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

TL;DR

This paper explores the 'crossing with the circle' conditions in Dijkgraaf-Witten topological quantum field theory, linking quantum invariants of manifolds to excitations in topological phases of matter, and formalizing the formation of loop-like excitations from string-like ones.

Contribution

It computes the 'crossing with the circle' conditions for 4-3-2-1 Dijkgraaf-Witten theory and connects these to the formation of loop-like excitations in topological phases.

Findings

Derived the 'crossing with the circle' conditions for Dijkgraaf-Witten theory.

Linked quantum invariants to defect and bulk excitations.

Formalized the creation of loop-like excitations from string-like excitations.

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.