Generalized Symmetries in Quantum Field Theory and Holography

本计画的目标是全面分析量子场论在不同维度上的高形式对称性。高等形式的对称性概括了在量子场论和数学物理中广泛研究的“普通”对称性的概念。普通对称有由局部算子和余维为1的拓扑电荷算子给出的带电对象。这个概念可以扩展到更高维度的带电物体来描述更高形式的对称。许多标准结果延续到这种推广，但更广泛的范围也允许通过研究理论的高级形式对称性来获得新的物理见解。我们将特别关注具有超对称性的共形不变理论。这些将从6d超共形理论（SCFTs）开始研究，该理论在降维和rg流的基础上产生更低维的SCFTs。该项目将重点研究更高形式的对称性与维度减少之间的关系，特别是从6d到5d/4d/3d。此外，高级对称的研究还需要分析这种全局对称的t - Hooft异常，以及高级对称之间的混合异常。为了补充这种场论方法，该项目还将分析scft全息对偶描述中更高形式对称性的印记。从反德西特（AdS）时空的双重引力理论来看，全息术已经成为研究强耦合scft的一个框架。对偶SCFT存在于AdS时空的边界上，并且SCFT的对称性等性质是根据重力解（在几何和通量方面）捕获的。项目的这一部分将探索在这些全息对偶性的背景下，规范群和其他更高形式对称结构的不同全局选择的表现形式。该框架的关键优势在于它允许我们在量子场论的强耦合状态下研究这种对称性。该方法将借鉴数学物理的各个领域：微扰量子场论方法，几何工程，以及全息对偶性。这既利用了物理学的见解，也利用了数学的见解，特别是几何和拓扑。令人惊讶的是，考虑到高等形式对称性的基本性质，它的历史相对较短，可以追溯到2014年Gaiotto、Kapustin、Seiberg和Willett发表的一篇开创性论文。高维广义对称的分析实际上是最近才开始的，是一个非常活跃和令人兴奋的研究领域。该项目属于EPSRC数学物理研究领域。

The goal of this project is the comprehensive analysis of higher form symmetries in quantum field theories in various dimensions. Higher form symmetries generalize the concept of `ordinary' symmetries studied widely in quantum field theory, and mathematical physics more broadly. Ordinary symmetries have charged objects given by local operators and charge operators that are topological of codimension one. This concept can be extended to higher dimensional charged objects to describe a higher form symmetry. Many standard results carry over to this generalization, but the broader scope also allows for new physical insights to be obtained by studying the higher form symmetries of a theory.A particular focus will be on conformally invariant theories with supersymmetry. These will be studied starting with the 6d superconformal theories (SCFTs), which upon dimensional reduction and RG-flow give rise to lower dimensional SCFTs. The project will focus on studying the relation between higher form symmetries and the dimensional reduction, in particular from 6d to 5d/4d/3d. Furthermore, the study of higher form symmetries will also require the analysis of 't Hooft anomalies of such global symmetries, as well as mixed anomalies between higher form symmetries. To complement this field theoretic approach, the project will also analyze the imprint of higher form symmetries in the holographic dual descriptions of SCFTs. Holography has emerged as a framework for studying strongly-coupled SCFTs, in terms of a dual gravitational theory in an anti-de Sitter (AdS) spacetime. The dual SCFT lives at the boundary of the AdS spacetime, and properties such as symmetries of the SCFT are captured in terms of the gravity solution (both in the geometry as well as the fluxes). This part of the project will explore the manifestation of the different global choices of gauge groups and other higher form symmetry structures in the context of these holographic dualities. The key advantage to this framework is that it allows us to study such symmetries in the strong coupling regime of quantum field theories. The methodology will draw from a variety of areas of mathematical physics: perturbative quantum field theory methods, geometric engineering, as well as holographic dualities. This utilizes both insights from physics as well as mathematics, in particular geometry and topology. Surprisingly, given their fundamental nature, higher form symmetries have a relatively short history dating back to the seminal paper in 2014, by Gaiotto, Kapustin, Seiberg and Willett. The analysis of generalized symmetries in higher dimensions has in fact only been initiated recently, and is a very active and exciting field of research. This project falls within the EPSRC Mathematical Physics research area.