alternative supertwistor description of the ${\cal N}$-extended AdS superspace in four dimensions, AdS$^{4|4\cal N}$, which corresponds to a realisation of the connected component $\mathsf{OSp}_0({\cal N}|4; {\mathbb R})$ of the AdS isometry supergroup as $\mathsf{SU}(2,2 |{\cal N}) \bigcap \mathsf{OSp} ({\cal N}| 4; {\mathbb C})$. The proposed realisation yields the following properties: (i) AdS$^{4|4\cal N}$ is an open domain of the compactified ${\cal N}$-extended Minkowski superspace, $\overline{\mathbb M}^{4|4\cal N}$; (ii) the infinitesimal ${\cal N}$-extended superconformal transformations naturally act on AdS$^{4|4\cal N}$; and (iii) the isometry transformations of AdS$^{4|4\cal N}$ are described by those superconformal transformations which obey a certain constraint. The obtained results for AdS$^{4|4\cal N}$ are then applied to develop a supertwistor formulation for an AdS flag superspace $ \text{AdS}^{4|8} \times {\mathbb F}_1(2)$ that we identify with the ${\cal N}=2$ harmonic/projective AdS superspace. This construction makes it possible to read off the superconformal and AdS isometry transformations acting on the analytic subspace of the harmonic superspace.