Crossing with the circle in Dijkgraaf–Witten theory and applications to topological phases of matter

Given a fully extended topological quantum field theory, the “crossing with circle” conditions establish that dimension, or categorification thereof, of invariant assigned to closed k-manifold Σ is equivalent ( k + 1)-manifold [Formula: see text]. We compute in this paper these for 4-3-2-1 Dijkgraaf–Witten theory. In context lattice Hamiltonian realization invariants circle and torus encode defect open string-like bulk loop-like excitations, respectively. The corresponding condition, thus, formalizes process by which excitations are formed out ones. Exploiting result, we revisit statement define representations linear necklace group as well loop braid group.