论文标题
量子场理论中的扩展功能场理论和异常
Extended Functorial Field Theories and Anomalies in Quantum Field Theories
论文作者
论文摘要
我们开发了一个通用框架，用于使用扩展功能性字段理论来描述异常，从而扩展了Freed和Monnier先前的工作。在此框架中，在较高的一维中可逆场理论描述了异常，并且在其边界上存在异常的场理论。我们为使用对称的单体生物游戏的语言提供了所有涉及的概念的精确数学定义。特别是，具有异常的现场理论将通过对称单体变换来描述。对于捕获异常情况的哈密顿图片，必须使用更高的分类概念。将详细解释与对路径积分和哈密顿对异常的描述的关系。此外，我们详细讨论了异常流入。我们将一般框架应用于奇数空间上的Fermionic Systems的平均异常。由于Loya和Melrose，我们使用Atiyah-Patodi-Singer索引定理的延伸来歧视，以明确构建编码异常的扩展可逆场理论。这使我们能够明确计算量规空间上规格对称性的柱子表示的2个循环。作为第二个应用程序，我们研究了Dijkgraaf-Witten理论的离散对称性及其测量。非阿布尔群体共同体用于描述离散的对称性，我们得出了这种对称性的具体条件，可以根据Lyndon-Hochschild-Serre光谱序列接受hooft异常。我们以“ T Hooft Anomaly为状态”在较高的Dijkgraaf-witten理论的边界上的状态中明确实现了离散仪表理论。
We develop a general framework for the description of anomalies using extended functorial field theories extending previous work by Freed and Monnier. In this framework, anomalies are described by invertible field theories in one dimension higher and anomalous field theories live on their boundaries. We provide precise mathematical definitions for all concepts involved using the language of symmetric monoidal bicategories. In particular, field theories with anomalies will be described by symmetric monoidal transformations. The use of higher categorical concepts is necessary to capture the Hamiltonian picture of anomalies. The relation to the path integral and the Hamiltonian description of anomalies will be explained in detail. Furthermore, we discuss anomaly inflow in detail. We apply the general framework to the parity anomaly in fermionic systems coupled to background gauge and gravitational fields on odd-dimensional spacetimes. We use the extension of the Atiyah-Patodi-Singer index theorem to manifolds with corners due to Loya and Melrose to explicitly construct an extended invertible field theory encoding the anomaly. This allows us to compute explicitly the 2-cocycle of the projective representation of the gauge symmetry on the quantum state space. As a second application, we study discrete symmetries of Dijkgraaf-Witten theories and their gauging. Non-abelian group cohomology is used to describe discrete symmetries and we derive concrete conditions for such a symmetry to admit 't Hooft anomalies in terms of the Lyndon-Hochschild-Serre spectral sequence. We give an explicit realization of a discrete gauge theory with 't Hooft anomaly as a state on the boundary of a higher-dimensional Dijkgraaf-Witten theory.
