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

TL;DR

This paper demonstrates that the Pope-Warner solutions on any Sasaki-Einstein manifold are perturbatively unstable due to specific eigenforms, extending known instability results beyond the seven-sphere case.

Contribution

It generalizes the known instability of Pope-Warner solutions to all Sasaki-Einstein manifolds with certain eigenforms, using harmonic analysis and Kaluza-Klein theory.

Findings

Pope-Warner solutions are perturbatively unstable on all Sasaki-Einstein manifolds with specific eigenforms.

The instability is linked to coupling with Kaluza-Klein scalar modes in long supermultiplets.

Instability persists even for orbifolded Sasaki-Einstein manifolds.

Abstract

Given a Sasaki-Einstein manifold, M_7, there is the N=2 supersymmetric AdS_4 x M_7 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 M_7 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 Sasaki-Einstein manifold, M_7, admitting a basic, primitive, transverse (1,1)-eigenform of the Hodge-de Rham Laplacian with the eigenvalue in the range between 2(9-4\sqrt 3) and 2(9+4\sqrt 3). Existence of such (1,1)-forms on all homogeneous Sasaki-Einstein manifolds can be shown explicitly using the Kahler 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.