ICTP lectures on (non-)invertible generalized symmetries
ICTP lectures on (non-)invertible generalized symmetries
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DOI：
10.1016/j.physrep.2024.01.007
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发表时间：
2024-04
期刊：
影响因子：
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通讯作者：
S. Schäfer-Nameki
中科院分区：
文献类型：
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作者：
S. Schäfer-Nameki
在过去的10年里，量子场论(QFT)的整体对称性已经得到了极大的扩展，不仅包括作用于更高维缺陷的对称性，而且最近还包括没有逆的对称性。使这一推广成为可能的原则是确定QFT中具有拓扑缺陷的对称性。在这些讲座中，我们介绍了广义对称，重点是不可逆对称。我们首先简要概述可逆广义对称，包括高形对称和高群对称，然后讨论不可逆对称。许多不可逆对称的构造背后的主要思想是将QFT与拓扑QFT(TQFT)叠加，然后度量对角非反常的全局对称性。TQFT成为规范理论中的拓扑缺陷，称为(扭曲的)theta缺陷，并包括一大类不可逆对称，包括凝聚缺陷、自对偶缺陷和规范理论的具有不连通规范群的不可逆对称。我们将解释一般原则，并提供许多具体的例子。在对称生成元的这种广泛的刻画之后，我们接着讨论它们在更高电荷上的作用，即扩展的物理算符。正如我们将解释的那样，即使对于可逆的更高形式的对称，这些也不仅是p-形式对称群的表示，而且更一般地被称为更高形式的表示。最后，我们介绍了对称拓扑场论(SymTFT)及其在刻画对称性、对称性度量和广义电荷方面的应用。
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, ie 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.