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Anomaly of $(2+1)$-Dimensional Symmetry-Enriched 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: 2023-05-02
Date submitted: 2023-03-03 17:06
Submitted by: Ye, Weicheng
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Theory
  • High-Energy Physics - Theory
  • Mathematical Physics
Approach: Theoretical


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.

Author comments upon resubmission

We thank the editor for dealing with our draft, and all referees for their constructive comments and suggestions. We respond to the referees' reports in the comments. We list changes we made in our new manuscript below.

List of changes

1. As suggested by Referee 2, we change the notation of the bordism group when $G$ contains anti-unitary symmetry from $\Omega_4^{O}(BG, q)$ to $\Omega_4^{O}((BG)^{q-1})$, 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)

Reports on this Submission

Anonymous Report 1 on 2023-3-6 (Invited Report)


The authors have satisfactorily addressed all the comments in my previous report. I recommend this paper for publication on SciPost.

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Anonymous on 2023-03-08  [id 3456]

I would like to thank the authors for making these changes. I believe I can recommend it for publication.