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Classification of chiral fermionic CFTs of central charge $\le 16$
by Philip Boyle Smith, Ying-Hsuan Lin, Yuji Tachikawa, Yunqin Zheng
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Submission summary
| Authors (as registered SciPost users): | Yuji Tachikawa |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202311_00013v1 (pdf) |
| Date submitted: | 2023-11-09 05:00 |
| Submitted by: | Tachikawa, Yuji |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We classify two-dimensional purely chiral conformal field theories which are defined on two-dimensional surfaces equipped with spin structure and have central charge less than or equal to 16, and discuss their duality webs. This result can be used to confirm that the list of non-supersymmetric ten-dimensional heterotic string theories found in the late 1980s is complete and does not miss any exotic example.
Author comments upon resubmission
List of changes
Reply to Referee 1:
1. We added a footnote 4 for $Spin(4k)/\mathbb{Z}_2$. We also added a paragraph called "Conventions" at the end of Sec.~1 to explain our bad convention of using capital $G$ for Lie algebras, and our convention on the symmetry group of the fermionic theory.
2. We tried to improve the typesetting of the two main Tables, so that the fact that the explanations below the table are to be considered part of it is clearer.
3. We followed Referee 1's suggestion.
Reply to Referee 2:
Weakness 1: Our intention was and still to make this as a physics paper, and we think our level of rigor is standard among (or actually slightly better than) other papers in hep-th.
Weakness 2: In the modern understanding of QFTs which has become standard in hep-th, a quantum field theory requires the specification of the spacetime structure on which it depends, with its gravitational anomaly specified. A modular invariant CFT in 2d in the traditional sense is a special case when the spacetime structure is the orientation with no gravitational anomaly. We added a footnote 2 concerning this gradual historical change of terminologies.
Weakness 3: We would be delighted to know more references, so that we could learn more and could make this paper more complete. Could Referee 2 give a list?
Requested changes:
1. We followed the suggestion.
2. We added the adjective `unitary' in the first sentence of the Introduction, and expanded the footnote 1 attached to it. We hope that this would be acceptable for Referee 2.
3. We added a phrase "at the physical level of rigor".
4. We think Referee 2 might be confused between the purely left-moving theory of a single $\psi$ with $c_L=1/2$ and $c_R=0$ we are discussing here, and the fermionic minimal model with $c_L=c_R=1/2$. In the latter, the Hilbert space in the R sector has the character $2\chi_{1/16} \overline{\chi_{1/16}}$, one bosonic and one fermionic, acted on by $\psi_0$ and $\overline{\psi_0}$.
In the former, the quantization of the theory on the R-sector circle requires the quantization of a single Majorana zero mode $\psi_0$, which leads to the usual subtleties as indicated in the fourth bullet point in our "Explanation of the Table". We added a reference to this bullet point to make it more clear. We also tried to improve the typesetting of Table 1 so that it stands out more clearly from the rest of the text.
5. We added a short explanation with a reference to a pedagogical article in the new footnote 9.
6. We followed Referee 2's suggestion.
7. We added a definition.
8. As for the first point, gauging a trivially-acting global symmetry group of a trivial theory can result in a nontrivial theory.
For example, 4d pure $SU(N)$ gauge theory is obtained by gauging an global $SU(N)$ symmetry acting on a completely trivial theory, and this is a highly nontrivial operation.
We added a footnote 12 about this.
As for the second point, we added a footnote 13 to say that the commutativity of the dimension-0 operators in the NS sector follows from unitarity via the spin-statistics theorem.
As for the third point and the fourth point, we are assuming that $n_{NS}=n_{R}=1$ at this point of the discussion, so the issue of the commutativity of R-sector dimension-0 states does not arise. We simply use the unique dimension-0 state in the R-sector to establish a 1-to-1 mapping between the R-sector states and the NS-sector states. We enlarged the discussions in the last paragraph of Sec.2.2.
9. References are added.
Current status:
Reports on this Submission
Report #2 by Anonymous (Referee 1) on 2023-12-11 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202311_00013v1, delivered 2023-12-11, doi: 10.21468/SciPost.Report.8258
Strengths
1. This gives a clear physics motivation for the classification results announced. I think it is readily readable by physicists and explains material that is otherwise rather inaccessible in the mathematics literature.
Weaknesses
1. I agree with the authors that I was confused about the status of the Ramond sector in the theory $\psi$, but I think that the status of the $\psi$ theory as a "theory" is a little problematic. Since $(-1)^F$ cannot be defined in the $R$ sector (I was indeed confused in my previous report), it is hard to say it is "fermionic" since the modular transform of the characters is $(\chi_0 + \chi_{1/2})(-1/\tau) = \sqrt2 \chi_{1/16}(\tau)$, it is not clear how to define the partition functions $Z_{NS}^R$ and $Z_R^{NS}$ in such a way that they both count states and the anomaly is a phase, and makes the state-field correspondence in the Ramond sector also look difficult, as the usual OPE of the fermion with the spin field is a disorder field. This does not affect its mathematical properties, but makes its physical interpretation hard. The authors might consider commenting on that.
Report
I think that with the changes as implemented, the paper is acceptable for publication, but I would suggest the authors think about the status of $\psi$ as a theory.
The other points have been all addressed by the authors. It is just this one that I have left.
I think the authors misunderstood my weakness 3 - I was simply saying that it is a weakness that the paper essentially repeats results in the maths literature - while at the same time being a strength that it explains them well.
Requested changes
1- I would suggest the authors think about the status of $\psi$ as a theory and comment on it.