The anti-de Sitter supergeometry revisited

TL;DR

This paper explores the geometry of anti-de Sitter superspaces in four dimensions, relating supergravity and group-theoretic frameworks, and presents new conformally flat realizations with applications to superparticle models and superconformal theories.

Contribution

It provides explicit conformally flat realizations of AdS superspaces for general extended supersymmetry, extending previous results for specific cases, and discusses their applications in superparticle and superconformal theories.

Findings

Explicit conformally flat realizations for general ${ m N}$

Deformation of AdS superspace intervals and superparticle models

Implications for superconformal higher-spin multiplets and anomalies

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) $\kappa$-symmetry of the massless AdS superparticle.