Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 342 / 1 / a / Solution

At a boundary parallel to lattice links, define each star as the product over links of the retained lattice incident on the vertex,A boundary vertex has three rather than four retained incident links. For a plaquette adjacent to the boundary, retainincluding its boundary link. A boundary link belongs to only one bulk plaquette, so applying $X$ on that link flips one $B_p$ rather than two. A magnetic string made from $X$ operators can therefore terminate at the boundary and its endpoint can be created or removed by a local boundary operator. This is precisely an $m$-condensing, or magnetic, boundary. By contrast, the same local operation does not permit an isolated electric endpoint.