Ground-state degeneracy in the Levin-Wen model for topological phases

We study the properties of topological phases by calculating the ground-state degeneracy (GSD) of the two-dimensional Levin-Wen (LW) model. Here it is explicitly shown that the GSD depends only on the spatial topology of the system. Then we show that the ground state on a sphere is always nondegenerate. Moreover, we study an example associated with a quantum group, and show that the GSD on a torus agrees with that of the doubled Chern-Simons theory, which is consistent with the conjectured equivalence between the LW model associated with a quantum group and the doubled Chern-Simons theory.