Spectrum and stability of compactifications on product manifolds

We study the spectrum and perturbative stability of Freund-Rubin compactifications on M-p x M-Nq, where M-Nq is itself a product of N q-dimensional Einstein manifolds. The higher-dimensional action has a cosmological term A and a q-form flux, which individually wraps each element of the product; the extended dimensions M-p can be anti-de Sitter, Minkowski, or de Sitter. We find the masses of every excitation around this background, as well as the conditions under which these solutions are stable. This generalizes previous work on Freund-Rubin vacua, which focused on the N = 1 case, in which a q-form flux wraps a single q-dimensional Einstein manifold. The N = 1 case can have a classical instability when the q-dimensional internal manifold is a product-one of the members of the product wants to shrink while the rest of the manifold expands. Here, we will see that individually wrapping each element of the product with a lower-form flux cures this cycle-collapse instability. The N = 1 case can also have an instability when A > 0 and q >= 4 to shape-mode perturbations; we find the same instability in compactifications with general N, and show that it even extends to cases where A <= 0. On the other hand, when q = 2 or 3, the shape modes are always stable and there is a broad class of AdS and de Sitter vacua that are perturbatively stable to all fluctuations.