Higher representations for extended operators

TL;DR

<3-5 sentence high-level summary>Higher representations for extended operators develops a comprehensive framework for how extended operators in quantum field theory transform under finite higher-group symmetries. The authors formulate and classify actions of 1-, 2-, and 3-representations across local, line, and surface defects using elementary, induction, and categorical viewpoints, tying these to gapped boundary conditions and higher TQFTs. They provide concrete data characterizing irreducible representations: for 1-representations via subgroups and 2-cocycles $(H,u)$, for 2-representations via $(H,u,\lambda)$ and their induction from subgroups, and for 3-representations via $(H,\rho,\kappa,\mu)$ with related categorical constructions; these are illustrated with gauge theories and mixed anomalies. The framework unifies symmetry actions on extended operator content, clarifies how anomalies and higher-form symmetries enter, and offers a solid toolset for analyzing discrete 't Hooft anomalies and related phenomena in diverse dimensions.

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.

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