Notes on generalized global symmetries in QFT

TL;DR

This paper explores the mathematical structure of generalized global symmetries in quantum field theories, extending the concept to higher groups and discussing their implications for topological defects and anomalies.

Contribution

It provides a mathematical framework for understanding higher-form symmetries as higher groups and discusses their physical implications in quantum field theories.

Findings

Higher-form symmetries are special cases of 2-groups and higher groups.

Examples of quantum field theories with higher group symmetries are discussed.

A potential link between anomalies and higher group symmetries is proposed.

Abstract

It was recently argued that quantum field theories possess one-form and higher-form symmetries, labelled `generalized global symmetries.' In this paper, we describe how those higher-form symmetries can be understood mathematically as special cases of more general 2-groups and higher groups, and discuss examples of quantum field theories admitting actions of more general higher groups than merely one-form and higher-form symmetries. We discuss analogues of topological defects for some of these higher symmetry groups, relating some of them to ordinary topological defects. We also discuss topological defects in cases in which the moduli `space' (technically, a stack) admits an action of a higher symmetry group. Finally, we outline a proposal for how certain anomalies might potentially be understood as describing a transmutation of an ordinary group symmetry of the classical theory into a 2-group or higher group symmetry of the quantum theory, which we link to WZW models and bosonization.