Fourier-transformed gauge theory models of three-dimensional topological orders with gapped boundaries

TL;DR

This work extends the Fourier-transform framework from 2D to 3D topological orders, showing that a 3D gauge theory with input data $G$ admits a gapped boundary described by a Frobenius algebra in $\\mathcal{R}ep(G)$ and that the Fourier transform maps the 3D GT model to the Walker-Wang model with input $\\mathcal{R}ep(G)$. The boundary data become encoded by the Frobenius algebra $A_{G,K}$, clarifying charge condensation and the boundary spectrum in 3D topological orders. Moreover, the transform yields a systematic construction of the gapped boundary for the Walker-Wang model and establishes a precise correspondence between extended Dijkgraaf-Witten and extended Crane-Yetter theories in three dimensions. The results unify bulk and boundary perspectives across two TFT frameworks, enabling explicit, lattice-based realizations of boundary condensation and facilitating the mapping between GT and WW models. This provides a concrete, constructive bridge between 3D topological orders and their boundary theories, with potential implications for exactly solvable models and higher-categorical topological phases.

Abstract

In this paper, we apply the method of Fourier transform and basis rewriting developed in arXiv:1910.13441 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 $\mathcal Rep(G)$ of $G$, which also describes the 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 $\mathcal Rep(G)$ 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.

Figures (9)

• Figure 1: A portion of an oriented trivalent lattice, on which the three-dimensional GT model with gapped boundaries is defined. Each edge of the lattice is graced with a group element of a finite gauge group $G$. The thick lines comprise the boundary while the dashed lines comprise the bulk.
• Figure 2: Charge excitation (red dot) and string-like excitation (deep blue dashed line, consisting of a series of light blue plaquettes where the local flatness condition is violated) that terminat on the boundary in the three-dimensional GT model with gapped boundaries.
• Figure 3: The Fourier transform of a six-valent vertex in the bulk of the lattice $\Gamma$ on which the three-dimensional GT model is defined.
• Figure 4: Rewrite the rep-basis as defined on a trivalent lattice. In the first step, we fuse $\mu$ and $\lambda$ by contracting there indices $m_\mu$ and $n_\lambda$, which results in a linear combination of irreducible representations $\{\alpha\}$ with a free end and labeled by $m_\alpha$. Repeating this procedure and in the end, we fuse $\delta$ and $\nu$ and obtain a tail attached to the original vertex with an free end, labeled by $(s,m_s)$.
• Figure 5: A whole plaquette in the bulk, where the edges labeled by $\mu$ and $\nu$ are braided, as are edges labeleds by $\rho$ and $\sigma$.