Point-group symmetry enriched topological orders

TL;DR

This paper classifies two-dimensional topological orders enriched by point-group symmetries using a folding approach, analyzing boundary and junction properties to understand symmetry enrichment and obstructions.

Contribution

It introduces a novel framework for classifying point-group symmetry enriched topological orders via boundary and junction analysis, including new obstruction classifications.

Findings

Classification of symmetry-enriched topological orders using cohomology.

Identification of $H^1$ and $H^2$ obstructions at junctions.

Connection between boundary classifications and symmetry charges.

Abstract

We study the classification of two-dimensional (2D) topological orders enriched by point-group symmetries, by generalizing the folding appraoch which was previously developed for mirror-symmetry-enriched topological orders. We fold the 2D plane hosting the topological order into the foundamental domain of the group group, which is a sector with an angle $2\pi/n$ for the cyclic point group $C_n$ and a sector with an angle $\pi/n$ for the dihedral point group $D_{2n}$, and the point-group symmetries becomes onsite unitary symmetries on the sector. The enrichment of the point-group symmetries is then fully encoded at the boundary of the sector and the apex of the section, which forms a junction between the two boundaries. The mirror-symmetry enrichment encoded on the boundaries is analyzed by the classification theory of symmetric gapped boundaries, and the point-group-symmetry enrichment encoded on the junction is analyzed by a framework for classifying symmetric gapped junctions between boundaries which we develop in this work. We show that at the junction, there are two potential obstructions, which we refer to as an $H^1$ obstruction and an $H^2$ obstruction, respectively. When the obstruction vanishes, the junction, and therefore the point-group-symmetry-enriched topological orders, are classified by an $H^0$ cohomology class and an $H^1$ cohomology class, which can be understood as an additional Abelian anyon and a symmetry charge attached to the rotation center, respectively. These results are consistent with the classification of onsite-symmetry-enriched topological orders, where the $H^1$ and $H^2$ obstructions and the junction corresponds to the $H^3$ and $H^4$ obstructions for onsite symmetries, respectively.