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Topological Defect Lines in Two Dimensional Fermionic CFTs
by Chi-Ming Chang, Jin Chen, Fengjun Xu
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Submission summary
| Authors (as registered SciPost users): | Chi-Ming Chang · Jin Chen · Fengjun Xu |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2208.02757v4 (pdf) |
| Date submitted: | 2023-05-14 05:44 |
| Submitted by: | Chen, Jin |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We consider topological defect lines (TDLs) in two-dimensional fermionic conformal field theories (CFTs). Besides inheriting all the properties of TDLs in bosonic CFTs, TDLs in fermionic CFTs could host fermionic defect operators at their endpoints and junctions. Furthermore, there is a new type of TDLs, called q-type TDLs, that have no analog in bosonic CFTs. Their distinguishing feature is an extra one-dimensional Majorana fermion living on the worldline of the TDLs. The properties of TDLs in fermionic CFTs are captured in the mathematical language of the super fusion category. We propose a classification of the rank-2 super fusion categories generalizing the $\mathbb Z_8$ classification for the anomalies of $\mathbb Z_2$ symmetry. We explicitly solve the F-moves for all the nontrivial categories, and derive the corresponding spin selection rules that constrain the spectrum of the defect operators. We find the full set of TDLs in the standard fermionic minimal models and a partial set of TDLs in the two exceptional models, which give CFT realizations to the rank-2 super fusion categories. Finally, we discuss a constraint on the renormalization group flow that preserves a q-type TDL.
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Reports on this Submission
Report #2 by Anonymous (Referee 2) on 2023-10-6 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2208.02757v4, delivered 2023-10-06, doi: 10.21468/SciPost.Report.7906
Report
The manuscript investigates non-invertible topological lines in 1+1d fermionic CFTs. The manuscript discovers new structures compared to the topological lines in bosonic theories, such as decoration by Majorana zero mode. I recommend publication if the authors address the following mostly minor comments:
- In well-defined non-chiral fermionic theories one can gauge the fermion parity symmetry to obtain bosonic theories, and the topological lines will become the topological lines previously studied.
This method is extensively used to study fermionic SPT phases.
Can the author comment on such approach to understanding topological lines in fermionic theories?
- The manuscript discussed changing the normalization of the topological line operators. However, the fusion coefficients of topological operators are quantized (described by well-defined lower dimensional TQFTs), and such change of normalization does not seem compatible with the topological structure. Can the author clarify this?
Report #1 by Anonymous (Referee 1) on 2023-9-15 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2208.02757v4, delivered 2023-09-15, doi: 10.21468/SciPost.Report.7831
Report
The paper generalizes the concept of categorical symmetries, which are characterized by the presence of non-invertible topological defect lines (TDLs), to theories with fermions. Here are some of the new properties of TDLs in such theories that they discuss:
1) Because the lines can now end on fermionic operators, the usual pentagon identity is replaced by a super pentagon identity.
2) There is a new type of TDLs, called q-type TDLs, which does not exist in bosonic theories. One defining feature of this is that the identity term in the fusion rule comes with an extra factor of 2.
They give several examples of fermionic minimal models where such TDLs are present. They also present an application to constraining RG flows that preserve the TDL. Overall, I recommend this paper for publication because it introduces a novel framework for dealing with non-invertible symmetries in the presence of fermions. Even though the paper deals with theories in two dimensions, I expect some of the features to persist in higher dimensions.
Author: Jin Chen on 2023-10-18 [id 4044]
(in reply to Report 2 on 2023-10-06)We thank the referee's comments. Followed are our repies. More detailed updates can be found in the list of changes.
For the first question, we add a paragraph in the end of summary and discussion section, which explains the observed relations between TDLs in bosonic and fermionic CFTs, as well as connections to the fermionic anyon condensation widely used in the condense matter community.
For the second question, the change of the normalization in (2.50) is only a formal manipulation to make the fusion rules look like a group multiplication. However, the fusion coefficients of the new fusion rules do not reflect the relation to the lower dimensional TQFT anymore, as the referee pointed out. We emphasize this by adding the word ``formal" before (2.50).
We also add a sentence below eq.(2.47) to explain the "factor of 2" in the fusion rule can be interpreted as an 1d TFT gapped from two 1d Majorana fermions.