Crossing with the circle in Dijkgraaf-Witten theory and applications to topological phases of matter
Alex Bullivant, Clement Delcamp
TL;DR
This work develops and applies the crossing with the circle principle to the fully extended 4-3-2-1 Dijkgraaf-Witten TQFT, centering on a lattice Hamiltonian realisation to interpret circle and torus invariants as string-like and bulk loop-like excitations. By using loop groupoids, twisted groupoid algebras, and their categorified centres, the authors derive explicit dimension and decategorification relations across the manifolds $\mathbb{T}^3$, $\mathbb{T}^2$, and $\mathbb{S}^1$, showing that loop-like excitations arise from tracing string-like ones. Key results include braided monoidal equivalences between module categories and centres at successive levels: Mod$(\mathbb{C}[Λ^2 G]^{\mathsf{t}^2(π)})\simeq \mathscr{Z}(\mathsf{Vec}^{\mathsf{t}(π)}_{Λ G})$ and $\mathcal{Z}^{π}_G(\mathbb{S}^1) \simeq \mathsf{MOD}(\mathsf{Vec}^{\mathsf{t}(π)}_{Λ G})$, which together realize the crossing with the circle for all dimensions considered. Consequently, the circle-to-torus-to-circle chain equates the ground-state degeneracies and categorical dimensions with the structure of loop-like and string-like excitations, yielding representations of the linear necklace group and the loop braid group. The framework offers a unified higher-categorical approach to extended excitations in DW theory and suggests generalization to other 3+1D TQFTs built from spherical fusion bicategories.
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 $Σ\times \mathbb S^1$. 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.