Symmetry enrichment in three-dimensional topological phases

While two-dimensional symmetry-enriched topological phases (SETs) have been studied intensively and systematically, three-dimensional ones are still open issues. We propose an algorithmic approach of imposing global symmetry G(s) on gauge theories (denoted by GT) with gauge group G(g). The resulting symmetric gauge theories are dubbed symmetry-enriched gauge theories (SEG), which may be served as low-energy effective theories of three-dimensional symmetric topological quantum spin liquids. We focus on SEGs with gauge group G(g) = Z(N1) x Z(N2) x ... and onsite unitary symmetry group G(s) = Z(K1) x Z(K2) x ... or G(s) = U(1) x Z(K1) x .... Each SEG(G(g), G(s)) is described in the path-integral formalism associated with certain symmetry assignment. From the path-integral expression, we propose how to physically diagnose the ground-state properties (i.e., SET orders) of SEGs in experiments of charge-loop braidings (patterns of symmetry fractionalization) and the mixed multiloop braidings among deconfined loop excitations and confined symmetry fluxes. From these symmetry-enriched properties, one can obtain the map from SEGs to SETs. By giving full dynamics to background gauge fields, SEGs may be eventually promoted to a set of new gauge theories (denoted by GT*). Based on their gauge groups, GT*s may be further regrouped into different classes, each of which is labeled by a gauge group G(g)*. Finally, a web of gauge theories involving GT, SEG, SET, and GT* is achieved. We demonstrate the above symmetry-enrichment physics and the web of gauge theories through many concrete examples.