Topological and nontopological degeneracies in generalized string-net models

Generalized string-net models have been proposed recently in order to enlarge the set of possible topological quantum phases emerging from the original string-net construction. In the present work we do not consider vertex excitations, and we restrict ourselves to plaquette excitations, or fluxons, that satisfy important identities. We explain how to compute the energy-level degeneracies of the generalized string-net Hamiltonian associated with an arbitrary unitary fusion category. In contrast to the degeneracy of the ground state, which is purely topological, the degeneracy of excited energy levels depends not only on the Drinfeld center of the category, but also on internal multiplicities obtained from the tube algebra defined from the category. For a noncommutative category, these internal multiplicities result in extra nontopological degeneracies. Our results are valid for any trivalent graph and any orientable surface. We illustrate our findings with nontrivial examples.