Generalized charges, part I: Invertible symmetries and higher representations

Abstract

qq-charges describe the possible actions of a generalized symmetry on qq-dimensional operators. In Part I of this series of papers, we describe qq-charges for invertible symmetries; while the discussion of qq-charges for non-invertible symmetries is the topic of Part II. We argue that qq-charges of a standard global symmetry, also known as a 0-form symmetry, correspond to the so-called (q+1)(q+1)-representations of the 0-form symmetry group, which are natural higher-categorical generalizations of the standard notion of representations of a group. This generalizes already our understanding of possible charges under a 0-form symmetry! Just like local operators form representations of the 0-form symmetry group, higher-dimensional extended operators form higher-representations. This statement has a straightforward generalization to other invertible symmetries: qq-charges of higher-form and higher-group symmetries are (q+1)(q+1)-representations of the corresponding higher-groups. There is a natural extension to higher-charges of non-genuine operators (i.e. operators that are attached to higher-dimensional operators), which will be shown to be intertwiners of higher-representations. This brings into play the higher-categorical structure of higher-representations. We also discuss higher-charges of twisted sector operators (i.e. operators that appear at the boundary of topological operators of one dimension higher), including operators that appear at the boundary of condensation defects.