Gauging (3+1)-dimensional topological phases: an approach from surface theories

We discuss several bosonic topological phases in (3+1) dimensions enriched by a global $\mathbb{Z}_2$ symmetry, and gauging the $\mathbb{Z}_2$ symmetry. More specifically, following the spirit of the bulk-boundary correspondence, expected to hold in topological phases of matter in general, we consider boundary (surface) field theories and their orbifold. From the surface partition functions, we extract the modular $\mathcal{S}$ and $\mathcal{T}$ matrices and compare them with $(2+1)$d toplogical phase after dimensional reduction. As a specific example, we discuss topologically ordered phases in $(3+1)$ dimensions described by the BF topological quantum field theories, with abelian exchange statistics between point-like and loop-like quasiparticles. Once the $\mathbb{Z}_2$ charge conjugation symmetry is gauged, the $\mathbb{Z}_2$ flux becomes non-abelian excitation. The gauged topological phases we are considering here belong to the quantum double model with non-abelian group in $(3+1)$ dimensions.