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Generalized Charges, Part I: Invertible Symmetries and Higher Representations
by Lakshya Bhardwaj, Sakura Schäfer-Nameki
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Submission summary
| Authors (as registered SciPost users): | Lakshya Bhardwaj |
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| Preprint Link: | https://arxiv.org/abs/2304.02660v4 (pdf) |
| Date accepted: | 2024-03-12 |
| Date submitted: | 2024-03-05 07:47 |
| Submitted by: | Bhardwaj, Lakshya |
| Submitted to: | SciPost Physics |
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| Academic field: | Physics |
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Abstract
$q$-charges describe the possible actions of a generalized symmetry on $q$-dimensional operators. In Part I of this series of papers, we describe $q$-charges for invertible symmetries; while the discussion of $q$-charges for non-invertible symmetries is the topic of Part II. We argue that $q$-charges of a standard global symmetry, also known as a 0-form symmetry, correspond to the so-called $(q+1)$-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: $q$-charges of higher-form and higher-group symmetries are $(q+1)$-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.
Author comments upon resubmission
List of changes
1) This is an important distinction that we have clarified in footnote 3 on page 22.
2) We do not know of an example of a defect exhibiting this 2-charge, but we have provided a suggestion in Example 2.6 on how such a 2-charge may be produced.
Published as SciPost Phys. 16, 093 (2024)