发表机构
Institute of Theoretical Physics, Faculty of Mathematics and Physics, Charles University; Institute of Mathematics, Czech Academy of Sciences（查理大学数学物理学院理论物理研究所; 捷克科学院数学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对高维Kerr-NUT-(A)dS空间的黎曼张量结构，发现无法仅从黎曼张量协变构造非平凡2-形式，进而通过代数与微分条件刻画了对应曲率的黎曼对称张量族，为该度规族的IDEAL刻画提供了关键基础。
AI 中文摘要
受最终目标是找到这类度规族的IDEAL（内蕴、演绎、显式且算法化）刻画的驱动，我们研究高维Kerr-NUT-(A)dS空间的黎曼曲率张量的代数与微分结构。这类空间的特殊几何由主张量$h_{ab}$（非退化闭共形 Killing-Yano 2-形式）支配，其存在性区分出Kerr-NUT-(A)dS族；因此我们聚焦于黎曼曲率$R_{abcd}$与$h_{ab}$之间的代数关系。研究中我们遇到一个障碍，它导出了一个不可能定理：仅从不带导数的黎曼张量本身，无法协变构造出任何非平凡2-形式，包括$h_{ab}$自身。尽管如此，我们仍通过代数条件和微分条件刻画了可作为Kerr-NUT-(A)dS度规曲率的黎曼对称张量族，这些条件源于$h_{ab}$定义方程的可积性及第二比安基恒等式。我们的计算适用于所有高维情形，这通过明智地应用与稳定$h_{ab}$的半直积群$U(1)^n\rtimes S_n$相关的表示论技术得以实现。
英文摘要
We study the algebraic and differential structure of the Riemann curvature tensor of the higher-dimensional Kerr-NUT-(A)dS spaces, motivated by the eventual goal of finding an IDEAL (Intrinsic, Deductive, Explicit and ALgorithmic) characterization of this family of metrics. The special geometry of these spaces is governed by the principal tensor $h_{ab}$, a non-degenerate closed conformal Killing-Yano $2$-form, whose existence singles out the Kerr-NUT-(A)dS family; we therefore focus on the algebraic relationship between the Riemann curvature $R_{abcd}$ and $h_{ab}$. In our investigations we encountered an obstruction, which results in a no-go theorem: no non-trivial $2$-form, including $h_{ab}$ itself, can be constructed covariantly from the undifferentiated Riemann tensor alone. Despite that, we do characterize the family of Riemann-symmetric tensors that could be the curvature of a Kerr-NUT-(A)dS metric by an algebraic and a differential condition, stemming from the integrability of the defining equation of $h_{ab}$ and the second Bianchi identity. Our calculations apply in all higher dimensions, which becomes feasible by a judicious application of representation theoretic techniques related to a semi-direct product group $U(1)^n\rtimes S_n$ that stabilizes $h_{ab}$.
Comments56 pages
