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String correlators on AdS$_3$: analytic structure and dual CFT
by Andrea Dei and Lorenz Eberhardt
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Submission summary
| Authors (as registered SciPost users): | Andrea Dei · Lorenz Eberhardt |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202204_00038v1 (pdf) |
| Date submitted: | 2022-04-22 14:46 |
| Submitted by: | Dei, Andrea |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We continue our study of string correlators on Euclidean $\text{AdS}_3$ with pure NS-NS flux. The worldsheet and spacetime correlators have a rich analytic structure, which we analyse completely for genus 0 four-point functions. We show that correlators exhibit a simple behaviour near their singularities. The spacetime correlators are meromorphic functions in the $\mathrm{SL}(2,\mathbb{R})$-spins, whose pole structure is shown to agree with the prediction of a recent proposal for the dual $\text{CFT}_2$. Moreover, we also compute the residues of the spacetime correlators for some of the poles exactly and find again a perfect match with the proposal for the dual $\text{CFT}_2$, thereby checking the duality for some non-trivial four-point functions exactly. Our computations simplify drastically in the tensionless limit of $\mathrm{AdS}_3 \times \mathrm{S}^3 \times \mathbb{T}^4$ where the behaviour near the poles gives in fact the exact answer. This paper is the third in a series with several installments.
Current status:
Reports on this Submission
Report #2 by Anonymous (Referee 2) on 2022-7-23 (Contributed Report)
- Cite as: Anonymous, Report on arXiv:scipost_202204_00038v1, delivered 2022-07-23, doi: 10.21468/SciPost.Report.5445
Strengths
1- Impressive technical results on worldsheet correlators
2- Detailed match with the boundary CFT2
3- Very good exposition in spite of the nature of the material
Weaknesses
1- Probably not so easy to follow for an outsider, in spite of the author's efforts
Report
This paper is part of a series on the AdS3/CFT2 correspondence. The string background under consideration only has NSNS flux, and can be described using worldsheet techniques. Most of the paper is dedicated to computing certain worldsheet correlators; at the end the results are compared with those expected from the CFT2.
The worldsheet computations are an impressive tour de force, and lead to beautiful agreement with AdS/CFT. The use of the worldsheet (rather than supergravity) allows for a more thorough match than would be possible in most instances of the correspondence. While unfortunately these techniques cannot be easily exported to other, perhaps more famous dual pairs, they are likely to provide insights that could be useful more broadly.
The authors have done their best to explain their very technical results in such a way that they can be followed to a reasonable degree. Overall the paper is very impressive.
Requested changes
Two typos:
- p. 11, "lower codimension loci as"
- p. 18, "literately"
Report #1 by Sylvain Ribault (Referee 1) on 2022-5-10 (Invited Report)
- Cite as: Sylvain Ribault, Report on arXiv:scipost_202204_00038v1, delivered 2022-05-10, doi: 10.21468/SciPost.Report.5050
Report
This article is the third in a series on string correlators in $AdS_3$, which studies the $AdS_3/\text{CFT}_2$ relation in unprecedented detail. The previous two articles have established explicit formulas for three- and four-point functions from the worldsheet CFT: starting from these formulas, the task is now to study their singularities, and compare them with singularities of the corresponding spacetime CFT correlators.
The formulas for the string correlators have rich geometrical structures, and their detailed study requires a wealth of technical results, many of which are given in the appendices. The main text focuses on the general arguments while minimizing technical details. The resulting article is quite clear and concise. The price to pay is that checking any one of the many nontrivial results can require quite a lot of unpacking. It would not be realistic for a reader or reviewer to check all the results, and neither would it be realistic for the authors to write all their derivations in detail. The question is rather whether the motivated reader has enough indications for understanding the gist of the calculations, and for being able to reconstruct any one argument should the need arise. The submitted text is already quite good in this respect, although I think that some improvements are possible and desirable.
Before making specific suggestions, let me discuss one of the article's main results: the formula (4.1) for the poles of the spacetime four-point function. This very simple result has a complicated derivation, which involves considering quite a few different cases. To find a simpler derivation (or at least a test), one might remember that the vertex operators $V^w_{j,h,\bar h}(x;z)$ should in principle be linear combinations of the fields of the WZW model. These fields are in one-to-one correspondence with states in the spectrum, and they do not depend on $x$: this $x$ may be interpreted as determining the direction in which we perform spectral flow, but we might choose it to be the same for all fields. Then we might use [17], which relates our $N$-point function with a correlation function of Liouville theory of the type
$$
\left<\prod_{i=1}^N V_{\alpha_i}(z_i) \prod_{j=1}^{N-2+r} V_{-\frac{1}{2b}}(y_j)\right>
$$
where $\alpha_i=b(1-j_i)+\frac{1}{2b}$, and the spectral flow violation $r=\sum w_i$ obeys $|r|\leq N-2$.
From standard Coulomb gas arguments, this Liouville correlation function is singular if $\sum_i \alpha_i -\frac{N-2+r}{2b} + mb+nb^{-1} = b+b^{-1}$ for $m,n\in\mathbb{Z}_{\geq 0}$. In terms of the $AdS_3$ spins and level, this condition is
$$
\sum_{i=1}^N j_i = (n+a)(k-2) + m + N-1
$$
where $a=\frac{r}{2}$ is half-integer such that $|a|\leq \frac{N-2}{2}$. For $N=4$ this agrees with Eq. (4.1), with the only difference that $a$ is now half the spectral flow violation instead of the more elaborate (4.2). The relaxation of spectral flow violation due to what is in principle just a change of bases is a mysterious feature of the results of [23, 24], maybe due to the fact that we are not dealing with a bona fide change of bases -- in other words, not just with algebra, but with analysis. In any case, this relaxation takes a rather simple form. It might be possible to guess the generalization of (4.1) to $N$-point functions on arbitrary Riemann surfaces, with the help of [19].
However, the fact that (4.1) now seems to emerge from unintegrated worldsheet correlators is at odds with the idea that $z$-integration plays a role.
Requested changes
1. The abstract mentions a series of papers in several installments, without specifying what the next installment will be about or whether it is even planned. The conclusion lists a number of open questions, without saying which are the most difficult or important ones. I would be interested to know more about the authors' opinions and plans, in order to properly understand the place of this article in their work and more generally in the subject.
2. The abundance of footnotes, and references to appendices, makes the reading less smooth than it could be. There are even two references to previous footnotes: a sure sign that material therein was not so accessory. I am not sure anything can be done about the appendices without overburdening the main text. But most footnotes could be integrated into the text.
3. In the partial list [1-25] in the introduction, many of the cited articles are outdated or irrelevant. When citing an article it would be helpful to indicate why the reader might be interested to consult it.
4. I am not sure I understand what is meant by the "times dependency" of the background, given that it is invariant under time translations.
5. "cross-ratio" seems more standard than "cross ratio" or "crossratio". At least, the authors should choose one convention.
6. The claim on page 3 that CFT correlators usually only have singularities in $z$ when vertex operators collide is dubious. Having extra singularities seems to be the norm when the symmetry algebra is larger than just conformal symmetry, as was realized in the case of the algebra $\widehat{\mathfrak{sl}}_2$ as early as [61]. (For an example with the $W_3$ algebra see arXiv:1109.6764.) A more cautious formulation of this interesting discussion would be welcome.
7. Is $P_w$ (2.9) integer?
8. Is the prefactor $S_w$ (2.10) symmetric under permutations? Does is really not have a simpler expression?
9. A short discussion of the behaviour of the four-point function (2.13a) under permutations of the four fields might be interesting. In fact, it might be useful to make permutation symmetry manifest by restoring arbitrary positions and isospins $z_1,\dots,z_4$ and $x_1,\dots,x_4$.
10. Right after (2.14), it would be useful to write $X_\emptyset$ explicitly, rather than (or in addition to) having it in (3.2).
11. I do not understand why we need chiral correlators in (3.7). The suggestion of taking one chiral term in a conformal block expansion apparently makes the determination of singularities dependent on the choice of an expansion (s-channel, t-channel or u-channel). But such choices are apparently not needed in subsequent calculations.
12. "lower codimension loci as" $\rightarrow$ "lower codimension loci than"?
13. Section 3.1 is hard to understand in the absence of a precise definition of an exceptional intersection. The main statement of this section about the existence of exceptional intersections (3.8), is implied to be "trivial" (compared to the statement that there are no other exceptional intersections).
However, the present text does not convey this simplicity to the reader.
If I understand correctly, the idea is that $X_I$ is a polynomial in the $y_i$s, and that there is an exceptional intersection whenever one of its coefficients vanishes. And this point does not depend on the specific form of the $P_w$s.
It would be nice to have a simple explanation, together with a more explicit treatment of one example.
By now the authors seem to consider $P_w(x;z)$ and $X_I$ as elementary objects just like the sine function, but they should remember that readers may not be so familiar with these objects.
14. To make (3.8) precise, it is necessary to explicitly replace $I$ with $\{i|\epsilon_i\neq 0\}$ or something of the sort.
15. I am puzzled that the left-hand side of (3.8) only depends on the sum $w+\epsilon$, whereas the right-hand side seems to depend on $w,\epsilon$ independently. A comment would be welcome.
16. On page 13, the discussion of the singularity $P_w(x;z)=0$ could benefit from reminders that $X_\emptyset = P_w(x;z) \propto c-z$. (See (3.4).)
17. On page 14 I do not see the point of mentioning that (3.12) gives the leading term and that there are subleading terms: this seems obvious enough.
18. On page 15, it is nice to show explicitly that the second critical exponent vanishes. It would be even better to say what this means and whether this was expected. This seems to be a common (if not universal) occurrence for the singularities where operators do not coincide.
19. The formula (3.20) is a free-floating expression, not an equation. It forces the authors to later refer to "(3.16) and (3.20)". It would be better to write the full result. One option is to insert (3.20) into (3.16), and later to write that the factor (3.20) results from the integral over $y_i$.
20. Typo in (3.26): $\epsilon =\sum_i j_i-k+1$ misses $\sim 0$.
21. Footnote 12 is not very enlightening, in the absence of a more detailed explanation.
22. On page 18, the expression "external variables" is not clear.
23. Section 4.1 is a rare case of an explanation that is overly detailed and complicated. It is in principle a good idea to give the general mechanism that leads to (4.7), but why not write $F_i = \rho f_i$ where $\rho$ is the distance to the codimension $m$ subvariety, and quickly conclude? It is not clear to me that the case of multiple zeros needs to be mentioned, since the $F_i$s need not be distinct.
24. On page 14, we have one of many mentions of reflection symmetry: it would be useful to properly define reflection symmetry with the reflection equation for vertex operators, rather than drop the reflection coefficient in (3.27) without explanation. This would also help in the run-up to Eq. (5.4), which is a bit obscure as it is now.
25. It would be nice to explain in more detail why only two of the conditions (4.12) are independent, and whether this relies on properties of $P_w(x;z)$.
26. In Section 4.2, if we were to impose the vanishing of an arbitrary subset of the $X_I$s, would we always obtain a singularity that is related to the cases under study by a reflection? There are $2^{16}$ such subsets, and it is not quite clear why it is enough to focus on the few cases under study.
27. In the codimension 4 case page 20, more details on "going to zero at twice the speed" would be welcome.
28. Page 21, the trivialization of $X_\emptyset$ under condition (4.25) is not obvious: its geometrical origin could be explained.
29. In Section 4.4, two statements that would deserve more explanations are "the corresponding codimension 4 singularity does not exist" and "$X_\emptyset = 0$ in this case".
30. In Section 5.2, it would be helpful to review the screening operators and their dimensions, possibly by reproducing relevant formulas from [25], for instance (2.34) and (2.47). The dimension of $\sigma_2$ and the fact that $e^{-\frac{\phi}{b}}$ has the momentum $-\frac{1}{2b}$ (which shows up in Eq. (5.8)) should be made clearer. Is it true that the dual screening has momentum $Q+\frac{1}{2b}$ with $Q=b^{-1}-b$, and that the $\ell$ term in (5.8) has contributions from both screenings? Is the integer $\ell$ positive? If not, what is the $\ell$-th order in perturbation theory?
31. In Appendix A.6, the limit to be computed "will turn out to be useful in the main text": give a more precise reference? More generally, references from the appendices to the main text could be useful.
32. In Equation (E.3), it would be preferable to have more explicit notations, rather than apologizing that the notations are misleading. Actually, the notations make it seem that the dependence on $w_i$ is simple, whereas (if I understand correctly) there is a non-trivial dependence hidden in the second line.
Author: Andrea Dei on 2022-07-25 [id 2685]
(in reply to Report 1 by Sylvain Ribault on 2022-05-10)We would like to thank the referee for his careful reading of the manuscript.
We think that the suggestion of the referee for the derivation of spacetime poles in the correlators is a good one and we did not think of it previously. It is true that what is suggested here is much simpler than what we did (assuming of course the highly non-trivial result of [19]). However, we do not think that it captures all the singularities and we think that one needs additional arguments to turn this into a full derivation. We see some problems:
$\bullet$ The H3^+/ Liouville correspondence gives the worldsheet singularities in the m-basis. As the referee says, the x-basis correlators could conceivably be obtained from the m-basis correlators by considering also correlators of descendants in the m-basis and summing an infinite number of them. This sum may or may not converge. Conversely, the m-basis correlators can be obtained from the x-basis correlators by sending $N-1$ $x_i$’s to zero and one $x_i$ to $\infty$. Generically this limit is singular and hence there is not always an m-basis correlator to each x-basis correlator. An m-basis correlator can be recovered precisely if the old `spectral flow violation rules’ are satisfied, i.e. when $|\sum_i w_i| \le N-2$. We discussed this limit in detail in our previous paper 2107.01481, see Section 4.5. We also want to mention that this feature of more non-vanishing x-basis correlators was not a mysterious feature of our previous papers, but a phenomenon that is already known for a long time. The spectral flow violation rules were established in hep-th/0111180, see their Appendix D for a precise derivation. It is also a feature in their derivation that the x-basis correlators satisfy weaker bounds than the m-basis correlators that are precisely consistent with what we find. The main point to mention this again here is that the x-basis correlators are more general than the m-basis correlators. It can (and does) happen that the x-basis correlators have more poles than the m-basis correlators which are just a particular limit of the x-basis correlators. Thus the formula suggested by the referee captures only a subset of poles in the x-basis worldsheet correlators that survive the limit described above. Moreover it of course only applies whenever the m-basis correlator is non-zero, i.e. when the m-basis spectral flow violation bound is satisfied. This argument says nothing about the x-basis correlator with, say, $w_1=100$, $w_2=110$, $w_3=120$, $w_4=140$.
$\bullet$ The second problem with the suggested derivation is that it misses the integral over $z$. This integral clearly leads to more poles, which can already be seen at the simplest example. For the correlation function of four unflowed vertex operators, the worldsheet correlator behaves as $(x-z)^{k-j_1-j_2-j_3-j_4} \times \text{c.c.}$ as the two cross-ratios approach. This behaviour clearly leads to additional poles when we integrate over $z$. Thus we maintain that it is very important to the problem that we integrate over $z$ and this changes the set of poles.
$\bullet$ One should be careful when predicting the poles from the H3+/Liouville correspondence and also take into account that the integrals over y_i in e.g. equation (3.29) of [19] may introduce additional zeroes. We would of course be more than happy to find a simpler derivation of our result and also think that this should probably be possible. Unfortunately we were not able to find such a derivation. Perhaps it is possible to turn this suggested line of attack into a proof, but we do not immediately see how this would work.
In order to highlight the fact that the integration over $z$ does matter for the set of poles in the string correlator, we also included the result for the poles of the worldsheet correlators right before Section 4.1. Hence the reader can now easily see which poles arise from integration over $z$ and which don’t.