A Representation Theory for Categorical Symmetries

Abstract

Symmetries play a fundamental role in our understanding of nature. While traditionally described by the theory of groups and their representations, recent years have seen a vast generalisation of the notion of symmetry, leading to so-called generalised (or categorical) symmetries in quantum field theory. In this thesis, we examine the mathematical structure that underlies such generalised symmetries and develop a representation theory that captures their action on physical observables. We focus on the case of finite bosonic symmetries in low spacetime dimensions, where the appropriate mathematical framework is given by the theory of (higher) fusion categories. We construct higher-dimensional analogues of Ocneanu’s tube algebra and classify their higher representations using the so-called sandwich construction (or Symmetry TFT) for categorical symmetries. We provide explicit examples that include both anomalous group-like symmetries as well as non-invertible symmetries.