Cardy states, defect lines and chiral operators of coset CFTs on the lattice

We construct Cardy states, defect lines and chiral operators for rational coset conformal field theories on the lattice. The bulk theory is obtained by taking the overlap between tensor network representations of different string-nets, while the primary fields emerge from using the topological superselection sectors of the anyons in the original topological theory. This mapping provides an explicit manifestation of the equivalence between conformal field theories in two dimensions and topological field theories in three dimensions: their groundstates and elementary excitations are represented by exactly the same tensors.