Symmetry Enrichment in Three-Dimensional Topological Phases

Abstract

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 ${\mathsf{GT}}^{\ast }$). Based on their gauge groups, ${\mathsf{GT}}^{\ast }$s may be further regrouped into different classes each of which is labeled by a gauge group $G_g^{\ast }$. Finally, a web of gauge theories involving $\mathsf{GT}$, $\mathsf{SEG}$, $\mathsf{SET}$ and ${\mathsf{GT}}^{\ast }$ is achieved. We demonstrate the above symmetry-enrichment physics and the web of gauge theories through many concrete examples.

24 pages, 5 figures

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• Continuum field theory of 3D topological orders with emergent fermions and braiding statistics