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 Hamiltonian. 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), we construct a gapped boundary Hamiltonian using {K, α}, with a subgroup K ⊆ G and a 3-cochain α of K over U(1), which satisfies the generalized Frobenius condition. The Hamiltonian is invariant under the topological renormalization group flow (via Pachner moves). Each solution {K, α} 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}.
Highlights
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 Hamiltonian
Given the bulk Hamiltonian defined by a gauge group G and a fourcocycle ω in the fourth cohomology group of G over U(1), we construct a gapped boundary Hamiltonian using {K, α}, with a subgroup K ⊆ G and a 3-cochain α of K over U(1), which satisfies the generalized Frobenius condition
Summary
We briefly review the TGT model of topological orders on closed 3-manifold. The TGT model is defined by a low-energy effective Hamiltonian HG,ω on a triangulation Γ (see figure 1) of a closed, orientable, 3-manifold, e.g., a 3-sphere and a 3-torus. Where vi means removing the point vi We take this notation in order to figure out the triangles required to be flat (defined in eq (2.7)) when we use Pachner moves. One first reads off a list of the vertices from any of the four tetrahedra of the defining graph of the 4-cocycle, e.g., the v2v3v4v5 from figure 2(a) and v3v2v4v5 from figure 2(b). Over the operators Agv specified by a group element g ∈ G acting on the same vertex. Since the new vertex v4′ is set to be slightly off the 3d space of the tetrahedron v1v2v3v4, and since every newly created vertex bears a label slightly less than that of the original vertex acted on by the vertex operator, one can always choose the convention such that ǫ(v1v2v3v4′ v4) = ǫ(v1v2v3v4)sgn(v4′ , v1, v2, v3, v4). Unlike the 2 + 1-dimensional TQD model on a torus, the ground states and quasiexcitations don’t remain in one-to-one correspondence for 3 + 1-dimensional TGT models on a 3-torus
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