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On Exotic Consistent Anomalies in (1+1)$d$: A Ghost Story
by ChiMing Chang, YingHsuan Lin
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Submission summary
As Contributors:  ChiMing Chang · YingHsuan Lin 
Arxiv Link:  https://arxiv.org/abs/2009.07273v3 (pdf) 
Date submitted:  20201216 04:36 
Submitted by:  Lin, YingHsuan 
Submitted to:  SciPost Physics 
Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
We revisit ’t Hooft anomalies in (1+1)d nonspin 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 ChernSimons actions. On the one hand, we prove that certain exotic anomalies can only be realized by nonreflectionpositive or noncompact theories; on the other hand, without insisting on reflectionpositivity, 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 ChernSimons 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.
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Reports on this Submission
Anonymous Report 1 on 2021116 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2009.07273v3, delivered 20210116, doi: 10.21468/SciPost.Report.2416
Strengths
1 Detailed analysis of anomalies in 1+1dimensional nonunitary theories.
2 Relatively notwell studied but important subject.
3 The logic and derivation are clear.
Report
The authors studied the perturbative anomaly in possibly nonunitary 1+1dimensional 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 nonunitary 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 nonunitary 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), "reflectivepositivity" > "reflectionpositivity"
Author: YingHsuan Lin on 20210211 [id 1225]
(in reply to Report 1 on 20210116)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.