A remark on gapped domain walls between topological phases

We give a mathematical definition of a gapped domain wall between topological phases and a gapped boundary of a topological phase. We then provide answers to some recent questions studied by Lan, Wang and Wen in condensed matter physics based on works of Davydov, M"uger, Nikshych and Ostrik. In particular, we identify their tunneling matrix and a coupling matrix of Rehren, and show that their conjecture does not hold.