Anomalies in mirror symmetry enriched topological orders

TL;DR

This paper classifies mirror symmetry enriched topological orders by analyzing gapped boundaries in a folded bilayer system, revealing obstructions related to cohomology and constructing states on 3D mirror SET surfaces.

Contribution

It introduces a classification framework for mirror symmetry enriched topological orders using anyon condensation and cohomology obstructions, extending to 3D surface states.

Findings

Derived an $H^2$ obstruction function for gapped boundaries

Connected $H^2$ obstructions to $H^3$ obstructions in time-reversal cases

Constructed surface states with nontrivial $H^2$ obstructions on 3D mirror SETs

Abstract

Two-dimensional mirror symmetry enriched topological (SET) orders can be studied using the folding approach: it can be folded along the mirror axis and turned into a bilayer system on which the mirror symmetry acts as a $\mathbb Z_2$ layer-exchange symmetry. How mirror symmetry enriches the topological order is then encoded at the mirror axis, which is a gapped boundary of the folded bilayer system. Based on anyon-condensation theory, we classify possible $\mathbb Z_2$-symmetric gapped boundaries of the folded system. In particular, we derive an $H^2$ obstruction function, which corresponds to an $H^3$ obstruction for topological orders enriched by the time-reversal symmetry instead of mirror symmetry. We demonstrate that states with a nontrivial $H^2$ obstruction function can be constructed on the surface of a three-dimensional mirror SET order.