Anomalies in mirror symmetry enriched topological orders

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.