Partition function of the Kitaev quantum double model

We compute the degeneracy of energy levels in the Kitaev quantum double model for any discrete group $G$ on any planar graph forming the skeleton of a closed orientable surface of arbitrary genus. The derivation is based on the fusion rules of the properly identified vertex and plaquette excitations, which are selected among the anyons, i.e., the simple objects of the Drinfeld center $\mathcal{Z}({\mathrm{V}\mathrm{e}\mathrm{c}}_G)$. These degeneracies are given in terms of the corresponding $S$-matrix elements and allow one to obtain the exact finite-temperature partition function of the model, valid for any finite-size system.