Topological symmetry in quantum field theory

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.Comment: 65 pages, 37 figures. v2 adds references and corrects topological misstatements in section 4.4.2. v3 is a major revision with improved exposition throughout, a new section 2.5 on the passage from local to global defects, substantially expanded sections 4.4 and A.