Higher categorical symmetries and gauging in two-dimensional spin systems

TL;DR

The paper develops a systematic framework to study higher categorical symmetries in 2D spin systems by modeling symmetries with fusion 2-categories and module 2-categories. It shows how dualities arise from Morita duals and twisted gauging, and provides explicit lattice constructions of symmetry operators and duality maps. Through concrete finite-group examples (Z2, Klein four, S3), it demonstrates both invertible and non-invertible surface operators and their line content, revealing rich symmetry structures beyond conventional groups. The work connects abstract higher representation theory with tangible lattice models, enabling controlled exploration of dualities, condensations, and boundary phenomena in two-dimensional quantum systems.

Abstract

We present a framework to systematically investigate higher categorical symmetries in two-dimensional spin systems. Though exotic, such generalised symmetries have been shown to naturally arise as dual symmetries upon gauging invertible symmetries. Our framework relies on an approach to dualities whereby dual quantum lattice models only differ in a choice of module 2-category over some input fusion 2-category. Given an arbitrary two-dimensional spin system with an ordinary symmetry, we explain how to perform the (twisted) gauging of any of its sub-symmetries. We then demonstrate that the resulting model has a symmetry structure encoded into the Morita dual of the input fusion 2-category with respect to the corresponding module 2-category. We exemplify this approach by specialising to certain finite group generalisations of the transverse-field Ising model, for which we explicitly define lattice symmetry operators organised into fusion 2-categories of higher representations of higher groups.