On the action of non-invertible symmetries on local operators in 3+1d

TL;DR

This paper investigates how non-invertible symmetries act on local operators in 3+1 dimensions, revealing that without topological line operators, their action is necessarily invertible, and explores conditions for anomaly-freeness.

Contribution

It demonstrates that non-invertible symmetries without topological line operators act invertibly on local operators and provides criteria for their anomaly-freeness in 3+1d.

Findings

Non-invertible symmetries without topological line operators act invertibly on local operators.

Decomposition of non-invertible symmetry actions into invertible parts and gauging interfaces.

Necessary conditions for anomaly-freeness of non-invertible symmetries in 3+1d.

Abstract

Most of the known non-invertible symmetries of quantum field theories in three and four spacetime dimensions act invertibly on local operators. An exception is coset symmetries, which can be constructed from gauging a non-normal subgroup of an invertible symmetry. In this paper, we study the action of a general finite non-invertible symmetry on local operators in four dimensions. We show that non-invertible symmetries without topological line operators necessarily act invertibly on local operators. Using this result, we argue that the action of a general non-invertible symmetry in 3+1d on local operators can be decomposed into the invertible action of some operators composed with the action of a gauging interface. We use this result to study when such a symmetry is anomaly-free. We find a necessary condition for a finite non-invertible symmetry in 3+1d to be anomaly-free, and show that anomaly-free non-invertible symmetries without topological line operators are non-intrinsically non-invertible.