Anomaly of $(2+1)$-Dimensional Symmetry-Enriched Topological Order from $(3+1)$-Dimensional Topological Quantum Field Theory

Symmetry acting on a (2+1)$D$ topological order can be anomalous in the sense that they possess an obstruction to being realized as a purely (2+1)$D$ on-site symmetry. In this paper, we develop a (3+1)$D$ topological quantum field theory to calculate the anomaly indicators of a (2+1)$D$ topological order with a general finite group symmetry $G$, which may contain anti-unitary elements and/or permute anyons. These anomaly indicators are partition functions of the (3+1)$D$ topological quantum field theory on a specific manifold equipped with some $G$-bundle, and they are expressed using the data characterizing the topological order and the symmetry actions. Combined with the relative anomaly formalism, our framework actually enables us to calculate the anomaly of a given topological order with a fully general symmetry. Our framework is applied to derive the anomaly indicators for various symmetry groups, including $mathbb{Z}_2 imesmathbb{Z}_2$, $mathbb{Z}_2^T imesmathbb{Z}_2^T$, etc, where $mathbb{Z}_2$ and $mathbb{Z}_2^T$ denote a unitary and anti-unitary order-2 group, respectively.