Symmetry Enrichment in Three-Dimensional Topological Phases

While two-dimensional symmetry-enriched topological phases $mathsf{SET}$s 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 $mathsf{GT}$ with gauge group G_g. The resulting symmetric gauge theories are dubbed "symmetry-enriched gauge theories" $mathsf{SEG}$, which may be served as low-energy effective theories of three-dimensional symmetric topological quantum spin liquids. We focus on mathsf{SEG}s with gauge group G_g=mathbb{Z}_N1timesmathbb{Z}_N2timescdots and on-site unitary symmetry group G_s=mathbb{Z}_K1timesmathbb{Z}_K2timescdots or G_s=U(1)times mathbb{Z}_K1timescdots. Each mathsf{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., $mathsf{SET}$ orders of mathsf{SEG}s in experiments of charge-loop braidings (patterns of symmetry fractionalization) and the mixed multi-loop braidings among deconfined loop excitations and confined symmetry fluxes. From these symmetry-enriched properties, one can obtain the map from mathsf{SEG}s to mathsf{SET}s. By giving full dynamics to background gauge fields, mathsf{SEG}s may be eventually promoted to a set of new gauge theories denoted by $mathsf{GT}^*$. Based on their gauge groups, mathsf{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 mathsf{GT}, mathsf{SEG}, mathsf{SET} and mathsf{GT}^* is achieved. We demonstrate the above symmetry-enrichment physics and the web of gauge theories through many concrete examples.