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

• URL: http://arxiv.org/abs/2103.12717v1
• Date: Tue, 23 Mar 2021 17:37:24 GMT
• Title: Crossing with the circle in Dijkgraaf-Witten theory and applications to topological phases of matter
• Authors: Alex Bullivant, Clement Delcamp
• Abstract summary: We compute these conditions for the 4-3-2-1 Dijkgraaf-Witten theory. In the context of the lattice Hamiltonian realisation of the theory, the quantum invariants assigned to the circle and the torus encode the defect open string-like and bulk loop-like excitations.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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 $\Sigma$ is equivalent to that assigned to the ($k$+1)-manifold $\Sigma \times \mathbb S^1$. We compute in this manuscript these conditions for the 4-3-2-1 Dijkgraaf-Witten theory. In the context of the lattice Hamiltonian realisation 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 formalises 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.

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