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While two-dimensional symmetry-enriched topological phases ($\mathsf{SET}$s)have been studied intensively and systematically, three-dimensional ones arestill open issues. We propose an algorithmic approach of imposing globalsymmetry $G_s$ on gauge theories (denoted by $\mathsf{GT}$) with gauge group$G_g$. The resulting symmetric gauge theories are dubbed "symmetry-enrichedgauge theories" ($\mathsf{SEG}$), which may be served as low-energy effectivetheories of three-dimensional symmetric topological quantum spin liquids. Wefocus on $\mathsf{SEG}$s with gauge group$G_g=\mathbb{Z}_{N_1}\times\mathbb{Z}_{N_2}\times\cdots$ and on-site unitarysymmetry 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 associatedwith certain symmetry assignment. From the path-integral expression, we proposehow to physically diagnose the ground state properties (i.e., $\mathsf{SET}$orders) of $\mathsf{SEG}$s in experiments of charge-loop braidings (patterns ofsymmetry fractionalization) and the \emph{mixed} multi-loop braidings amongdeconfined loop excitations and confined symmetry fluxes. From thesesymmetry-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 maybe further regrouped into different classes each of which is labeled by a gaugegroup ${G}^*_g$. Finally, a web of gauge theories involving $\mathsf{GT}$,$\mathsf{SEG}$, $\mathsf{SET}$ and $\mathsf{GT}^*$ is achieved. We demonstratethe above symmetry-enrichment physics and the web of gauge theories throughmany concrete examples.

Symmetry Enrichment in Three-Dimensional Topological Phases