Bianchi恒等式的两种表述及其对模稳定化的意义
Revisiting the two formulations of Bianchi identities and their implications on moduli stabilization
摘要：
在非几何II型orientifold紧化的背景下，有两个公式来表示各种NS-NS Bianchi恒等式。在第一个公式中，标准三形式通量（$H_3$），几何通量（$ω$）和非几何通量（$Q$和$R$）通过使用实六维指数（例如$H_{ijk}, ω_{ij}{}^k, Q_i{}^{jk}$和$R^{ijk}$）来表示，并且该公式已被大量用于简化环形定向褶皱中的标量势。另一方面，有关的研究超出环形背景，第二个配方是利用所有的通量分量的各种对合的偶数/奇数$(2,1)$ -和$(1,1)$ -共同调的复杂的三倍。根据最近的模型构建兴趣和arXiv：0705.3410和arXiv：0709.2186中的一些观察，在本文中，我们详细回顾了两个最常研究的环形例子，以说明这两个公式的当前形式并不完全等价。为了证明这一点，我们将第一个公式的所有身份翻译成上同调成分，并且在对随后的约束进行繁琐的重新洗牌之后，有趣的是，我们发现第二个公式的所有身份都嵌入到第一个公式中，其中有一些额外的约束。此外，我们还对这些Bianchi恒等式的可能解进行了详细的分析，发现有些解可以非常显著地减小标量势的大小，有些解的强度太大而不能完全打破无标度结构.最后，我们还评论的影响，强加的一些解决方案的Bianchi恒等式在研究模稳定。
英文摘要：
In the context of non-geometric type II orientifold compactifications, there have been two formulations for representing the various NS-NS Bianchi-identities. In the first formulation, the standard three-form flux ($H_3$), the geometric flux ($ω$) and the non-geometric fluxes ($Q$ and $R$) are expressed by using the real six-dimensional indices (e.g. $H_{ijk}, ω_{ij}{}^k, Q_i{}^{jk}$ and $R^{ijk}$), and this formulation has been heavily utilized for simplifying the scalar potentials in toroidal-orientifolds. On the other hand, relevant for the studies beyond toroidal backgrounds, a second formulation is utilized in which all flux components are written in terms of various involutively even/odd $(2,1)$- and $(1,1)$-cohomologies of the complex threefold. In the lights of recent model building interests and some observations made in arXiv:0705.3410 and arXiv:0709.2186, in this article, we revisit two most commonly studied toroidal examples in detail to illustrate that the present forms of these two formulations are not completely equivalent. To demonstrate the same, we translate all the identities of the first formulation into cohomology ingredients, and after a tedious reshuffling of the subsequent constraints, interestingly we find that all the identities of the second formulation are embedded into the first formulation which has some additional constraints. In addition, we look for the possible solutions of these Bianchi identities in a detailed analysis, and we find that some solutions can reduce the size of scalar potential very significantly, and in some cases are too strong to break the no-scale structure completely. Finally, we also comment on the influence of imposing some of the solutions of Bianchi identities in studying moduli stabilization.
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