The Osp(8|4) singleton action from the supermembrane

TL;DR

This paper derives the explicit supermembrane action on an $AdS_4 imes S^7$ background, demonstrating its superconformal invariance and showing it reduces to a free singleton field theory in flat space.

Contribution

It provides the first explicit form of the supermembrane action coupled to an $AdS_4 imes S^7$ background and shows its invariance under superconformal transformations.

Findings

Derived the explicit supermembrane action on $AdS_4 imes S^7$

Demonstrated superconformal invariance of the action

Showed the singleton reduces to a free field theory in Minkowski space

Abstract

Our goal is to study the supermembrane on an $AdS_4 \times \cal M_7$ background, where $\cal M_7$ is a 7--dimensional Einstein manifold with $N$ Killing spinors. This is a direct way to derive the $Osp(N|4)$ singleton field theory with all the additional properties inherited from the geometry of the internal manifold. As a first example we consider the maximally supersymmetric $Osp(8|4)$ singleton corresponding to the choice $\cal M_7 = S^7$. We find the explicit form of the action of the membrane coupled to this background geometry and show its invariance under non--linearly realised superconformal transformations. To do this we introduce a supergroup generalisation of the solvable Lie algebra parametrisation of non--compact coset spaces. We also derive the action of quantum fluctuations around the classical configuration, showing that this is precisely the singleton action. We find that the singleton is simply realised as a free field theory living on flat Minkowski space.