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On Exotic Consistent Anomalies in (1+1)$d$: A Ghost Story

by Chi-Ming Chang, Ying-Hsuan Lin

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Authors (as registered SciPost users): Chi-Ming Chang · Ying-Hsuan Lin
Submission information
Preprint Link:  (pdf)
Date submitted: 2020-12-16 04:36
Submitted by: Lin, Ying-Hsuan
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • High-Energy Physics - Theory
Approach: Theoretical


We revisit ’t Hooft anomalies in (1+1)d non-spin quantum field theory, starting from the consistency and locality conditions, and find that consistent U(1) and gravitational anomalies cannot always be canceled by properly quantized (2+1)d classical Chern-Simons actions. On the one hand, we prove that certain exotic anomalies can only be realized by non-reflection-positive or non-compact theories; on the other hand, without insisting on reflection-positivity, the exotic anomalies present a caveat to the inflow paradigm. For the mixed U(1) gravitational anomaly, we propose an inflow mechanism involving a mixed U(1)×SO(2) classical Chern-Simons action with a boundary condition that matches the SO(2) gauge field with the (1+1)d spin connection. Furthermore, we show that this mixed anomaly gives rise to an isotopy anomaly of U(1) topological defect lines. The isotopy anomaly can be canceled by an extrinsic curvature improvement term, but at the cost of creating a periodicity anomaly. We comment on a subtlety regarding the anomaly of finite subgroups of U(1), and end with a survey of the holomorphic bc ghost system which realizes all the exotic consistent anomalies.

Current status:
Has been resubmitted

Reports on this Submission

Anonymous Report 1 on 2021-1-16 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2009.07273v3, delivered 2021-01-16, doi: 10.21468/SciPost.Report.2416


1- Detailed analysis of anomalies in 1+1-dimensional non-unitary theories.
2- Relatively not-well studied but important subject.
3- The logic and derivation are clear.


The authors studied the perturbative anomaly in possibly non-unitary 1+1-dimensional QFTs with U(1) symmetry.
The careful analysis reveals the properties that are previously not appreciated and opens the door to study more general anomalies in non-unitary QFTs. Therefore I recommend publishing the manuscript in SciPost Physics.

A few questions/comments:
1- Is there any relation among the anomaly coefficients? Is there a linear combination of them that should vanish modulo something, or are they independent?
2- I could not understand what is the conclusion in section 4. When $\kappa_F$ is odd, the topological lines describing the $Z_3$ lines cannot satisfy some of the axioms of fusion category?
3- (Just a comment) The inflow action involving the Euler characteristic exemplifies that the classification of the anomalies in non-unitary QFTs cannot be given by a *stable* homotopy theory. Here, to write down the Euler characteristic, the tangent bundle of $M_2$ had to be extended into the bulk as a $SO(2)$ bundle, while in a unitary theory the anomaly does not care much about $d$ of $SO(d)$. It would be nice to find and study the unstable homotopy that could describe these anomalies.

Requested changes

Just typos
1- Above (5.4), "anomalies coefficients" -> "anomaly coefficients"
2- Below (6.1), "reflective-positivity" -> "reflection-positivity"

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

Author:  Ying-Hsuan Lin  on 2021-02-11  [id 1225]

(in reply to Report 1 on 2021-01-16)

We thank the referee for the comments, and have made corrections and adjustments accordingly in our updated version. Our responses are below:

1- It is an interesting question. We did suspect that $\kappa_{F^2}$ might be odd iff $4\kappa_{FR}$ is odd, but we could not prove it.

2- That’s right. When $\kappa$ is odd, the anomalous phase depends on extra information that is not captured in the framework of background $\mathbb{Z}_N$ gauge transformations (or equivalently manipulations in $\mathbb{Z}_N$ fusion categories). We rewrote that section to clarify the discussion.

3- We thank the referee for this insightful comment. While the subject of stable homotopy is beyond the expertise of the authors, we added footnote 21 and thanked the referee.

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