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Diagonal fields in critical loop models
by Sylvain Ribault
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Submission summary
| Authors (as registered SciPost users): | Sylvain Ribault |
| Submission information | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/2209.09706v1 (pdf) |
| Date submitted: | 2022-11-16 13:58 |
| Submitted by: | Ribault, Sylvain |
| Submitted to: | SciPost Physics Core |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
In critical loop models, there exist diagonal fields with arbitrary conformal dimensions, whose $3$-point functions coincide with those of Liouville theory at $c\leq 1$. We study their $N$-point functions, which depend on the $2^{N-1}$ weights of topologically inequivalent loops on a sphere with $N$ punctures. Using a numerical conformal bootstrap approach, we find that $4$-point functions decompose into infinite but discrete linear combinations of conformal blocks. We conclude that diagonal fields belong to an extension of the $O(n)$ model.
Current status:
Reports on this Submission
Report #1 by Anonymous (Referee 1) on 2022-12-7 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2209.09706v1, delivered 2022-12-07, doi: 10.21468/SciPost.Report.6275
Strengths
1- interesting subject
2-new application of bootstrap approach
Weaknesses
1- the presentation of the main results is very abstract
2- references are minimal
Report
The author studied partition function of ensembles of loops on punctured sphere which could be interpreted correlation functions in a CFT. By making some assumptions on the spectrum, the author then bootstrap to solve some 4 point functions with varying parameters. The results includes curious solutions such as the diagonal fields conformal data given by that of the time-like Liouville theory. The author then conclude that corresponding $N$-point function belong to an extension of $O(n)$ CFT.
While the subject is interesting, and the results are curious, I find some of the presentation not completely clear. I propose the following changes to improve the quality of the paper.
Requested changes
1- the author claimed in table (15) that the CFT under investigation involves degenerate field $V_{\langle1,3\rangle}$ (although this is not part of the spectrum). Did the author use the interchiral block resulting from such degeneracy in the boostrap? This is not clearly stated in the paper.
2- the bootstrap was carried out by choosing a series of parameters $P_s, P_t, P_u$. What are the choices used for obtaining the results?
3- result 1 claims that the system has a one-dimensional space of solutions for a given choice of $P_s, P_t, P_u$. How is this one-dimensional space parametrized? The author could plot some conformal data for example to illustrate this.
4- above table (15) the author claims that structure constants in the Potts model obey shift equations, despite lacking the degenerate field. I believe this point is not true. In 2005.07258, it was obtained that the Potts structure constants obey a modified version of the shift relation (dubbed renormalized Liouville recursion there) rather than that results from the degeneracy of $V_{\langle2,1\rangle}$.