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

Gianguido Dall'Agata, Davide Fabbri, Christophe Fraser, Pietro Fre', Piet Termonia, Mario Trigiante

TL;DR

The paper derives the $OSp(8|4)$ singleton action from the $D=11$ supermembrane on $AdS_4\times S^7$, clarifying how boundary conformal dynamics emerge from a non-linear membrane action. It employs a rheonomic, κ-fixed formulation together with a supersolvable parametrization to realize the full superconformal symmetry on the worldvolume and to organize the geometry in terms of the $OSp(8|4)$ algebra. The fluctuation analysis around a classical boundary solution yields the singleton as a free 3D conformal field on Minkowski space, with explicit bosonic and fermionic content and SUSY transformations that realize the boundary conformal symmetry. The work also lays out a framework to extract flavor-group representations for $G/H$ branes and to perform direct holographic tests beyond the maximally supersymmetric case.

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.

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

TL;DR

The paper derives the singleton action from the supermembrane on , clarifying how boundary conformal dynamics emerge from a non-linear membrane action. It employs a rheonomic, κ-fixed formulation together with a supersolvable parametrization to realize the full superconformal symmetry on the worldvolume and to organize the geometry in terms of the algebra. The fluctuation analysis around a classical boundary solution yields the singleton as a free 3D conformal field on Minkowski space, with explicit bosonic and fermionic content and SUSY transformations that realize the boundary conformal symmetry. The work also lays out a framework to extract flavor-group representations for branes and to perform direct holographic tests beyond the maximally supersymmetric case.

Abstract

Our goal is to study the supermembrane on an background, where is a 7--dimensional Einstein manifold with Killing spinors. This is a direct way to derive the 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 singleton corresponding to the choice . 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.

Paper Structure

This paper contains 17 sections, 163 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The algebraic basis of the holographic correspondence
  3. $G/H$--branes
  4. The qualitative difference between the round $\hbox{\fam\msbmfam S}^7$ case and the lower supersymmetry cosets $G/H$
  5. The singleton problem in $G/H$ M2 branes
  6. The singleton from the supermembrane
  7. The organization of the paper
  8. The near--horizon geometry
  9. The supermembrane and $\kappa$--supersymmetry
  10. The first order "Polyakov" action of the supermembrane from rheonomy
  11. The bosonic equations of motion
  12. The supermembrane on the $AdS_4 \times S^7$ background
  13. The $AdS_4 \times S^7$ splitting
  14. The supersolvable algebra and the $\kappa$--symmetry gauge--fixing
  15. The superconformal action in second order "Nambu-Goto" formalism
  16. ...and 2 more sections

Figures (2)

  • Figure 1: The root diagram of $SO(2,3)$. The bosonic weights are represented by circles, and the fermionic weights by squares. The dilatation charge of horizontal planes in the diagram are on the left, while the worldvolume theory interpretation of the planes of generators are labelled to the right. The supersolvable algebra is the boxed subalgebra.
  • Figure 2: The root diagram of $SO(2,3)$, with simple roots $\epsilon_1$ and $\epsilon_2$. The Weyl reflection $W_{\epsilon_2}$ is shown in the picture.