[论文解读] ICTP Lectures on (Non-)Invertible Generalized Symmetries

Sakura Schäfer‐Nameki|arXiv (Cornell University)|May 29, 2023

Noncommutative and Quantum Gravity Theories被引用 20

一句话总结

介绍广义全局对称性，聚焦于 d≥3 的量子场论中的不可逆对称性（非可逆对称性），通过与拓扑量子场论（TQFT）叠加构造、theta缺陷/凝缩缺陷，以及对称性TFT框架来实现。

ABSTRACT

What comprises a global symmetry of a Quantum Field Theory (QFT) has been vastly expanded in the past 10 years to include not only symmetries acting on higher-dimensional defects, but also most recently symmetries which do not have an inverse. The principle that enables this generalization is the identification of symmetries with topological defects in the QFT. In these lectures, we provide an introduction to generalized symmetries, with a focus on non-invertible symmetries. We begin with a brief overview of invertible generalized symmetries, including higher-form and higher-group symmetries, and then move on to non-invertible symmetries. The main idea that underlies many constructions of non-invertible symmetries is that of stacking a QFT with topological QFTs (TQFTs) and then gauging a diagonal non-anomalous global symmetry. The TQFTs become topological defects in the gauged theory called (twisted) theta defects and comprise a large class of non-invertible symmetries including condensation defects, self-duality defects, and non-invertible symmetries of gauge theories with disconnected gauge groups. We will explain the general principle and provide numerous concrete examples. Following this extensive characterization of symmetry generators, we then discuss their action on higher-charges, i.e. extended physical operators. As we will explain, even for invertible higher-form symmetries these are not only representations of the $p$-form symmetry group, but more generally what are called higher-representations. Finally, we give an introduction to the Symmetry Topological Field Theory (SymTFT) and its utility in characterizing symmetries, their gauging and generalized charges. Lectures prepared for the ICTP Trieste Spring School, April 2023.

研究动机与目标

• 动机化并将广义全局对称性在量子场论中定义为拓扑算子。
• 解释可逆与不可逆对称性结构，包括融合规则和更高范畴语言。
• 给出通过叠加 TQFT 和积分（gauging）在 d=2、d=3、d=4 中构造不可逆对称性的方法。
• 介绍 theta-defects、凝缩缺陷以及更高维度的对称作用的概念。
• 讨论 Symmetry Topological Field Theory (SymTFT) 作为编码对称性和广义电荷的框架。

提出的方法

• 将对称性描述为不同余维度的拓扑算子及其融合规则。
• 讨论高形式对称性和0-形式对称性的辩认（gauging）以及不可逆结构的产生。
• 解释在进行 gauging 之前将理论与 S 对称的 TQFTs 叠加以产生 theta-defects。
• 以 2d RCFTs 和 3d–4d 规范理论作为不可逆对称性生成的具体例子。
• 引入拓扑缺陷的层次结构及由此产生的融合高范畴框架。
• 将 SymTFT (d+1) 作为组织对称性和广义电荷的工具。

实验结果

研究问题

• RQ1 通过叠加和 gauging，在高维量子场论中如何产生不可逆对称性？
• RQ2 theta-defects 和凝缩缺陷在产生不可逆对称性中起到什么作用？
• RQ3 如何将高形式对称性和高群对称性推广到非群结构？
• RQ4 在编码对称性和广义电荷方面，Symmetry Topological Field Theory 的作用是什么？

主要发现

• 通过将理论与 TQFT 叠加并对对角对称性进行 gauging 来产生 theta-defects，可以生成不可逆对称性。
• theta-defects 显示不可逆的融合关系，例如在 3d 例子中 D2(Z2) ⊗ D2(Z2) = 2 D2(Z2)。
• 凝缩缺陷等同于 theta-defects，并提供普适的不可逆对称性机制。
• 不可逆对称性存在于 2d RCFTs、高维度的规范理论，以及具有分离的规范群的量子场论中。
• 广义对称性对高阶电荷的作用涉及高表示范畴，而非简单的群表示。
• SymTFT 提供一个跨维度编码对称性和广义电荷的普遍框架。