# Does the surface topological order on the boundary of 3D topological insulator also have topological ground state degeneracy?

The boundary of a 3D topological insulator can be fully gapped (under strong interaction) by the surface topological order without breaking the symmetry (see Fidkowski-Chen-Vishwanath, Metlitski-Kane-Fisher, Bonderson-Nayak-Qi, Wang-Potter-Senthil). Suppose we make a solid torus shape of 3D topological insulator, and turn on the interaction to gap out the boundary (which is now a torus) by the surface topological order. For 2D topological ordered state placed on a torus, we expect a topological ground state degeneracy equal to the number of anyon types. However does the surface topological ordered 3D topological insulator also have the topological ground state degeneracy? If so how to understand the topological ground state degeneracy emerging from a symmetry protected trivial state which is not expected to have the degeneracy? Also, is the surface topological ground state degeneracy still related to the number of anyon types?

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