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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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Submission summary

Authors (as registered SciPost users): Chi-Ming Chang · Ying-Hsuan Lin
Submission information
Preprint Link: scipost_202102_00020v1  (pdf)
Date accepted: 2021-04-16
Date submitted: 2021-02-11 14:07
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)$\times$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 survey the holomorphic $bc$ ghost system which realizes all the exotic consistent anomalies, and end with comments on a subtlety regarding the anomalies of finite subgroups of U(1).

Author comments upon resubmission

We thank the referee for comments, and have updated accordingly.

List of changes

1- We rewrote the section "On the 'embedding' of anomalies of finite subgroups", and moved it later to improve the flow of the paper. The abstract and the introduction section are edited accordingly.

2- We added footnote 21 to mention a comment by the Referee.

Published as SciPost Phys. 10, 119 (2021)

Reports on this Submission

Anonymous Report 1 on 2021-4-4 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:scipost_202102_00020v1, delivered 2021-04-04, doi: 10.21468/SciPost.Report.2755


1. Clear and self-contained presentation
2. Complementary perspective on the quantization conditions of anomalies in QFTs compared to the inflow picture
3. Detailed discussions about anomalies in non-unitary Euclidean CFTs and manifestations on line defects


An important question in QFT is to understand the structure of anomalies. It is generally believed that the (parity-odd) 't Hooft anomalies take quantized values, based on anomaly inflow from an invertible topological field theory in one higher dimension. This paper makes an interesting attempt in two spacetime dimensions to derive such quantization conditions directly from general consistency conditions, such as the Wess-Zumino (WZ) consistency conditions and unitarity (reflection-positivity), on the QFT without assuming anomaly inflow. The authors find that with just WZ condition, the quantization conditions derived are strictly weaker than those from anomaly inflow, which give rise to what they call "exotic consistent anomalies" (which also include certain anomalies that are altogether forbidden in unitary theories), and are realized in non-unitary Euclidean bc CFTs which are studied in detail in this paper. After incorporating certain unitarity constraints, the authors find that the quantization conditions on some anomalies improve and match with the inflow prediction, while the others remain strictly weaker. It would be interesting to pursue a more complete analysis of unitarity constraints which may settle these discrepancies.

The paper is very well written and provides a fresh perspective on the recently well-studied anomalies in QFTs. I recommend this paper for publication.

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

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