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arXiv 1310.6360hep-th
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乘积流形上紧化的稳定性与谱

Stability and Spectrum of Compactifications on Product Manifolds

Adam R. Brown, Alex Dahlen

备注：
52 pages, 4 figures; v2: footnote and references added
发表信息：
Phys. Rev. D 90, 044047 (2014)
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摘要：

研究了$M_p \times M_{Nq}$上Freund-Rubin紧化的谱和扰动稳定性，其中$M_{Nq}$是$N$ $q$维Einstein流形的乘积.高维作用量有一个宇宙学项$Λ$和一个$q$形式的通量，它们分别包裹乘积的每个元素;扩展维$M_p$可以是反德西特、闵可夫斯基或德西特。我们找到了这个背景下的每个激发的质量，以及这些解决方案是稳定的条件。这推广了以前关于Freund-Rubin真空的工作，它集中在$N=1$的情况下，其中一个$q$形式的通量包裹一个单一的$q$维爱因斯坦流形。当$q$维内部流形是一个乘积时，$N=1$的情形可能具有经典不稳定性--乘积的一个成员想要收缩，而流形的其余部分扩张。在这里，我们将看到，单独地用较低形式的通量包裹乘积的每个元素可以治愈这种循环坍缩不稳定性。$N=1$的情况下，也可以有一个不稳定时，$Λ>0$和$q\ge4$的形状模式扰动，我们发现同样的不稳定性与一般$N$紧化，并表明，它甚至延伸到情况下，$Λ\le0$。另一方面，当$q=2$或3时，形状模式总是稳定的，并且存在对所有波动都微扰稳定的广泛的AdS和de Sitter真空类。

英文摘要：

We study the spectrum and perturbative stability of Freund-Rubin compactifications on $M_p \times M_{Nq}$, where $M_{Nq}$ 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 $q\ge4$ to shape-mode perturbations; we find the same instability in compactifications with general $N$, and show that it even extends to cases where $Λ\le0$. 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.

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