ICTP Lectures on (Non-)Invertible Generalized Symmetries

TL;DR

This paper introduces generalized symmetries in Quantum Field Theories, focusing on non-invertible symmetries, their construction via topological quantum field theories, and their implications for higher-charges and symmetry classification.

Contribution

It provides a comprehensive overview of non-invertible symmetries, including their construction, examples, and the role of Symmetry Topological Field Theory in their analysis.

Findings

Classification of non-invertible symmetries via topological defects

Construction of non-invertible symmetries using TQFTs and gauging

Introduction of higher-representations for higher-form symmetries

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.