[Paper Review] Generalized Charges, Part I: Invertible Symmetries and Higher Representations

Lakshya Bhardwaj, Sakura Schäfer‐Nameki|arXiv (Cornell University)|Apr 5, 2023

Molecular spectroscopy and chiralityChemistry72 references16 citations

TL;DR

The paper develops the concept that q-charges of invertible generalized symmetries are (q+1)-representations, extending charges from local to extended operators via higher-category theory, with explicit treatment for 0- and 1-form symmetries and twisted sectors.

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.

Motivation & Objective

• Motivate and define generalized charges for invertible symmetries across form degrees (0-form, higher-form, and higher-groups).
• Show that q-charges of a G^{(p)} p-form symmetry are (q+1)-representations of the associated (p+1)-group, generalizing ordinary representations.
• Illustrate the framework with explicit examples and set up higher-categorical structures for genuine and non-genuine charges.
• Discuss twisted generalized charges and the role of anomalies in shaping these charges.
• Lay groundwork for Part II, which will address non-invertible (categorical) symmetries.

Proposed method

• Argue physically that q-charges arise as (q+1)-representations of the symmetry structure (groups or higher-groups).
• Relate charges of 0-form symmetries to 2-representations, and generalize to higher q via higher-representations (appendix B).
• Describe the layering of genuine and non-genuine operators and map it to (q+1)-categories and morphisms.
• Analyze twisted sectors arising from symmetry generators and condensation defects through twisted higher-representations (e.g., D_{d-1}^{(g)} boundaries).
• Use examples such as 4d Maxwell theory to illustrate 0-, 1-, and higher-charges under 0-form symmetries.
• Outline the connection to the Drinfeld center and preparation for non-invertible symmetry generalizations in Part II.

Experimental results

Research questions

• RQ1 What is the appropriate higher-categorical generalization of charges for q-dimensional operators under invertible symmetries?
• RQ2 How do q-charges of a 0-form symmetry relate to (q+1)-representations of G^{(0)} and, more generally, to (p+1)-representations of higher groups for p-form symmetries?
• RQ3 How do twisted sectors and ’t Hooft anomalies modify generalized charges, particularly for twisted charges associated with symmetry generators and condensation defects?
• RQ4 What is the structure and interpretation of non-genuine charges within this higher-categorical framework?
• RQ5 How can these ideas be extended to non-invertible (categorical) symmetries via the Drinfeld center in Part II?

Key findings

• q-Charges of a G^{(0)} 0-form symmetry are (q+1)-representations of G^{(0)} for all q, with 0-charges recovering ordinary representations.
• 1-charges for 0-form symmetries are 2-representations of G^{(0)}, realized via the induced 0-form symmetry on a line operator and its stabilizer with a possible ’t Hooft anomaly.
• Higher q-charges under G^{(0)} correspond to higher-representations, a natural higher-categorical generalization of group representations.
• For p-form and higher-group symmetries, q-charges are (q+1)-representations of the associated (p+1)-group, capturing the actions on extended operators and their interactions.
• Twisted generalized charges arise from twisted sectors of symmetry generators and condensation defects, described by twisted higher-representations or their homological data.
• The layered structure of genuine and non-genuine charges maps to a (q+1)-category framework, linking charges to higher-morphisms and intertwiners.