Generalized Symmetries in Quantum Field Theories

这个博士项目将探索量子场论（QFTs）中高形式对称性的数学和物理含义，以及其扩展，包括高群结构和其他分类对称性。高形式对称具有高维带电物体，QFT中的缺陷算子，例如杨-米尔斯理论中的线算子。它们在拓扑算子下带电，形成所谓的高形式对称群。当给定理论的高形式对称不是简单地形成积群，而是形成“广义扩展”时，就会出现高群结构。这种对称性直到最近才在数学物理中被发现。与此相补充的是，它们的数学结构最近在代数拓扑/范畴论方面取得了许多令人兴奋的进展。本项目旨在协同这些发展，并探索这种对称性对四维和低维规范理论的物理含义：具体来说，理解高群在约束量子场动力学和真空结构中的作用，特别是N=1超对称规范理论。目标是获得对4d中所有高群和广义对称结构的全面理解。具体目标是：1。确定具有物质和颤振规范理论的四维规范理论的全局味对称群、一形式对称和这些高形式对称形成的二群结构或异常。2. 确定对这种量子场的真空结构的影响，特别是考虑到约束。3. 在QFT对偶上匹配这些广义对称性（例如Seiberg-like对偶）。4. 发展了在量子场的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（在凝聚态物质中的应用）以及非超对称理论。该方法以数学物理为基础，在代数拓扑和范畴论中有很强的成分。范畴对称是最近才出现的，该项目打算将它们与量子傅立波中的现代工具结合起来，如对偶性、弦理论中的几何实现以及低维理论（如凝聚态物质）的应用。这与EPSRC在数学物理、代数拓扑以及物理方面的凝聚态物质方面的研究策略一致。这个项目作为一个跨学科的研究项目脱颖而出，它与英国在上述学科中充满活力的研究环境相联系。

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.