The brief answer is yes, they do have topological degeneracy. My understanding is that if you only focus on the surface topological order with some symmetry given by the bulk symmetry protected topological (SPT) state, there is nothing wrong with that symmetry enriched topological order (SET) states. That means they also have all the properties as the usual SET states in pure 2D. However, if you try to gauge the surface SET, you will find the obstruction to define a consistent gauged theory. More precisely, the gauged theory won't satisfy the pentagon equation. Those obstructions are classified by the forth cohomology group, $H^{4}(G,U(1))$, where $G$ is the symmetry group of the bulk SPT. Usually, people say those SETs are anomalous. More details can be found in this paper http://arxiv.org/abs/1403.6491

This post imported from StackExchange Physics at 2015-02-05 10:19 (UTC), posted by SE-user Sheng-Jie Huang