Stability of AdS_p x M_q Compactifications Without Supersymmetry

TL;DR

The authors analyze the linear stability of Freund-Rubin backgrounds of the form $AdS_p\times M_q$ in gravity coupled to a $q$-form with no cosmological term. Using harmonic analysis on the internal Einstein space and the Breitenlohner-Freedman bound, they show that the generic $AdS_p\times S^q$ vacua are perturbatively stable without requiring supersymmetry, while stability can fail for product spaces $M_n\times M_{q-n}$ when $q<9$, and they identify a nonsupersymmetric unstable case $AdS_4\times S^6$ in massive IIA. The work also discusses holographic interpretations via operator dimensions, the behavior of vector and tensor KK modes, and implications for extremal black branes and negative energy configurations, providing a comprehensive framework for assessing non-supersymmetric AdS flux vacua. Overall, the paper delineates precisely which compactifications are perturbatively stable and elucidates the structure of their fluctuation spectra, with implications for proposed bosonic M-theory and potential CFT duals.

Abstract

We study the stability of Freund-Rubin compactifications, AdS_p x M_q, of p+q-dimensional gravity theories with a q-form field strength and no cosmological term. We show that the general AdS_p x S^q vacuum is classically stable against small fluctuations, in the sense that all modes satisfy the Breitenlohner-Freedman bound. In particular, the compactifications used in the recent discussion of the proposed bosonic M-theory are perturbatively stable. Our analysis treats all modes arising from the graviton and the q-form, and is completely independent of supersymmetry. From the masses of the linearized perturbations, we obtain the dimensions of some operators in possible holographic dual CFT's. Solutions with more general compact Einstein spaces need not be stable, and in particular AdS_p x S^n x S^{q-n} is unstable for q < 9 but is stable for q >= 9. We also study the AdS_4 x S^6 compactification of massive type IIA supergravity, which differs from the usual Freund-Rubin compactification in that there is a cosmological term already in ten dimensions. This nonsupersymmetric vacuum is unstable.