This paper extends a standard algebraic tool called the tube algebra so it can describe point-like defects that sit on boundaries or domain walls in two-dimensional gapped topological phases. The authors work in the setting of Turaev–Viro–Barrett–Westbury topological quantum field theories and Levin–Wen string-net models. Their main claim is that such codimension-2 defects are captured by a “defect tube algebra” that is a comodule algebra over the usual weak Hopf tube algebra used for bulk excitations. Each defect type then appears as a representation of the corresponding defect tube algebra.

At a high level, a tube algebra is built by surrounding a localized excitation with a small annulus or “tube” and defining multiplication by gluing tubes together. For bulk excitations this construction produces a weak Hopf algebra with both multiplication and a coproduct (a kind of algebraic operation that encodes fusion). The key difference the authors emphasize is that codimension-2 defects are not fusion-closed: fusing two defects does not in general give another defect. Because of that, the coproduct present for bulk excitations disappears in the defect sector. Instead, the defect tube algebra carries a compatible coaction of the bulk tube algebra. In plain terms, defects can absorb or emit bulk excitations, and this interaction is encoded by the comodule structure (a coaction is like an action but for the coalgebra side).

The paper applies this idea to both boundary defects and domain-wall defects. For a domain wall joining two bulk phases, the domain-wall defect tube algebra can be seen as a generalized Drinfeld double of the two boundary defect tube algebras that lie on either side. More generally, when a point defect joins N different bulk phases, the associated algebra becomes an N-tuple algebra with a multicomodule structure. The authors give a general theorem summarizing these results and present explicit constructions, including a lattice Hamiltonian formulation for defects and examples such as the toric code and finite-group models.