量子场论中反常的教学导论
A Pedagogical Introduction to Anomalies in Quantum Field Theory
- Laboratoire de Physique de l’École Normale Supérieure PSL University, CNRS, Sorbonne Université, Université Paris Cité（巴黎高等师范学院物理实验室、PSL大学、法国国家科学研究中心、索邦大学、巴黎西岱大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文提供量子场论中反常的教学导论，涵盖其起源、费曼图计算、Wess-Zumino条件与BRST上同调求解，并联系指标定理及IIB超引力中的反常消除。
AI中文摘要：
这些笔记旨在提供一个相对简洁且自成体系的量子场论中反常的导论。第一部分回顾了量子场论的必要概念，特别是我们通篇使用的泛函积分表述。然后我们详细讨论了反常如何以及为何作为泛函积分测度的非不变性或由于缺乏保持所有对称性的正则化子而出现，以及它们如何表现为反常圈图和非反常Ward恒等式中的流不守恒，或等效地表现为有效作用的非不变性。具体来说，我们处理了在手征变换下的所谓阿贝尔反常，以及所谓的在存在手征费米子时破坏非阿贝尔规范不变性的手征反常。我们对这两种情况都进行了详细的三角形费曼图计算。然后我们简要概述了反常的一些重要性质（局域性、有限性、单圈精确性），以及它们如何一般地由Wess-Zumino一致性条件来表征。我们展示了该条件如何用BRST上同调重新表述，以及如何通过下降方程在任意（偶数）时空维度中求解。最后，我们简要讨论了任意偶数维度2n中的反常与2n+2维度中的指标定理之间的一般关系，并将其推广到引力反常。我们以展示十维IIB超引力中高度非平凡的反常消除作为结论。
英文摘要：
These notes are aimed at providing a relatively concise and self-contained introduction to anomalies in quantum field theory. A first part reviews the necessary notions of quantum field theory, in particular its functional integral formulation which we use throughout. Then we discuss in some detail how and why anomalies arise as non-invariance of the functional integral measure or through the absence of a regulator preserving all symmetries, and how they manifest themselves as current non-conservation in anomalous loop diagrams and anomalous Ward identities, or equivalently as a non-invariance of the effective action. Specifically, we deal with the so-called abelian anomaly under chiral transformations, and the so-called chiral anomaly which is a violation of non-abelian gauge invariance in the presence of chiral fermions. We perform a detailed triangle Feynman diagram computation for both cases. Then we move on to provide a brief overview of some important properties of anomalies (locality, finiteness, one-loop exactness), and how they are characterized in general by the Wess-Zumino consistency condition. We show how this condition can be reformulated in terms of BRST cohomology and how it can be solved, for arbitrary (even) space-time dimensions, by the descent equations. Finally, we briefly discuss the general relation between anomalies in arbitrary even dimensions 2n and index theorems in 2n+2 dimensions, generalizing also to gravitational anomalies. We conclude with the exhibition of the highly non-trivial anomaly cancellation in ten-dimensional IIB supergravity.
