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

by Nikita Nemkov, Sylvain Ribault

This is not the current version.

Submission summary

As Contributors: Sylvain Ribault
Arxiv Link: (pdf)
Date submitted: 2021-07-22 14:05
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:
Has been resubmitted

Submission & Refereeing History

Reports on this Submission

Anonymous Report 2 on 2021-8-29 (Invited Report)


1. Contains novel conjectures regarding the computation of degenerate conformal blocks that may find other applications.
2. Very thorough in its presentation of technical results.


1. Rather technical subject matter.


In this paper the authors study correlation functions of local operators in the Ashkin-Teller model, the $\mathbb{Z}_2$ orbifold of the $c=1$ free boson CFT.

The authors begin with a review of correlation functions in the unorbifolded free boson theory. This includes an extended discussion of correlation functions of degenerate Virasoro primary operators that are not affine primaries. Although the conformal data of the latter are completely determined by those of the current algebra primaries via the affine symmetry, extracting this conformal data in this way in practice is highly tedious. The authors circumvent this by imposing crossing symmetry of the correlators of these Virasoro primaries directly, although this still turns out to be a rather technical exercise due to analytic subtleties of the conformal blocks when the dimensions correspond to those of degenerate representations.

The authors then proceed to study correlation functions in the orbifolded theory. This includes a, to the best of my knowledge, novel determination of structure constants involved in mixed four-point functions of affine primaries in the untwisted sector and twist fields. Together with the known four-point functions of affine primaries in the untwisted sector and of twist fields, this establishes consistency of the theory on the sphere. The authors also study correlation functions of degenerate Virasoro primaries and current algebra descendants of the twist fields, encountering similar subtleties as in the degenerate correlators of the free boson.

In studying the correlators of degenerate Virasoro primaries, the authors must compute the corresponding degenerate conformal blocks (or fusion kernels). In practice, the computation of degenerate blocks is subtle. The reason for this is that methods for the computation of generic conformal blocks view the blocks as meromorphic functions of (say) the central charge, with poles appearing when internal states correspond to degenerate representations of the Virasoro algebra (signalling the presence of null states). The authors propose a concrete method to determine degenerate conformal blocks as limits of generic conformal blocks in the form of two conjectures. These proposals might be of broader interest than the results on crossing symmetry in the Ashkin-Teller model.

The paper concerns rather technical subject matter but is well-written and very thorough in the presentation of its results. I recommend the publication of this paper in SciPost.

Requested changes

1. The notation $k_{123}$ and $k^1_{23}$ are undefined when used in (3.49). Although they are defined after (5.42), these definitions should be moved up to their first use in the main text.

2. The authors should discuss the extent to which they have checked their conjectures regarding degenerate conformal blocks as limits of generic conformal blocks.

  • validity: high
  • significance: ok
  • originality: good
  • clarity: high
  • formatting: perfect
  • grammar: perfect

Anonymous Report 1 on 2021-8-17 (Invited Report)


1. Very thorough and detailed.
2. Contains a conjecture which might be of interest also to readers who are not necessarily interested in the specific model.


1. Very very technical.
2. Presentation could be improved.


In this papers the authors study the Ashkin-Teller model. This has been solved from the point of view of the affine algebra, meaning that the conformal data in terms of the affine primaries is known. Conformal data for the Virasoro primaries can then be obtained, in principle. However, this becomes rather technical; the authors study directly correlation functions of Virasoro primaries. By imposing crossing symmetry of four point functions, the three point function coefficients are found.
The paper also contains a pair of conjectures on how to use Zamolodchikov’s recursion relation for the Virasoro blocks in case of degenerate operators. Naively, this is ill defined. The authors instead propose a precise way of deforming the theory so that the correct Virasoro blocks can be obtained. This might be of interest also to readers who are not necessarily interested in the Ashkin-Teller model.
I find the paper very thorough and rather technical. Most of the edits I suggest concern notation and presentation of the paper. Exceptions are the more important points 5 and 6, which deal with the precise recipe of Conjecture 2.

Requested changes

1. I have some remarks about the notation of section 2: starting from (2.11), the summation variable $s$ should be renamed since $s$ appears also as the letter indicating that we are expanding in the $s$-channel. (2.19) could be made leaner by replacing $\Delta_j \to j$, as is done in most of the section. Should (2.20) include $\Delta_t,\bar \Delta_t \in \text{spectrum} $ or something of the kind? Does (2.26) need a $\pm$ in front?

2. I find the minus sign difference between (3.39) and (3.46) confusing. Since the sign is not determined, can we choose it to be the same in these two expressions?

3. A clearer explaination of the notation of (3.41) would help (I mean which operator is at which coordinate in $F_{0,\epsilon} \big[\ldots\big]$)

4. I imagine that (4.19) could also be zero for some values of the $(n_i,m_j)$, given equation (4.10). Is that right? If so, it should be indicated in (4.19).

5. The recipe of Conjecture 2 doesn't specify which degenerate representation we should consider. There are several $\mathcal{R}^{(c)}$ who have the same number of states as $\mathcal{R}$ at generic central charge. The question of which $\mathcal{R}^{(c)}$ one must choose should be addressed, but it isn't. For example, above eq (5.16), the authors choose one of the two possible structures, but it's not explained why.

6. Probably related to the previous point, in which cases did the authors check the validity of the conjectures? Which conformal blocks can be computed explicitly and then compared to the conjecture? Is the check up to some level? It would be good to include this.

7.It should be explained what the pentagon diagram of (5.32) symbolizes so that a reader not familiar with it could understanding without looking at reference [9].

  • validity: good
  • significance: good
  • originality: ok
  • clarity: good
  • formatting: perfect
  • grammar: excellent

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