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Anomaly of $(2+1)$Dimensional SymmetryEnriched Topological Order from $(3+1)$Dimensional Topological Quantum Field Theory
by Weicheng Ye and Liujun Zou
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Submission summary
Authors (as registered SciPost users):  Weicheng Ye · Liujun Zou 
Submission information  

Preprint Link:  scipost_202212_00004v2 (pdf) 
Date accepted:  20230502 
Date submitted:  20230303 17:06 
Submitted by:  Ye, Weicheng 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
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)$D$ onsite 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 nonAbelian, contain antiunitary 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 antiunitary order2 group, respectively, and $O(N)^T$ denotes a symmetry group $O(N)$ such that elements in $O(N)$ with determinant $1$ are antiunitary. In particular, we demonstrate that some anomaly of $O(N)^T$ and $SO(N)\times \mathbb{Z}_2^T$ exhibit symmetryenforced gaplessness, i.e., they cannot be realized by any symmetryenriched topological order. As a byproduct, for $SO(N)$ symmetric topological orders, we derive their $SO(N)$ Hall conductance.
Author comments upon resubmission
List of changes
1. As suggested by Referee 2, we change the notation of the bordism group when $G$ contains antiunitary symmetry from $\Omega_4^{O}(BG, q)$ to $\Omega_4^{O}((BG)^{q1})$, to emphasize the choice of choosing a $q$twisted orientation of $\mc{M}$.
2. In Sec. IIIA and Sec. IVB, we emphasize that the anomaly indicators are numbers which serve as coefficients in front of a certain basis of the relevant cohomology or cobordism group.
3. We expand the point 3 in the discussion section to explain in more detail how our formalism can be generalized to fermionic systems and obtain partition functions and anomaly indicators thereof.
Published as SciPost Phys. 15, 004 (2023)
Anonymous on 20230308 [id 3456]
I would like to thank the authors for making these changes. I believe I can recommend it for publication.