Stability ofAdSp×Mqcompactifications without supersymmetry

We study the stability of Freund-Rubin compactifications, ${\mathrm{AdS}}_p\times {\mathrm{M}}_q,$ of $\mathrm{}(p+q)\mathrm{}$-dimensional gravity theories with a q-form field strength and no cosmological term. We show that the general ${\mathrm{AdS}}_p\times {\mathrm{S}}^q$ 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 ${\mathrm{AdS}}_p\times {\mathrm{S}}^n\times {\mathrm{S}}^{q-n}$ is unstable for $q<\mathrm{}9\mathrm{}$ but is stable for $q>~9.$ We also study the ${\mathrm{AdS}}_4\times {\mathrm{S}}^6$ 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.