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

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 $\omega$ 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 $\alpha$ 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, $\{G,\omega ,K,\alpha \}$.