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RG flows on two-dimensional spherical defects
by Tom Shachar, Ritam Sinha, Michael Smolkin
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Submission summary
| Authors (as registered SciPost users): | Ritam Sinha |
| Submission information | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/2212.08081v2 (pdf) |
| Date submitted: | 2023-02-16 12:10 |
| Submitted by: | Sinha, Ritam |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We study two-dimensional spherical defects in d-dimensional Conformal Field Theories. We argue that the Renormalization Group (RG) flows on such defects admit the existence of a decreasing entropy function. At the fixed points of the flow, the entropy function equals the anomaly coefficient which multiplies the Euler density in the defect's Weyl anomaly. Our construction demonstrates an alternative derivation of the irreversibility of RG flows on two-dimensional defects. Moreover in the case of perturbative RG flows, the entropy function decreases monotonically and plays the role of a C-function. We provide a simple example to explicitly work out the RG flow details in the proposed construction.
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Reports on this Submission
Report
This paper adapts an argument due to [53] to provide a new proof of the irreversibility of RG flows on two-dimensional defects in CFTs. The same result had been obtained before in [55].
The paper is clear and technically solid. It is interesting and non-trivial that the method of [53] generalizes to two-dimensional defects. The main claim is of sufficient importance to warrant publication of this paper.
Report #1 by Anonymous (Referee 1) on 2023-7-19 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2212.08081v2, delivered 2023-07-19, doi: 10.21468/SciPost.Report.7525
Strengths
1. Interesting topic
2. Pedagogical and clear writing
3. It contains an important non-perturbative result, which was known (ref [55]), but whose validity the authors prove in a new way.
4. It discusses in detail the perturbative case, motivating a possible extension of the known non-perturbative result.
Weaknesses
1. The proof is essentially identical to the one presented in ref. [53], which applied those ideas to a line defect.
Report
This is a well written paper on an interesting topic, which deserves publication on SciPost. The paper derives a c-function for renormalization group flows on surface defects. Recently, renormalization group flows on defects have received a lot of attention, and a proof of irreversibility is a general powerful tool.
The importance of this paper is somewhat diminished by the fact that this result was known, the c-theorem having been proven in ref [55]. On the other hand, ref [55] relies on some correct but subtle facts about effective field theory for the dilaton, and it is useful to have a different path to the same result.
Then again, the new path was in fact already found in ref. [53], which treated line defects. There, new ideas were strictly necessary, since the interpretation of the c-function as an anomaly was missing (and so was the anomaly-matching condition).
The present work also points out that the entropy function provides a candidate for a monotonically decreasing function on the RG. The authors test this claim at leading order in conformal perturbation theory.
Overall, the new results are sufficient to warrant publication, after the authors consider the minor observations listed below.
Requested changes
- In the introduction, the authors cite [53] as "motivating" their definitions, and this risks underrepresenting the debt of the present paper to that work. The authors should emphasise that the method of proof closely follows the one presented in [53].
- There is a notational ambiguity in section 2, which might be confusing. In 2.2, the defect contributions are integrated in d(sigma), and so the background fields, including the metric, depend on sigma through the spacetime coordinate x. However, in 2.6, the variation under reparametrization is taken to vanish. A consistent treatment requires to take the dependence on sigma into account. This should affect a number of equation, including the unnamed equation after 2.7, (hopefully) without affecting any of the physical consequences.
- After 3.16, there is a wrong reference to eq. 4.10 (it is more natural to refer to 3.15).
- The symbol C is used with two different meanings in sections 3.1 and 4, which leads to a mismatch between eq. 3.18 and 4.16. The authors explain the mismatch in a footnote, but it would be better to use consistent definitions for C and either change the coefficient in 4.8 to C x (explicit normalization factor), or use a different symbol, e.g. \tilde{C}.
- After 3.19, the authors claim that the entropy function is monotonic at all orders in the coupling, but they only proved the statement for the leading order. There is a difference between the two statements: what if the coupling is only marginally relevant? The authors should prove the statement more generally in the case where the beta function starts at higher orders, or they must limit their claim appropriately.
- In appendix A, the description of the defect in terms of a system of coordinates sigma is chosen. Then, the tangent and normal vectors and the curvatures of the defect are functions of these coordinates and only defined at the defect. Therefore, their derivatives in directions orthogonal to the defect are not defined, and cannot be taken to vanish as after A.4. The authors can for instance compare with refs [23] and [29], where these higher order terms are taken into account, to rephrase their discussion when necessary. They can also switch to a definition of defect geometry in terms of an adapted foliation of the full spacetime, in which case the normal derivatives mentioned above make sense and must be included in the computation.