Generalized Symmetries in Quantum Field Theories

这个博士项目将探索量子场论(QFT)中高阶对称的数学和物理含义，以及它的扩展包括高群结构和其他范畴对称。更高形式的对称具有带电对象，这些对象是QFT中的高维亏损算符，例如杨-Mills理论中的线算符。这些是在拓扑运算符下充电的，它形成了所谓的更高形式的对称群。当给定理论的高阶对称不是简单地形成一个乘积群，而是一个“广义推广”时，高群结构就出现了。这种对称性直到最近才在《数学物理》中被发现。作为补充，它们的数学结构最近在代数拓扑/范畴理论方面有了许多令人兴奋的发展。这个项目的目的是协同这些发展，并探索这种对称性对四维和低维规范理论的物理意义：具体地说，理解高基团在约束QFTs的动力学和真空结构中的作用，特别是N=1超对称规范理论。目标是在4d内对所有高群和广义对称结构有一个全面的了解。具体的目标是：1.对于含物质的四维规范理论和箭形规范理论，确定整体风味对称群、1-形式对称性以及这些高形式对称性形成的2-群结构或反常。2.特别是从禁闭的角度确定这类量子傅里叶变换对真空结构的影响。3.在QFT对偶(例如Seiberg-like对偶)上匹配这些广义对称性。4.发展了在QFT弦理论构造中实现这些广义对称性的方法。5.推广到低维QFT(在凝聚态中的应用)以及非超对称理论。方法论建立在数学物理中，在代数拓扑和范畴理论中具有很强的成分。范畴对称性的出现是最近才出现的，该项目打算将这些与QFT中的现代工具相结合，如对偶性、弦理论中的几何实现以及对低维理论的应用，如凝聚态。这与EPSRC关于数学物理、代数拓扑学以及物理学方面的凝聚态物质的研究策略是一致的。这个项目作为一个跨学科的研究项目脱颖而出，它与英国充满活力的上述主题的研究环境相联系。

This PhD project will explore the mathematical and physical implications of higher-form symmetries in Quantum Field Theories (QFTs), and extensions thereof includ-ing higher-group structures and other categorical symmetries. Higher-form sym-metries have charged objects that are higher-dimensional, defect operators in the QFT, e.g. line operators in Yang-Mills theory. These are charged under topological operators, which form a so-called higher-form symmetry group. The higher-group structure arises whenever higher-form symmetries of a given theory do not simply form a product group, but a "generalized extension". Such symmetries have only recently been uncovered in Mathematical Physics. Complementing this, their mathematical structure has recently seen a lot of exciting developments in algebraic topology/category theory. This project aims to synergize these developments and to explore the physical im-plications of such symmetries upon 4d and lower dimensional gauge theories: specifically, understanding the role of higher-groups in constraining the dynamics and vacuum structure of QFTs, and in particular N=1 supersymmetric gauge theo-ries. The goal is to obtain a comprehensive understanding of all higher-group and gen-eralized symmetry structures in 4d. Concretely the objectives are: 1. Determining the global flavor symmetry group, 1-form symmetries and the 2-group structures or anomalies that these higher-form symmetries form, for 4d gauge theories with matter as well as quiver gauge theories. 2. Determining the implications on the vacuum structure of such QFTs, in par-ticular in view of confinement. 3. Matching these generalized symmetries across QFT dualities (e.g. Seiberg-like dualities). 4. Developing methods to realize these generalized symmetries in string theo-ry constructions of QFTs. 5. Generalizing to lower-dimensional QFTs (with applications in condensed matter) as well as non-supersymmetric theories.The methodology is founded in Mathematical Physics, with a very strong compo-nent in algebraic topology and category theory. The emergence of categorical symmetries is very recent and the project intends to combine these with modern tools in QFT such as dualities, geometric realization in string theory and applications to lower dimensional theories, such as condensed matter. This aligns with the EPSRC research strategy for Mathematical Physics,algebraic topology, and, on the Physics side, condensed matter. This project stands out as an interdisciplinary research program, which connects to the vibrant re-search environment in the UK in said subjects.