Anomaly of (2+1)-dimensional symmetry-enriched topological order from (3+1)-dimensional topological quantum field theory

Symmetry acting on a (2+1)DD topological order can be anomalous in the sense that they possess an obstruction to being realized as a purely (2+1)DD on-site symmetry. In this paper, we develop a (3+1)DD topological quantum field theory to calculate the anomaly indicators of a (2+1)DD topological order with a general symmetry group GG, which may be discrete or continuous, Abelian or non-Abelian, contain anti-unitary elements or not, and permute anyons or not. These anomaly indicators are partition functions of the (3+1)DD topological quantum field theory on a specific manifold equipped with some GG-bundle, and they are expressed using the data characterizing the topological order and the symmetry actions. Our framework is applied to derive the anomaly indicators for various symmetry groups, including \mathbb{Z}_2\times\mathbb{Z}_2&Z;2×&Z;2, \mathbb{Z}_2^T\times\mathbb{Z}_2^T&Z;2T×&Z;2T, SO(N)SO(N), O(N)^TO(N)T, SO(N)\times \mathbb{Z}_2^TSO(N)×&Z;2T, etc, where \mathbb{Z}_2&Z;2 and \mathbb{Z}_2^T&Z;2T denote a unitary and anti-unitary order-2 group, respectively, and O(N)^TO(N)T denotes a symmetry group O(N)O(N) such that elements in O(N)O(N) with determinant -1−1 are anti-unitary. In particular, we demonstrate that some anomaly of O(N)^TO(N)T and SO(N)\times \mathbb{Z}_2^TSO(N)×&Z;2T exhibit symmetry-enforced gaplessness, i.e., they cannot be realized by any symmetry-enriched topological order. As a byproduct, for SO(N)SO(N) symmetric topological orders, we derive their SO(N)SO(N) Hall conductance.