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 \(G_g=\mathbb{Z}_{N_1}\times\mathbb{Z}_{N_2}\times\cdots\) and on-site unitary symmetry group \(G_s=\mathbb{Z}_{K_1}\times\mathbb{Z}_{K_2}\times\cdots\) or \(G_s=\mathrm{U(1)}\times \mathbb{Z}_{K_1}\times\cdots\). 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 \emph{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.