标题： Weyl几何与量子反常中的课题 显示英文标题

标题： Topics in Weyl Geometry and Quantum Anomalies

摘要： 本文的第一部分侧重于全息对称性的 Weyl 协变性质。我们将 Fefferman-Graham (FG) 环境构造从共形几何推广到相应的 Weyl 几何构造。通过 Weyl-环境构造，我们研究了 Weyl 流形上的 Weyl 协变量，并定义了 Weyl 障碍张量。我们展示了在偶数边界维度下，Weyl 障碍张量出现在 ALAdS 基础度规的 Fefferman-Graham 展开中的极点位置。在 Weyl-Fefferman-Graham 规范下的全息重整化过程中，我们在多维情况下计算了边界理论的 Weyl 异常，并证明 Weyl 障碍张量可以作为对偶量子场论 (QFT) 的 Weyl 异常的基本构建块。具有背景 Weyl 几何的全息计算还暗示了 Weyl 异常的一个潜在几何解释。 本文的第二部分致力于理解 BRST 形式主义和量子异常的几何性质。使用李代数的语言，BRST 复合可以从规范理论主丛衍生的 Atiyah 李代数丛的外代数中编码。我们证明了平凡化下的 Atiyah 李代数丛上同调导致了 BRST 上同调。然后我们应用李代数上同调来研究量子异常，并演示了手征和洛伦兹-Weyl 异常的计算。特别是，我们注意到李代数提供的几何直觉（在传统 BRST 复合中不存在）为一个更深层次的画面提供了线索，该画面同时将一致性形式和协变形式的异常几何化。在代数构造中，一致性异常和协变异常之间的差异仅仅是基的选择不同。 显示英文摘要

摘要： The first part of this thesis focuses on the Weyl-covariant nature of holography. We generalize the Fefferman-Graham (FG) ambient construction for conformal geometry to a corresponding construction for Weyl geometry. Through the Weyl-ambient construction, we investigate Weyl-covariant quantities on the Weyl manifold and define Weyl-obstruction tensors. We show that Weyl-obstruction tensors appear as poles in the Fefferman-Graham expansion of the ALAdS bulk metric for even boundary dimensions. Under holographic renormalization in the Weyl-Fefferman-Graham gauge, we compute the Weyl anomaly of the boundary theory in multiple dimensions and demonstrate that Weyl-obstruction tensors can be used as the building blocks for the Weyl anomaly of the dual quantum field theory (QFT). The holographic calculation with a background Weyl geometry also suggests an underlying geometric interpretation of the Weyl anomaly. The second part of this thesis is devoted to understanding the geometric nature of the BRST formalism and quantum anomalies. Using the language of Lie algebroids, the BRST complex can be encoded in the exterior algebra of an Atiyah Lie algebroid derived from the principal bundle of the gauge theory. We showed that the cohomology of an Atiyah Lie algebroid in a trivialization gives rise to the BRST cohomology. We then apply the Lie algebroid cohomology in studying quantum anomalies and demonstrate the computation for chiral and Lorentz-Weyl anomalies. In particular, we pay close attention to the fact that the geometric intuition afforded by the Lie algebroid (which was absent in the traditional BRST complex) provides hints of a deeper picture that simultaneously geometrizes the consistent and covariant forms of the anomaly. In the algebroid construction, the difference between the consistent and covariant anomalies is simply a different choice of basis.