Three-dimensional anomalous twisted gauge theories with global symmetry: Implications for quantum spin liquids

Topological spin liquids can be described by topological gauge theories with global symmetry. Due to the presence of both nontrivial bulk deconfined gauge fluxes and global symmetry, topological spin liquids are examples of the so-called “symmetry enriched topological phases” (SETs). In this paper, we find that, in some twisted versions of topological gauge theories (with discrete Abelian gauge group $G_g$), implementing a global symmetry (denoted by $G_s$) is anomalous although symmetry charge carried by topological pointlike excitations is normally fractionalized and classified by the second cohomology group. To demonstrate the anomaly, we fully gauge the global symmetry, rendering a new gauge theory that is not gauge invariant. Therefore, the SET order of the ground state is anomalous, which cannot exist in the three-dimensional system alone. Such an anomalous state construction generalizes the “2D surface topological order” to three dimensions. A concrete example with $G_g={\mathbb{Z}}_2\times {\mathbb{Z}}_4$ and $G_s={\mathbb{Z}}_2$ is calculated.