Towards All Categorical Symmetries in 2+1 Dimensions

TL;DR

This paper develops a comprehensive framework for classifying all categorical symmetries in 2+1 dimensional oriented field theories with finite symmetry groups, using advanced mathematical structures like fusion categories and topological quantum field theories.

Contribution

It introduces a rigorous classification scheme for all possible categorical symmetries in 2+1D theories via twisted crossed extensions and anomaly cancellation, capturing all such symmetries.

Findings

Framework captures all categorical symmetries in 2+1D theories.

Classification achieved through enumeration of topological quantum field theories.

Uses advanced mathematical tools like fusion categories and 3-representations.

Abstract

We investigate the most general gauging operations in 2+1 dimensional oriented field theories with finite symmetry groups, which correspond to gapped boundary conditions in 3+1 dimensional Dijkgraaf-Witten theory. The classification is achieved by enumerating 2+1 dimensional oriented topological quantum field theories that cancel the 't Hooft anomaly associated with the symmetry. This framework is rigorously formulated using twisted crossed extensions of modular fusion categories and projective 3-representations. Additionally, we explore the resulting fusion 2-category symmetries and argue that this framework captures all possible categorical symmetries in 2+1 dimensional oriented field theories.