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

TL;DR

This paper introduces a new geometric framework for ${ m AdS}^{4|4 ext{N}}$ superspace using a novel realization of the supergroup, enabling a coset construction, an atlas with charts, and a superparticle model with conformal supergeometry.

Contribution

It develops a new realization of the ${ m OSp}( ext{N}|4; ext{R})$ supergroup and applies it to construct the geometry and models of ${ m AdS}^{4|4 ext{N}}$ superspace.

Findings

New coset construction for ${ m AdS}^{4|4 ext{N}}$

Atlas with two charts for superspace

Invariant superparticle model

Abstract

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 ${\rm AdS}^{4|4\mathcal N}$.