Topological symmetry in quantum field theory

TL;DR

This paper develops a comprehensive framework for internal topological symmetries in quantum field theory, including noninvertible and categorical symmetries, and explores their mathematical structure and physical implications.

Contribution

It introduces a new calculus for topological defects in quantum field theory, emphasizing finite symmetries and providing tools for their analysis and generalization.

Findings

Framework for noninvertible and categorical symmetries

Calculus of topological defects in quantum field theory

Connections to algebraic topology and homotopy theories

Abstract

We introduce a framework for internal topological symmetries in quantum field theory, including "noninvertible symmetries" and "categorical symmetries". This leads to a calculus of topological defects which takes full advantage of well-developed theorems and techniques in topological field theory. Our discussion focuses on finite symmetries, and we give indications for a generalization to other symmetries. We treat quotients and quotient defects (often called "gauging" and "condensation defects"), finite electromagnetic duality, and duality defects, among other topics. We include an appendix on finite homotopy theories, which are often used to encode finite symmetries and for which computations can be carried out using methods of algebraic topology. Throughout we emphasize exposition and examples over a detailed technical treatment.