Gapped Boundary Theory of the Twisted Gauge Theory Model of Three-Dimensional Topological Orders

TL;DR

This work formulates a complete, exactly solvable framework for 3D topological orders with gapped boundaries by extending the twisted gauge theory (TGT) model. The bulk is specified by a finite group $G$ and a 4-cocycle $\omega\in H^4[G,U(1)]$, while boundaries are encoded by a subgroup $K\subseteq G$ and a 3-cochain $\alpha\in C^3[K,U(1)]$, constrained by a generalized Frobenius condition to ensure gapped, topology-preserving boundary dynamics. The authors establish a one-to-one correspondence between boundary data and the cohomology group $H^3[K,U(1)]$, derive explicit ground-state wavefunctions on a 3-ball, and obtain a closed-form ground-state degeneracy formula on a 3-cylinder expressed solely in terms of input data $\{G,\omega,K,\alpha\}$. A detailed example with $G=\mathbb{Z}_2\times\mathbb{Z}_2\times\mathbb{Z}_2$ illustrates the boundary-classification structure and demonstrates the concrete computation of allowed boundary conditions via $H^3[K,U(1)]$ and $H^4[G,U(1)]$ data. This work advances understanding of bulk-boundary condensation, Pachner-move invariance, and holographic perspectives in higher-dimensional topological phases.

Abstract

We extend the twisted gauge theory model of topological orders in three spatial dimensions to the case where the three spaces have two dimensional boundaries. We achieve this by systematically constructing the boundary Hamiltonians that are compatible with the bulk Hamoltonian. Given the bulk Hamiltonian defined by a gauge group $G$ and a four-cocycle $ω$ in the fourth cohomology group of $G$ over $U(1)$, a boundary Hamiltonian can be defined by a subgroup $K$ of $G$ and a three-cochain $α$ in the third cochain group of $K$ over $U(1)$. The boundary Hamiltonian to be constructed must be gapped and invariant under the topological renormalization group flow (via Pachner moves), leading to a generalized Frobenius condition. Given $K$, a solution to the generalized Frobenius condition specifies a gapped boundary condition. We derive a closed-form formula of the ground state degeneracy of the model on a three-cylinder, which can be naturally generalized to three-spaces with more boundaries. We also derive the explicit ground-state wavefunction of the model on a three-ball. The ground state degeneracy and ground-state wavefunction are both presented solely in terms of the input data of the model, namely, $\{G,ω,K,α\}$.

Figures (10)

• Figure 1: A portion of a graph that represent the basis vectors in the Hilbert space. Each edge carries an arrow and is assigned a group element denoted by $[ab]$ with $a<b$.
• Figure 2: (a) The defining graph of the $4$-cocycle $[v_1v_2 v_3v_4v_5]$. (b) Of $[v_1v_2v_3v_4v_5]^{-1}$.
• Figure 3: The topology of the action of $A_{v_4}^g$.
• Figure 4: A $3$-cylinder, i.e., a solid torus with a solid torus removed from interior. This is an example of a $3$-manifold with two boundaries, the inner boundary (dashed line) $\partial_1\Gamma$ and the outter boundary (solid line) $\partial_2\Gamma$. A portion of the triangulation $\Gamma$ of this $3$-cylinder is shown, where $v_1$ and $v_2$ are two vertices in the bulk.
• Figure 5: The boundary is the grey plane, above which is the bulk. A bulk tetrahedron and a boundary face $f=123$ is explicitly shown. The dashed lines represent the rest of the graph that is now drawn.