Generalized Symmetries in Quantum Field Theories

这个博士项目将探索量子场论（QFT）中更高形式对称性的数学和物理含义，及其扩展，包括更高的群结构和其他分类对称性。高阶形式的对称性有带电的对象，它们是QFT中的高维亏损算子，例如杨-米尔斯理论中的线算子。它们在拓扑算子下带电，形成所谓的高阶对称群。当一个给定理论的高阶形式对称性不是简单地形成一个乘积群，而是一个“广义扩张”时，高阶群结构就出现了。这种对称性直到最近才在数学物理学中被发现。补充这一点，他们的数学结构最近看到了很多令人兴奋的发展，代数拓扑/范畴理论。这个项目的目的是协同这些发展，并探索这种对称性在4d和低维规范理论上的物理意义：特别是，理解更高群在约束QFT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。目标是全面了解4d中所有的高级群和一般化对称结构。具体的目标是：1。确定整体味对称群，1-形式对称性和2-群结构或这些更高形式对称性形成的异常，对于有物质的4d规范理论和无物质规范理论。2.确定这种QFT的真空结构的含义，特别是在考虑约束。3.在QFT对偶中匹配这些广义对称性（例如Seiberg-like对偶）。4.发展在QFT弦理论结构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。该方法建立在数学物理学中，在代数拓扑学和范畴论中有很强的组成部分。范畴对称的出现是最近的，该项目打算将这些与QFT中的现代工具相结合，如对偶性，弦论中的几何实现以及对低维理论的应用，如凝聚态。这与EPSRC的数学物理，代数拓扑学的研究策略相一致，在物理学方面，凝聚态物质。该项目突出作为一个跨学科的研究计划，它连接到充满活力的研究环境在英国在上述科目。

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.