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

Abstract

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