高自旋引力的几何工具箱（第三部分）：纤维丛上联络的一般理论导论
Geometric tool kit for higher spin gravity (part III): An introduction to the general theory of connections on fibre bundles
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中文总结 AI 辅助
本文介绍纤维丛上联络的各类概念，重点关注对高自旋引力有重要意义的Cartan联络等，可填补相关教材空白，适用于引力及全息术等领域。
中文摘要 AI 辅助
这些讲义对当代数学文献中出现的各类联络概念进行了介绍。除了主丛上熟知的Ehresmann联络、向量丛上的Koszul联络外，还有大量可能不太为人们所熟悉的联络概念，其中最值得关注的当属主丛上的Cartan联络及其在牵引丛（tractor bundle）上的对应形式，这些联络概念对于进一步阐明高自旋相互作用与对称性可能具有重要意义。本综述的研究动机既源于此前这些联络概念在高自旋引力中的（显性或隐性）应用，也源于它们在该领域未来的潜在应用。本文涵盖的大部分内容也与引力领域的高级数学主题相关，例如共形几何的严格处理（这对全息术具有重要意义），并可能填补标准教材留下的空白——大多数标准教材通常不涉及Cartan联络、牵引演算（tractor calculus）及相关主题。
英文摘要
These notes provides an introduction to a wide variety of notions of connection that appear in the contemporary mathematical literature. Besides the familiar Ehresmann connections on principal bundles and Koszul connections on vector bundles, there is a large zoo of perhaps less familiar notions, most notably Cartan connections on principal bundles and their avatars on tractor bundles, which may deserve more attention for further elucidating higher-spin interactions and symmetries. This review is motivated both by the previous (explicit or implicit) use of these notions of connection in higher-spin gravity and by their potential future applications in this context. Much of the material covered here is also relevant to advanced mathematical topics in gravity, such as the rigorous treatment of conformal geometry (which is of interest for holography) and may help fill gaps left by standard textbooks (most of which typically do not address Cartan connections, tractor calculus, and related topics).
