The anti-de Sitter supergeometry revisited

TL;DR

This work clarifies the relation between the embedding and conformal-superspace descriptions of ${AdS}^{4|4\mathcal{N}}$ in four dimensions, showing that the AdS geometry can be realised as a conformally flat superspace with covariantly constant torsion $S^{ij}$. It provides two explicit conformally flat realizations (stereographic and Poincaré) and demonstrates how degauging to ${\mathsf SU}}({\cal N})$ superspace yields a tractable AdS supergeometry, with the AdS radius determined by $|S|$. The paper also introduces a two-parameter deformation of the AdS interval and a corresponding superparticle model, and discusses applications to superconformal higher-spin multiplets and the ${\cal N}=2$ super-Weyl anomaly, highlighting practical implications for field theory in AdS superspace. Overall, it presents a unified geometric framework for AdS superspace that connects embedding and conformal approaches and clarifies κ-symmetry structures of massless AdS superparticles.

Abstract

In a supergravity framework, the $\cal N$-extended anti-de Sitter (AdS) superspace in four spacetime dimensions, $\text{AdS}^{4|4\cal N} $, is a maximally symmetric background that is described by a curved superspace geometry with structure group $\mathsf{SL}(2, \mathbb{C}) \times \mathsf{U}({\cal N})$. On the other hand, within the group-theoretic setting, $\text{AdS}^{4|4{\cal N}} $ is realised as the coset superspace $\mathsf{OSp}({\cal N}|4;\mathbb{R}) /\big[ \mathsf{SL}(2, \mathbb{C}) \times \mathsf{O}({\cal N}) \big]$, with its structure group being $\mathsf{SL}(2, \mathbb{C}) \times \mathsf{O}({\cal N})$. Here we explain how the two frameworks are related. We give two explicit realisations of $\text{AdS}^{4|4{\cal N}} $ as a conformally flat superspace, thus extending the ${\cal N}=1$ and ${\cal N}=2$ results available in the literature. As applications, we describe: (i) a two-parameter deformation of the $\text{AdS}^{4|4{\cal N}} $ interval and the corresponding superparticle model; (ii) some implications of conformal flatness for superconformal higher-spin multiplets and an effective action generating the $\mathcal{N}=2$ super-Weyl anomaly; and (iii) $κ$-symmetry of the massless AdS superparticle.