Title: Moduli space of spin connections on three-dimensional homogeneous spaces Show Chinese title

Title: 三维齐次空间上的自旋联络模空间

Abstract: In this manuscript, we aim to classify and characterize the moduli space of homogeneous spin connections and homogeneous SU(2) connections on three-dimensional Riemannian homogeneous spaces. An analysis of the topology of the associated moduli spaces reveals that they are finite-dimensional topological manifolds (possibly with boundary) possessing trivial homotopy groups. Owing to their deep connection with cosmological models in the Ashtekar-Barbero-Immirzi formulation of General Relativity, this study offers a mathematically rigorous interpretation of the Ashtekar-Barbero-Immirzi-Sen connection within a cosmological context. In particular, we show that a correct formulation of the theory relies crucially on identifying the moduli space of homogeneous spin connections, thereby emphasizing the essential role of the spin structure in ensuring consistency with the physical content of the theory. The favorable topological properties of these moduli spaces circumvent many of the usual difficulties associated with singularities and the definition of regular measures in Quantum Field Theory and Quantum Gravity. As a result, they provide a solid foundation for the rigorous implementation of quantum theory in the cosmological setting. Show Chinese abstract

Abstract: 在本文中，我们旨在对三维黎曼齐性空间上的齐性自旋连接和齐性SU(2)连接的模空间进行分类和表征。 对相关模空间拓扑的分析表明，它们是有限维的拓扑流形（可能带有边界），并且具有平凡的同伦群。 由于它们与广义相对论的Ashtekar-Barbero-Immirzi公式中的宇宙学模型有深刻的联系，这项研究为在宇宙学背景下Ashtekar-Barbero-Immirzi-Sen连接提供了数学上严格的解释。 特别是，我们表明该理论的正确表述关键在于识别齐性自旋连接的模空间，从而强调了自旋结构在确保与理论物理内容一致中的重要作用。 这些模空间有利的拓扑性质避免了许多通常与奇点以及量子场论和量子引力中正则测度定义相关的困难。 因此，它们为在宇宙学环境中严格实现量子理论提供了坚实的基础。