Gapped Domain Walls, Gapped Boundaries, and Topological Degeneracy

Gapped domain walls, as topological line defects between $(2+1)\mathrm{D}$ topologically ordered states, are examined. We provide simple criteria to determine the existence of gapped domain walls, which apply to both Abelian and non-Abelian topological orders. Our criteria also determine which $(2+1)\mathrm{D}$ topological orders must have gapless edge modes, namely, which $(1+1)\mathrm{D}$ global gravitational anomalies ensure gaplessness. Furthermore, we introduce a new mathematical object, the tunneling matrix $\mathcal{W}$, whose entries are the fusion-space dimensions ${\mathcal{W}}_{ia}$, to label different types of gapped domain walls. By studying many examples, we find evidence that the tunneling matrices are powerful quantities to classify different types of gapped domain walls. Since a gapped boundary is a gapped domain wall between a bulk topological order and the vacuum, regarded as the trivial topological order, our theory of gapped domain walls inclusively contains the theory of gapped boundaries. In addition, we derive a topological ground state degeneracy formula, applied to arbitrary orientable spatial 2-manifolds with gapped domain walls, including closed 2-manifolds and open 2-manifolds with gapped boundaries.

Figure 1

(a),(b) Tunneling channels. (c) Separated domain walls ${\mathcal{W}}^{(1)}$ and ${\mathcal{W}}^{(2)}$. (d) Composite domain wall ${\mathcal{W}}^{(2)}{\mathcal{W}}^{(1)}$.

Figure 2

Computing GSD by tensor contraction: Cut a complicated manifold (e) into simple segments, add oriented skeletons and anyon indices. Associate the segments with: (a) a cylinder with ${\delta }_{ab}$, (b) a domain wall with its tunneling matrix ${\mathcal{W}}_{ia}$, (c) a pair of pants with the fusion tensor ${\mathcal{N}}_{ij}^k$, and (d) a cap with ${\delta }_{1u}$. Finally, contract all the tensors.

Figure 3

Some 2-manifolds with gapped domain walls. (a) Several domain walls on the torus. (b) A cylinder with two gapped boundaries. (c) A pair of pants with three gapped boundaries.