Unitary Categorical Symmetries

TL;DR

This paper generalizes the concept of global invertible symmetries in quantum systems to non-invertible symmetries using higher fusion categories and introduces a framework involving tube algebras and higher S-matrices to classify their actions.

Contribution

It introduces a novel approach to describe non-invertible symmetries in quantum systems via $ ext{C}^*$-representations of tube algebras and higher S-matrices, extending symmetry theory.

Findings

Classified $ ext{C}^*$-representations of tube algebras for higher fusion categories.

Established a connection between higher braided fusion categories and symmetry actions.

Provided a framework for understanding non-invertible symmetries in quantum systems.

Abstract

Global invertible symmetries act unitarily on local observables or states of a quantum system. In this note, we aim to generalise this statement to non-invertible symmetries by considering unitary actions of higher fusion category symmetries $\mathcal{C}$ on twisted sector local operators. We propose that the latter transform in $\ast$-representations of the tube algebra associated to $\mathcal{C}$, which we introduce and classify using the notion of higher $S$-matrices of higher braided fusion categories.