On local fields invariant under the action of topological defects

In the context of rational conformal field theories (RCFT) we look into the problem of constructing and classifying pairs consisting of a local operator and a topological defect which commutes or anticommutes with it. We discuss the bulk and boundary versions of the problem. In the latter one considers a conformal boundary condition, a boundary operator on it and a junction with a topological defect. In the case of the charge conjugation modular invariant commuting configurations in each problem can be obtained when a certain restriction on the fusion rules in realised. We study the corresponding fusion rule problems in detail. While in the bulk case it reduces to realising the $a\times b = c$ fusion rule which was studied in arXiv:2012.14689 [hep-th], in the boundary it leads to a new type of problem. We obtain a full solution to this problem for the $\mathrm{SU(3)}$ WZW theory, thus constructing a class of commuting boundary operators and junctions in that theory, and suggest an approach to general WZW theories.