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Analytic conformal bootstrap and Virasoro primary fields in the Ashkin-Teller model

by Nikita Nemkov, Sylvain Ribault

Submission summary

As Contributors: Sylvain Ribault
Arxiv Link: (pdf)
Date accepted: 2021-09-13
Date submitted: 2021-09-02 10:12
Submitted by: Ribault, Sylvain
Submitted to: SciPost Physics
Academic field: Physics
  • High-Energy Physics - Theory


We revisit the critical two-dimensional Ashkin-Teller model, i.e. the $\mathbb{Z}_2$ orbifold of the compactified free boson CFT at $c=1$. We solve the model on the plane by computing its three-point structure constants and proving crossing symmetry of four-point correlation functions. We do this not only for affine primary fields, but also for Virasoro primary fields, i.e. higher twist fields and degenerate fields. This leads us to clarify the analytic properties of Virasoro conformal blocks and fusion kernels at $c=1$. We show that blocks with a degenerate channel field should be computed by taking limits in the central charge, rather than in the conformal dimension. In particular, Al. Zamolodchikov's simple explicit expression for the blocks that appear in four-twist correlation functions is only valid in the non-degenerate case: degenerate blocks, starting with the identity block, are more complicated generalized theta functions.

Current status:
Publication decision taken: accept

Editorial decision: For Journal SciPost Physics: Publish
(status: Editorial decision fixed and (if required) accepted by authors)

Author comments upon resubmission

We have added clarification following the reviewers' suggestions, in particular in Section 5.1.

List of changes

Answer to comments by Reviewer 1:

1. We have added a clarification about the notations after (2.12). In (2.19) and (2.20) we use the notation $\Delta_j$ in order to emphasize that the Virasoro blocks and fusion kernels are universal quantities that only depend on conformal dimensions, as we explain after (2.19). Eq. (2.20) should not include any restiction on $(\Delta_t,\bar\Delta_t)$, as we now clarify after that equation. We have added $\pm$ in Eq. (2.26) as suggested, in order to display the sign ambiguity explicitly.

2. As suggested, we have flipped the sign of (3.35), and consequently also of (3.39), (3.40), (4.42). This has allowed us to simplify the discussion of the sign ambiguity after (3.46), and to state after (4.60) that we have agreement with (4.40)-(4.43), and not just agreement up to signs.

3. We have tried to clarify this issue by consistently using the matrix notation for the fusion kernel, starting with (2.19). This change of noation has also affected Eqs. (B.1), (B.2), (B.17).

4. Our point of view is that we should not rely on the structure constants to impose fusion rules. In the OPE (2.8) and decomposition (2.11), fusion rules are imposed at the level of the summation instead. However, fusion rules were missing from the three-point function (2.6): we have now added them using an explicit prefactor, and added an explanation after that equation. This allows structure constants such as (4.19) to remain simple.

5. We believe that this choice does not matter. We have made this clearer in Conjecture 2 by using the word 'any' in 'Let $\mathcal{R}^{(c)}$ be any degenerate representation at generic $c$ that has the same number of states...'. In the text before (5.17), we have also stated explicitly that the choice of one of the two possible representations does not matter.

7. We have added explanations after the pentagon diagram, including the new diagram (5.34).

Answer to comments by Reviewer 2:

1. Following the suggestion, we have moved up the definitions of these notations.

Answer to both reviewers:

Following the convergent requests of both reviewers, we have expanded the paragraph 'Scope and validity of the conjectures' in Section 5.1, in order to discuss the evidence in more detail. In particular, the new Eq. (5.15) lists the conformal blocks for which we have evidence. We do not mention direct pedestrian tests of Conjecture 2, as they are limited to low levels, and therefore quite weak.

Answer to Editor's request:

We have changed the Bibtex style to the SciPost style, which displays journal references.

Submission & Refereeing History

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Resubmission 2106.15132v2 on 2 September 2021

Reports on this Submission

Anonymous Report 1 on 2021-9-8 (Invited Report)


With the expanded discussion on the scope and validity of the authors' conjectures regarding the computation of degenerate conformal blocks as limits of generic conformal blocks, I am happy to recommend the publication of this paper in SciPost.

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