Matrix product operator symmetries and intertwiners in string-nets with domain walls

We provide a description of virtual non-local matrix product operator (MPO) symmetries in projected entangled pair state (PEPS) representations string-net models. Given such PEPS representation, we show that the consistency conditions its MPO amount to set six coupled equations can be identified with pentagon bimodule category. This allows us classify all equivalent and build intertwiners between them, synthesising generalising wide variety tensor network topological phases. Furthermore, use this generalisation explicit realisations domain walls different phases as constructed by Kitaev Kong [Commun. Math. Phys. 313 (2012) 351-373]. While prevailing abstract categorical approach is sufficient describe structure phases, are required simulate these systems on computer, needed for calculating thresholds quantum error-correcting codes based string-nets boundaries. Finally, understood specific instances Turaev-Viro state-sum models field theory three-manifolds physical boundary, thereby putting constructions mathematically rigorous footing.