Finding diffeomorphism-invariant observables to characterize the properties of gravity and spacetime at the Planck scale is essential for making progress in quantum gravity. The holonomy and Wilson loop of the Levi-Civita connection are potentially interesting ingredients in the construction of quantum curvature observables. Motivated by recent developments in nonperturbative quantum gravity, we establish new relations in three and four dimensions between the holonomy of a finite loop and certain curvature integrals over the surface spanned by the loop. They are much simpler than a gravitational version of the nonabelian Stokes' theorem, but require the presence of totally geodesic surfaces in the manifold, which follows from the existence of suitable Killing vectors. We show that the relations are invariant under smooth surface deformations, due to the presence of a conserved geometric flux.

中文翻译：

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引力威尔逊环的几何通量公式

在普朗克尺度上寻找微分同胚不变的可观测量来表征引力和时空的特性对于在量子引力方面取得进展至关重要。Levi-Civita 连接的完整和威尔逊环是构建量子曲率可观测量的潜在有趣成分。受非微扰量子引力最新发展的推动，我们在有限环的完整度和环所跨越的表面上的某些曲率积分之间建立了三维和四维的新关系。它们比非阿贝尔斯托克斯定理的引力版本简单得多，但需要在流形中存在完全测地曲面，这是由于存在合适的杀伤向量。我们表明这些关系在光滑表面变形下是不变的，