On perturbative instability of Pope-Warner solutions on Sasaki-Einstein manifolds

Given a Sasaki-Einstein manifold, M7, there is the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ \mathcal{N}=2 $\end{document} supersymmetric AdS4 × M7 Freund-Rubin solution of eleven-dimensional supergravity and the corresponding non-supersymmetric solutions: the perturbatively stable skew-whiffed solution, the perturbatively unstable Englert solution, and the Pope-Warner solution, which is known to be perturbatively unstable when M7 is the seven-sphere or, more generally, a tri-Sasakian manifold. We show that similar perturbative instability of the Pope-Warner solution will arise for any regular Sasaki-Einstein manifold, M7, admitting a basic, primitive, transverse (1,1)-eigenform of the Hodge-de Rham Laplacian with the eigenvalue in the range between \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ 2\left( {9-4\sqrt{3}} \right) $\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ 2\left( {9+4\sqrt{3}} \right) $\end{document}. Existence of such (1,1)-forms on all homogeneous Sasaki-Einstein manifolds can be shown explicitly using the Kähler quotient construction or the standard harmonic expansion. The latter shows that the instability arises from the coupling between the Pope-Warner background and Kaluza-Klein scalar modes that at the supersymmetric point lie in a long Z-vector supermultiplet. We also verify that the instability persists for the orbifolds of homogeneous Sasaki-Einstein manifolds that have been discussed recently.