Higher representations for extended operators

Abstract

It is known that local operators in quantum field theory transform in representations of ordinary global symmetry groups. The purpose of this paper is to generalise this statement to extended operators such as line and surface defects. We explain that $(n-1)$-dimensional operators transform in $n$-representations of a finite invertible or group-like symmetry and thoroughly explore this statement for $n=1,2,3$. We therefore propose higher representation theory as the natural framework to describe the action of symmetries on the extended operator content in quantum field theory.

arXiv comment60 pages + appendix. v2: minor corrections

Cited by in corpus (15)

• Lecture Notes on Generalized Symmetries and Applications
• Anomalies of Non-Invertible Symmetries in (3+1)d
• Subsystem Non-Invertible Symmetry Operators and Defects
• Fusion Surface Models: 2+1d Lattice Models from Fusion 2-Categories
• Topological holography, quantum criticality, and boundary states
• Generalized Charges, Part II: Non-Invertible Symmetries and the Symmetry TFT
• Emergent generalized symmetries in ordered phases and applications to quantum disordering
• Higher-group symmetry of (3+1)D fermionic ${\mathbb{Z}}_2$ gauge theory: logical CCZ, CS, and T gates from higher symmetry
• Fermionic Non-Invertible Symmetries in (1+1)d: Gapped and Gapless Phases, Transitions, and Symmetry TFTs
• Boundary SymTFT
• Symmetry fractionalization, mixed-anomalies and dualities in quantum spin models with generalized symmetries
• Topological aspects of brane fields: solitons and higher-form symmetries
• Higher-Matter and Landau-Ginzburg Theory of Higher-Group Symmetries
• Subsystem Symmetry Fractionalization and Foliated Field Theory
• (2+1)d Lattice Models and Tensor Networks for Gapped Phases with Categorical Symmetry