Generalized Charges, Part I: Invertible Symmetries and Higher Representations

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.

77 pages, v4: minor revision

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• Gapped Phases with Non-Invertible Symmetries: (1+1)d
• Anomalies of Non-Invertible Symmetries in (3+1)d
• Fusion Surface Models: 2+1d Lattice Models from Fusion 2-Categories
• On the Symmetry TFT of Yang-Mills-Chern-Simons theory
• The Club Sandwich: Gapless Phases and Phase Transitions with Non-Invertible Symmetries
• Remarks on Geometric Engineering, Symmetry TFTs and Anomalies
• Generalized Charges, Part II: Non-Invertible Symmetries and the Symmetry TFT
• Emergent generalized symmetries in ordered phases and applications to quantum disordering
• Holographic duals of symmetry broken phases
• Hasse Diagrams for Gapless SPT and SSB Phases with Non-Invertible Symmetries
• Fermionic Non-Invertible Symmetries in (1+1)d: Gapped and Gapless Phases, Transitions, and Symmetry TFTs
• Finiteness and the Emergence of Dualities
• Non-Invertible Symmetry in Calabi-Yau Conformal Field Theories
• Boundary SymTFT
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• Symmetry fractionalization, mixed-anomalies and dualities in quantum spin models with generalized symmetries
• Generalized symmetries in singularity-free nonlinear $\sigma$ models and their disordered phases
• Fusion 3-Categories for Duality Defects
• Category of SET orders
• Topological aspects of brane fields: solitons and higher-form symmetries
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• Subsystem Symmetry Fractionalization and Foliated Field Theory
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• Projective Representations, Bogomolov Multiplier, and Their Applications in Physics
• Symmetry topological field theory and non-abelian Kramers-Wannier dualities of generalised Ising models
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• Tube Category, Tensor Renormalization and Topological Holography