Algebraic locality and non-invertible Gauss laws

TL;DR

This paper investigates algebraic locality principles in 2+1D lattice models with non-invertible symmetries, revealing conditions under which Haag duality and disjoint additivity are preserved or require modifications.

Contribution

It extends prior work on invertible symmetries to non-invertible symmetries, demonstrating how Haag duality and disjoint additivity behave in these more general settings.

Findings

Haag duality is preserved exactly for cuspless regions.

A weak form of Haag duality applies to cusped regions with a collar.

Disjoint additivity holds for group-based models, with a weakened form for Hopf algebra models.

Abstract

We study algebraic locality principles on a 2+1D closed lattice in the presence of a Gauss law for a non-invertible symmetry. Prior work in arXiv:2509.03589 showed that when enforcing the Gauss law of an invertible symmetry, the principle of "Haag duality" is preserved exactly, and "disjoint additivity" is preserved after appropriate treatment of discreteness artifacts. Here we show that for a large class of non-invertible on-site symmetries, Haag duality is preserved exactly only for sufficiently nice, "cuspless" regions. For cusped regions, we instead have a weak form of Haag duality that requires adding a collar. Our results apply to double models with a purely magnetic constraint, and to the more general framework of constraints induced by the on-site action of a Hopf algebra. In particular, we treat a class of extended string-net models explicitly. We also demonstrate disjoint additivity for double models based on a group, and a weakened form of disjoint additivity in the setting of a general Hopf algebra.