An Introduction to Higher-Form Symmetries

TL;DR

This paper provides a comprehensive pedagogical overview of higher-form symmetries, exploring their role in gauge theories, including Abelian and non-Abelian cases, and discussing symmetry breaking phenomena.

Contribution

It introduces the concept of higher-form symmetries, detailing their application in various gauge theories and explaining their physical implications and breaking mechanisms.

Findings

Higher-form symmetries are fundamental in understanding extended operators in gauge theories.

Wilson and 't Hooft operators serve as charged objects for these symmetries.

Spontaneous breaking of higher-form symmetries has significant physical consequences.

Abstract

These notes are intended to be a pedagogical introduction to higher-form symmetries, which are symmetries whose charged objects are extended operators supported on lines, surfaces, and etc. This subject has been one of the most popular and effervescent topics of theoretical physics in recent years. Gauge theories are central in the study of higher-form symmetries, with Wilson and 't Hooft operators corresponding to the charged objects. Along these notes, we discuss in detail some basic aspects, including Abelian Maxwell and Chern-Simons theories, and $SU(N)$ non-Abelian gauge theories. We also discuss spontaneous breaking of higher-form symmetries.