On the Virasoro fusion kernel at $c=25$

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.

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:

1. 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.

2. 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.

3. 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 time-like Liouville partition function. It is unclear to us whether there exists a well-defined 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