Bulk-Boundary Correspondence for Three-Dimensional Symmetry-Protected Topological Phases

Abstract

We derive a bulk-boundary correspondence for three-dimensional (3D) symmetry-protected topological phases with unitary symmetries. The correspondence consists of three equations that relate bulk properties of these phases to properties of their gapped, symmetry-preserving surfaces. Both the bulk and surface data appearing in our correspondence are defined via a procedure inwhichwe gauge the symmetries of the systemof interest and then study the braiding statistics of excitations of the resulting gauge theory. The bulk data are defined in terms of the statistics of bulk excitations, while the surface data are defined in terms of the statistics of surface excitations. An appealing property of these data is that it is plausibly complete in the sense that the bulk data uniquely distinguish each 3D symmetry-protected topological phase, while the surface data uniquely distinguish each gapped, symmetric surface. Our correspondence applies to any 3D bosonic symmetryprotected topological phase with finite Abelian unitary symmetry group. It applies to any surface that (1) supports onlyAbelian anyons and (2) has the property that the anyons are not permuted by the symmetries.