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On the Virasoro fusion kernel at $c=25$
by Sylvain Ribault, Ioannis Tsiares
Submission summary
Authors (as registered SciPost users):  Sylvain Ribault · Ioannis Tsiares 
Submission information  

Preprint Link:  https://arxiv.org/abs/2310.09334v2 (pdf) 
Date accepted:  20240807 
Date submitted:  20240603 09:57 
Submitted by:  Tsiares, Ioannis 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approaches:  Theoretical, Computational 
Abstract
We find a formula for the Virasoro fusion kernel at $c=25$, in terms of the connection coefficients of the Painlev\'e VI differential equation. Our formula agrees numerically with previously known integral representations of the kernel. The derivation of our formula relies on a duality $c\to 26c$ that is obeyed by the shift equations for the fusion and modular kernels. We conjecture that for $c<1$ the fusion and modular kernels are not smooth functions, but distributions.
Author indications on fulfilling journal expectations
 Provide a novel and synergetic link between different research areas.
 Open a new pathway in an existing or a new research direction, with clear potential for multipronged followup work
 Detail a groundbreaking theoretical/experimental/computational discovery
 Present a breakthrough on a previouslyidentified and longstanding research stumbling block
Current status:
Editorial decision:
For Journal SciPost Physics: Publish
(status: Editorial decision fixed and (if required) accepted by authors)
Reports on this Submission
Strengths
1 A new formula for the fusion kernel for c=25 Virasoro conformal blocks suggested using the introduced VirosoroWick transformation,
2 Numerical check of coincidence of new fusion kernel for c=25 with the TeschnerPonsot fusion kernel.
Weaknesses
1 There is no analytic proof for the coincidence of the proposed fusion kernel for c=25 with the TeschnerPonsot fusion kernel. It is not critical for publishing the paper, but it seems to be a good task for the future since it may reveal some important mathematical structures.
Report
The paper "On the Virasoro fusion kernel at c=25" by Sylvain Ribault and Ioannis Tsiares is devoted to studying the fusion kernels of Virasoro conformal blocks. The fusion kernels are kernels of integral transformations relating 4point Virasoro conformal blocks in s and t channels. These two channels correspond to two bases in the space of conformal blocks associated with different asymptotic regions in the space of positions of four fields. The fusion kernels relating these two bases can be considered as a deep generalization of connection matrices for hypergeometric series.
Teschner and Ponsot found an explicit formula for the fusion kernel valid for any real central charge $c>1$ (and for central charges with nonzero imaginary parts). Later, a formula for the fusion kernel for $c=1$ was proposed by Iorgov, Lisovyy, and Tykhyy relating the fusion kernel with the connection constant of Painleve VI equation.
These types of formulas for the fusion kernel are given for different values of cental charge and look very different.
Sylvain Ribault and Ioannis Tsiares in this paper propose a socalled VirosoroWick transformation for the fusion kernels relating central charges $c$ and $26c$.
It allowed them to suggest an alternative formula for the fusion kernel for $c=25$ using the fusion kernel for $c=1$ and to check numerically its coincidence with TeschnerPonsot formula.
For the future, it is interesting to find an analytic proof of this coincidence and to try to find a generalization of this story to some other central charges.
I recommend the paper "On the Virasoro fusion kernel at c=25" by Sylvain Ribault and Ioannis Tsiares for publication in SciPost.
Recommendation
Publish (easily meets expectations and criteria for this Journal; among top 50%)
Strengths
1. Wellwritten.
2. Propose a new formula for c=25 Virasoro fusion kernel. Very interesting results.
3. Clear derivation and proof.
Weaknesses
/
Report
See attached report.
Recommendation
Publish (easily meets expectations and criteria for this Journal; among top 50%)
Author: Ioannis Tsiares on 20240729 [id 4662]
(in reply to Report 1 on 20240722)
Dear referee,
Thank you for your time and your useful comments.
We address the three points you raised at the end of your report below:

In our (2.17), the integral factor indeed has tetrahedral symmetry, as it is invariant under any permutation of the edges that preserves the vertices. On the other hand, the prefactor singles out the $t$edge. With a different prefactor, we could obtain 6j symbols that respect tetrahedral symmetry, as in Eq. (2.27) of ref. [11]. Our prefactor follows from the natural normalization (1.10) of conformal blocks.

The scenario is that for generic $c$ the infinite sum is obtained from the integral expression by rewriting it as sum of residues of its poles. It is not straightforward to do this, as we cannot move the integration contour to infinity, but this may be possible with the help of the shift equations for the integrand. Then, for $c\to (\infty, 1)$, the integral expression no longer makes sense, and the infinite sum becomes divergent, and may be interpreted as a distribution. For $b^2\in \mathbb{Q}$, we indeed expect a similar picture as for $b=i$ and $b=1$. Again, this would require nontrivial analytic manipulations of the integral formulas.

Assuming that the physical kernels for $c\leq 1$ are distributions, there are, in principle, no new surprising implications for the inner product of conformal blocks: the inner product, just like the kernels, should always be thought of as integrated against a test function (i.e. a conformal block) to yield a physical observable and hence it is OK if it happens to be a distribution as a function of $P$ (the integrated momentum). On the other hand, in the recent work by Collier, Eberhardt, and Zhang it was shown that for $c>25$, the space of Liouville conformal blocks forms a Hilbert space of a 3d TQFT that corresponds to the quantization the Teichmuller space of Riemann surfaces. The corresponding inner product takes the form of an integral over Teichmuller space with a measure given by the bc ghosts and a timelike Liouville partition function. It is unclear to us whether there exists a welldefined TQFT along the lines of this work for $c\leq 1$, and whether the corresponding inner product would take an analogous form (i.e. as an integral over Teichmuller space) assuming that it's a distribution. Deriving rigorously the $c\leq 1$ kernels from the CFT perspective would definitely shed light on these interesting questions.
Kind regards, The authors
Author: Ioannis Tsiares on 20240802 [id 4675]
(in reply to Report 2 on 20240731)Dear referee,
Thank you for your time and the comments in your report.
Regarding the weakness comment, we'd like to mention that we have an upcoming publication that explains the analytic origin of the formula for the c=25 fusion kernel.
Kind regards,
The authors.