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

TL;DR

This paper explores anomalous twisted gauge theories with global symmetry in 3D topological spin liquids, revealing anomalies that prevent certain SET phases from existing independently in three dimensions.

Contribution

It introduces a framework for understanding anomalies in 3D twisted gauge theories with global symmetry, extending 2D surface topological order concepts to three dimensions.

Findings

Demonstrates that certain symmetry implementations are anomalous in 3D gauge theories.

Provides a concrete example with specific gauge and symmetry groups.

Shows that gauging the global symmetry leads to non-gauge-invariant theories, indicating anomalies.

Abstract

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" (SET). 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 point-like 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 3D system alone. Such anomalous state construction generalizes the "2D surface topological order" to 3D. A concrete example with $G_g=\mathbb{Z}_2\times\mathbb{Z}_4$ and $G_s=\mathbb{Z}_2$ is calculated.