TY  - JOUR
PB  - SciPost Foundation
DO  - 10.21468/SciPostPhys.15.1.004
TI  - Anomaly of $(2+1)$-dimensional symmetry-enriched topological order from $(3+1)$-dimensional topological quantum field theory
PY  - 2023/07/10
UR  - https://scipost.org/SciPostPhys.15.1.004
JF  - SciPost Physics
JA  - SciPost Phys.
VL  - 15
IS  - 1
SP  - 004
A1  - Ye, Weicheng
AU  - Zou, Liujun
AB  - 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 symmetry group $G$, 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)$D$ 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. Our framework is applied to derive the anomaly indicators for various symmetry groups, including $\mathbb{Z}_2\times\mathbb{Z}_2$, $\mathbb{Z}_2^T\times\mathbb{Z}_2^T$, $SO(N)$, $O(N)^T$, $SO(N)\times \mathbb{Z}_2^T$, etc, where $\mathbb{Z}_2$ and $\mathbb{Z}_2^T$ denote a unitary and anti-unitary order-2 group, respectively, and $O(N)^T$ denotes a symmetry group $O(N)$ such that elements in $O(N)$ with determinant $-1$ are anti-unitary. In particular, we demonstrate that some anomaly of $O(N)^T$ and $SO(N)\times \mathbb{Z}_2^T$ exhibit symmetry-enforced gaplessness, i.e., they cannot be realized by any symmetry-enriched topological order. As a byproduct, for $SO(N)$ symmetric topological orders, we derive their $SO(N)$ Hall conductance.
ER  -

@Article{10.21468/SciPostPhys.15.1.004,
	title={{Anomaly of $(2+1)$-dimensional symmetry-enriched topological order from $(3+1)$-dimensional topological quantum field theory}},
	author={Weicheng Ye and Liujun Zou},
	journal={SciPost Phys.},
	volume={15},
	pages={004},
	year={2023},
	publisher={SciPost},
	doi={10.21468/SciPostPhys.15.1.004},
	url={https://scipost.org/10.21468/SciPostPhys.15.1.004},
}