Stability and Spectrum of Compactifications on Product Manifolds

Adam R. Brown; Alex Dahlen

doi:10.1103/PhysRevD.90.044047

paper

Stability and Spectrum of Compactifications on Product Manifolds

arXiv:1310.6360 · doi:10.1103/PhysRevD.90.044047

Abstract

We study the spectrum and perturbative stability of Freund-Rubin compactifications on $M_{p} \times M_{N q}$ , where $M_{N q}$ is itself a product of $N$ $q$ -dimensional Einstein manifolds. The higher-dimensional action has a cosmological term $Λ$ 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 $Λ > 0$ and $q \geq 4$ to shape-mode perturbations; we find the same instability in compactifications with general $N$ , and show that it even extends to cases where $Λ \leq 0$ . 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.

52 pages, 4 figures; v2: footnote and references added

Stability and Spectrum of Compactifications on Product Manifolds

Abstract

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