Siyuan Wang, Yanyan Chen, Hongyu Wang, Yuting Hu, Yidun Wan
SciPost Phys. 19, 018 (2025) ·
published 17 July 2025
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In this paper, we apply the method of Fourier transform and basis rewriting developed in [H. Wang et al., J. High Energy Phys. 02, 030 (2020)] for the two-dimensional quantum double model of topological orders to the three-dimensional gauge theory model (with a gauge group $G$) of three-dimensional topological orders. We find that the gapped boundary condition of the gauge theory model is characterized by a Frobenius algebra in the representation category $Rep(G)$ of $G$, which also describes charge splitting and condensation on the boundary. We also show that our Fourier transform maps the three-dimensional gauge theory model with input data $G$ to the Walker-Wang model with input data $Rep(G)$ on a trivalent lattice with dangling edges, after truncating the Hilbert space by projecting all dangling edges to the trivial representation of $G$. This Fourier transform also provides a systematic construction of the gapped boundary theory of the Walker-Wang model. This establishes a correspondence between two types of topological field theories: the extended Dijkgraaf-Witten and extended Crane-Yetter theories.
Yu Zhao, Shan Huang, Hongyu Wang, Yuting Hu, Yidun Wan
SciPost Phys. Core 6, 076 (2023) ·
published 8 November 2023
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In this paper, we construct an exactly solvable lattice Hamiltonian model to investigate the properties of a composite system consisting of multiple topological orders separated by gapped domain walls. There are interdomain elementary excitations labeled by a pair of anyons in different domains of this system; This system also has elementary excitations with quasiparticles in the gapped domain wall. Each set of elementary excitations corresponds to a basis of the ground states of this composite system on the torus, reflecting that the ground-state degeneracy matches the number of either set of elementary excitations. The characteristic properties of this composite system lie in the basis transformations, represented by the $S$ and $T$ matrices: The $S$ matrix encodes the mutual statistics between interdomain excitations and domain-wall quasiparticles, and the $T$ matrix encapsulates the topological spins of interdomain excitations. Our model realizes a spatial counterpart of a temporal phase transition triggered by anyon condensation, bringing the abstract theory of anyon condensation into manifestable spatial interdomain excitation states.
