Introduction to Generalized Symmetries

TL;DR

This paper provides an accessible introduction to generalized symmetries, including higher-form, non-invertible, and categorical symmetries, with applications across various dimensions and physical theories.

Contribution

It offers a comprehensive, self-contained overview of generalized symmetries, emphasizing non-invertible symmetries and fusion categories in both 1+1 and 3+1 dimensions.

Findings

Introduction of non-invertible symmetries in (1+1)D systems

Discussion of fusion categories and their physical applications

Extension of symmetry concepts to (3+1)D theories

Abstract

These notes were prepared for a series of intensive lectures delivered at Hokkaido University, Nagoya University, Kyoto University, and Kyushu University. We begin with a brief review of higher-form symmetries, anomalies, and discrete gauge theories, before introducing non-invertible symmetries in $(1+1)$-dimensional systems. The basic structure of fusion categories is then discussed, including a discussion of categorical analogs of discrete gauging and representation theory. We subsequently turn to $(3+1)$-dimensional theories, where several physical applications of non-invertible symmetries are discussed. These notes are intended to be largely self-contained, and require no prior familiarity with subjects such as conformal field theory or lattice models.