On the Virasoro fusion and modular kernels at any irrational central charge

Crossing kernels are fundamental kinematical objects in conformal field theory. They describe how conformal blocks are mapped to each other under change of OPE channel, and have played a central role in recent analytic approaches to the conformal bootstrap in two dimensions. A formula for the crossing kernel for Virasoro sphere four-point conformal blocks with central charge $c$ in the region $c\in\mathbb{C}\setminus(-\infty,1]$ is remarkably known in closed form due to work of Ponsot and Teschner, but no general formula is known in the case $c\leq 1$ and the Ponsot-Teschner formula cannot be analytically continued to this region.

The main result of this paper is to propose an infinite series representation, with recursively determined coefficients, for the Virasoro sphere four-point (and torus one-point) crossing kernels. The author argues for this series representation by demonstrating that it solves a shift equation that is a renormalization of that satisfied by the Ponsot-Teschner kernel (which itself descends from the pentagon equation). This also leads to an unfamiliar infinite series representation for the crossing kernel in the unproblematic region $c\in\mathbb{C}\backslash (-\infty, 1]$, whose equality with the Ponsot-Teschner kernel the author verifies numerically. The author verifies their proposal by showing that it numerically reproduces the known formulas at special values of the central charge and external conformal weights, and essentially, by numerically checking that it implements the crossing transformation on the conformal blocks. They also say that the formula numerically satisfies a functional relation that implies crossing symmetry of timelike Liouville CFT based on Zamolodchikov’s proposal for the $c\leq 1$ structure constants. None of these checked properties are manifest from the series representation.



To my mind, these checks are strong indications of the proposal’s correctness. However the infinite sum representation obscures the analytic structure of the crossing kernel, which will play an important role in most applications. An obvious target for future work will be to find a more efficient representation of the kernel akin to the Ponsot-Teschner formula (although an obstacle is that it must have weaker analyticity properties than the latter). The paper is well-written and the main result is novel and should find applications in the future. I recommend it for publication in SciPost.


Before publication I have some comments/questions that the author may wish to comment on:
- In the discussion of the relation between the sphere four-point and torus one-point blocks, there should be a reference to the paper 0902.1331 by Fateev, Litvinov, Neveu and Onofri
- The errors presented in the tables on pages 6 and 8 would be more meaningful if they were normalized in some way. Relatedly, the author should be more explicit about what is meant by "accuracy" in these tables.
- I wonder if the combination $f^{(b)}_{P_s,P_t}/ f^{(-ib)}_{iP_s,iP_t}$ (omitting the external operator labels) simplifies in the same way as $K$ prefactors.
- I also wonder if the sphere four-point kernel simplifies in the case of pairwise identical external operators with exchange of the identity in the T-channel. In that case the Ponsot-Teschner kernel essentially reduces to the Liouville CFT structure constant.

Publish (meets expectations and criteria for this Journal)

1. Low significance of the obtained results.
2. Lack of a proof of convergence of the obtained series representation of the crossing kernel.

The integral kernel of the crossing transformation for the four-point correlation function studied in this work is an important object in two-dimensional conformal field theory. The work also contains a number of original results. However, I have several critical comments about it.

First and foremost, I would like to point out that the possibility of numerically, and therefore necessarily approximately, determining the values of the distribution represented by the studied object for specific argument values — which, in my opinion, is how the representation in the form of a series should be viewed — is not particularly significant. For proving the consistency of a conformal field theory model, what is important are the general analytical properties of this distribution, more precisely, the orthogonality relations it satisfies, which, by the way, the Author is aware of. These properties are known, and the reviewed work does not introduce anything new in this context.

A major drawback of the obtained result is the lack of proof of convergence for the series discussed in the work, as well as the lack of discussion on the range of arguments for which the series is convergent. One of the consequences of not providing such a proof is the inability to discuss the positions and types of singularities of the analyzed integral kernel.

The work should also be considered incomplete, as it contains — as the Author admits — statements that are merely hypotheses.

Reject

The Virasoro fusion and modular kernels are fundamental quantities in 2d conformal field theory. These quantities describe changes of bases of conformal blocks, and may be used for writing crossing symmetry and modular invariance equations -- the basic equations of the bootstrap approach.

These quantities are complicated functions of 7 and 4 parameters respectively, and obey many remarkable identities. It would be very useful to have explicit expressions of these quantities, although it is probably unrealistic to hope for a single expression that would lend itself to easy analytic manipulations, efficient numerical calculations, and have transparent symmetry properties. In the case of conformal blocks, we know various expressions of the same quantity, each one with its own advantages and limitations, including: a pedestrian sum over Virasoro descendants, two recursive formulas due to Zamolodchikov, and a sum over partitions from the AGT relation.

In the case of the fusion kernel, the situation greatly depends on the value of the central charge $c$: for $c\in \mathbb{C}-(-\infty, 1]$, we have 2 different integral formulas by Ponsot--Teschner and Teschner--Vartanov. And for $c\leq 1$, no explicit expression is known, except in special cases. However, many interesting CFTs have $c\leq 1$, starting with minimal models. In the case of the modular kernel, the situation is similar.

The article by Roussillon conjectures two series representations for the fusion kernel, valid for $c\in \mathbb{C}-(-\infty, 1]$ and $c\leq 1$. For $c\in \mathbb{C}-(-\infty, 1]$, this representation is numerically found to agree with the known integral representations. For $c\leq 1$, there is nothing to compare it with, so it is tested by directly checking the fusion relation for conformal blocks. The article also gives similar series representations for the modular kernel.

The series representations are obtained as solutions of certain shift equations that the kernels obey. These solutions are expected to be unique under certain analyticity assumptions. However, it is hard to prove convergence of the series representations, and therefore to control the analytic properties of the resulting kernels. This is why numerical tests are a crucial complement to solving the shift equations.

Additional evidence comes from the study of quite a few special cases where the kernels simplify, and from checks of remarkable identities that the kernels obey, but which are not manifest in the series representations. On balance, this article makes a very strong case for the conjectured series representations.

The article is generally well-organized and clearly written, although some local improvements are possible.

Remarks on substance:

1. The relation between torus 1pt blocks and sphere 4pt blocks is attributed to Poghossian, but the formulas (1.7), (1.8) look much more like the relation by Fateev et al https://arxiv.org/abs/0902.1331, which differs from Poghossian's, and appeared earlier.

2. In Section 1.4, it would be interesting to know more about the context and motivations of Ruijsenaars' work. In that work, how is $M$ defined, if not as a modular kernel? Why is this quantity interesting outside the context of CFT?

3. The discussion of convergence in Section 2.3 is welcome, but the statements could be clarified. A number of assumptions on $b$ and $P_i$ are made: are these necessary for convergence, or just convenient for making a plausible argument? Based on analytic arguments and numerical observations, are there simple conjectures for regions in parameter space where the series converges or diverges? And would it not be simpler to study convergence of the modular kernel?

Clarity and other formal issues:

1. In the introduction, when citing the uniqueness result of [4], both notations $b$ and $c$ are used, which is confusing.

2. Typos in (1.19) and (3.13): $Ps \to P_s$.

3. In the unnumbered table on page 6, what exactly is the "accuracy"? Rather than the real (or imaginary) part, why not compute the complex modulus of the difference?

4. At the beginning of Section 1.3.2, the numerical value of the Ponsot--Teschner formula $0.308+5.22\times 10^{-22}$ is mysterious. Should we understand that the exact value is $0.308$, and the second term a numerical artefact?

5. At the end of Section 1.3.2, the argument for the factor of 2 is obscure, a clarification would be welcome.

6. After Eq. (2.9), it is a bit confusing to delay the relation between $D$ and $H$ until Section 2.2, rather than just write (2.14)-(2.17) there. Also, it could be stated more explicitly that (2.10), (2.11) are equivalent to (2.4), (2.5).

7. At the end of Section 2.1, "to high order" could be made more quantitative.

8. At the end of Section 2.2, for the Virasoro--Wick rotation, it would be more convenient to cite Eq. (2.6) rather than ref. [23].

9. At the end of Section 2.3, it would be interesting to comment on the result (2.20), and explain why this result is useful or interesting.

10. In Eq. (3.8), it might be good to write the simple end result for the limit of $K^{(b)}$, in addition to the ratio of $G$ functions that will eventually simplify.

NB: As a matter of principle, I have no recommendation about publication in this or that journal. Since it is necessary to select a recommendation for submitting a report, I picked the first one in the list.

Publish (surpasses expectations and criteria for this Journal; among top 10%)