Higher representations for extended operators

TL;DR

This paper extends the concept of symmetry representations in quantum field theory to include higher-dimensional operators like lines and surfaces, proposing higher representation theory as the appropriate framework.

Contribution

It generalizes the notion of symmetry representations to extended operators and explores this in the context of 1-, 2-, and 3-dimensional operators using higher representation theory.

Findings

Extended operators transform in higher representations of symmetries.

Higher representation theory naturally describes symmetry actions on extended operators.

The framework applies to finite invertible or group-like symmetries.

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.