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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

Submission summary

Authors (as Contributors): Weicheng Ye · Liujun Zou
Submission information
Preprint link: scipost_202212_00004v1
Date submitted: 2022-12-01 21:27
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.

Current status:
In refereeing

Submission & Refereeing History

Reports on this Submission

Anonymous Report 2 on 2023-2-1 (Invited Report)


1) The work studies the interesting problem of diagnosing the ’t Hooft anomalies of G-symmetric 2+1d topological orders by constructing a 3+1d TQFT built from the data of the 2+1d topological order along with the G-action on it.

2) Several interesting and illustrative computations of the partition functions of the 3+1d anomaly TQFT have been described.

3) While various aspects of the work have appeared previously in the literature, in particular in Ref 34 and 39 cited in the paper, the paper under review makes the technical contribution of using handle-body decomposition of the 4-manifold in computing the partition function thereof.

4) Additionally the authors discuss anomalies of Lie group global symmetries which haven’t been explored much in the literature particularly, using the methods of the present work. The authors also show that certain anomalies of Lie groups can’t be saturated by any topological order.


Given the large amount technical methods used in the paper, it is not very pedagogically written at several places.


The discussions are quite detailed, and the referee thinks that the paper can be published after the points suggested below are taken care of:

Requested changes

1. There seems to be a typo in Equation (51). The anomaly action of $\mathbb Z_2^T$ global symmetry should be $I_0= (-1)^{\int w_2 \cup w_2}$ and $I_1= (-1)^{\int w_1^4}$. It is unclear, what is meant by $I_0^{w_2^2}$ and $ I_1^{t^4}$.

2. In Equation (50), should there be an additional factor of $d_a$ as in the expression for $I_0$, coming from the product over 2-handles in Equation (44)?

3. Similar to point 1 above, in (54), should the anomaly action for 2+1d topological orders with $\mathbb Z_2 \times \mathbb Z_2$ global symmetry be simply $I_1 \times I_2$. For instance, $I_1= (-1)^{ \int A_1^3 \cup A_2} = (-1)^{c_1^3 c_2}$, where $(A_1,A_2)$ is the $\mathbb Z_2\times \mathbb Z_2$ background gauge field etc.

4. (66) is obtained by generalising (44) to connected Lie groups. In general, as opposed to the case of finite groups, network of symmetry defects does not fully capture a G bundle for a Lie group G. Can the authors clarify how one can neglect possible contributions to the anomaly coming from the curvature of the G-bundle. For instance, from the Chern class of the bundle.

5. Similar to (70), what is the reason there aren’t additional SO(N) line bundles for N >= 6,8, etc.?

  • validity: high
  • significance: good
  • originality: good
  • clarity: high
  • formatting: excellent
  • grammar: reasonable

Anonymous Report 1 on 2023-1-15 (Contributed Report)


1- The paper gives a pedagogical introduction to anomaly indicators and the recipe for writing one down in bosonic theories.

2-The authors compute the relevant bordism groups associated to the anomaly and also give the manifold generators.

3-The authors were able to come up with intricate formulas for the anomaly indicators of different symmetries by studying the handle body decomposition of the manifold generators. They checked that the formula they give is gauge invariant, and also vertex basis transformations.


1-This is not a critique of the work that has been done: it just so happens that the formulas are a bit unwieldy and it is a bit hard to see how to apply them to theories aside from the simple ones where we know the F and R symbols.


I believe that this work is very well done and deserves publication. I do have a few questions for the authors:

How applicable are anomaly indicators for other continuous groups, or perhaps nonsimply connected Lie groups?

If I have a TQFT of the form U(N)_{N,2N} with T^2 = (-1)^F symmetry, can I use anomaly indicators to find the anomaly that this theory takes? It should be \pm 2 mod 16, as was shown in the work here:

It was known how to detect the anomalies for theories with T^2=(-1)^F, and then the anomaly indicator was reproduced on the generating manifold of Pin^+ bordism in degree 4, i.e. RP^4. Since this is also a spin theory, can the methods used here to obtain the anomaly indicator be used to find the anomaly indicators of other fermionic theories? By this, I mean a theory the couples to spin structure, and has a "fermionic symmetry" of the form of an extension of some bosonic symmetry by Z_2^F. If not, then what are the obstructions to writing an anomaly indicator for fermionic theories in general?

Requested changes

1- In appendix D, the relevant bordism group should be \Omega^{\SO}((BG)^{\sigma-1}) as the symmetry structure is a \sigma-twisted orientation of BG. The map q: Z_4-->Z_2 induces a line bundle \sigma-->BZ_4 by pulling the tautological line bundle back from BO_1. If P --> X is a principal Z_4-bundle, we let \sigma_R --> X be the associated line bundle; then w1(\sigma_P ) = x(P).

  • validity: top
  • significance: high
  • originality: high
  • clarity: high
  • formatting: excellent
  • grammar: excellent

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