The Virasoro fusion kernel and Ruijsenaars' hypergeometric function

This is a review of the preprint by Roussillon.

Context and general comments
Virasoro conformal blocks play a fundamental role in 2d CFT. The Virasoro fusion kernel describes the linear relation between s-channel and t-channel conformal blocks. This relation is essential for the conformal bootstrap approach to 2d CFT, as it underlies the crossing symmetry equation. Crossing symmetry is commonly written in terms of conformal blocks, but it is also possible to eliminate the blocks and write it in terms of the fusion kernel, which is a somewhat less complicated object.

This justifies the study of the Virasoro fusion kernel, and of its many symmetries and non-trivial properties. The present article shows that this kernel coincides with an already known special function called Ruijsenaars' hypergeometric function, provided the parameters are appropriately mapped, and appropriate normalization prefactors are included. This potentially opens the gate to flows of results from each side to the other, for example:
 * From the Ruijsenaars' side, the $$D_4$$ symmetry and symmetries from Lemma B.1.
 * From the Virasoro side, the pentagon equation and the symmetries that follow from the relation with the Liouville boundary three-point function.

Validity
The Virasoro fusion kernel and Ruijsenaars' hypergeometric function have integral representations that are manifestly of the same type. Given their complexity, it is a priori not obvious that they are related. To discover the relation, the author relies on the difference equations that these objects obey. The proof of the difference equations for the Virasoro fusion kernel in the Ponsot-Teschner representation boils down to elementary but complicated manipulations of trigonometric functions.

Once the relation is known, it can be proved using Lemma B.1: a nontrivial identity for hyperbolic Barnes integrals. If we accept this identity, the proof is short and simple, and can be easily checked. Therefore, there is little doubt that the main result is correct.

Interest
This article establishes an identity between two complicated objects that were not known to be related, although both were known for about 20 years. This brings together two different strands of the literature, one mathematical strand about special functions and one strand from mathematical physics about 2d CFT.

Although the result is purely technical, and although its proof is rather easy, it can potentially be interesting on both sides. Just translating known results from one side to the other would probably lead to nontrivial identities. Moreover, due to the interpretation of Ruijsenaars' hypergeometric function in terms of an integrable system, there is a potential for new insights on the role of integrability in 2d CFT.

Clarity
The article is clear and well-written, in particular each technical step is easy to follow.

Notations are not always optimal: calling the four external momentums $$(\theta_0,\theta_t,\theta_1,\theta_\infty)$$ makes symmetries less clear, differs a lot from the Ruijsenaars parameters $$(\gamma_0,\gamma_1,\gamma_2,\gamma_3)$$, and leads to the external momentum $$\theta_t$$ having the same index as the internal momentum $$\sigma_t$$. The overall structure of the reasoning is not always optimal: after devoting much effort to difference operators, the reader naturally expects them to play a role in proving the main result, but this is not the case.

Recommended minor changes

 * 1) In the Introduction, $$b$$ should be defined from the central charge before being used.
 * 2) The clause "From the conformal blocks viewpoint" could be eliminated.
 * 3) After discussing rank $$N$$ systems, it would be good to write explicitly that the article will consider the case $$N=1$$.
 * 4) Sequences of poles are called "increasing/decreasing" or "upper/lower": it would be better to adopt one terminology.
 * 5) The sentence before Eq. (2.4) does not make it clear that this is the definition of $$R$$. Anyway only $$R_\text{ren}$$ is used, maybe $$R$$ is not needed.
 * 6) After Eq. (2.7), it would be interesting to state the $$D_4$$ symmetry, at least by giving its action on $$\gamma$$ (if not the prefactors that make $$R_\text{ren}$$ invariant).
 * 7) At the very end of Section 2, the paragraph on integrable systems is rather vague: the author is writing either too little, or too much. It might be better to merge these considerations into point (1) of the Conclusion.
 * 8) The definition of the prefactor $$N$$ of chiral vertex operators is relegated to Section 5, although we need to know it for understanding the fusion kernel Eq. (4.1). If that kernel corresponds to $$N=1$$, this prefactor could be omitted from Eqs. (3.5b) and (3.7). Its appearance could wait until the definition of renormalized blocks (5.9).
 * 9) The mention that the AGT relation leads to closed-form expressions for (coefficients of) conformal blocks is slightly misleading, as these expressions are generally less useful, and hardly more explicit, than Zamolodchikov's recursion.
 * 10) "The linear span of four-point Virasoro conformal blocks" is vague: is this a space of functions of $$z$$ spanned by conformal blocks with fixed external dimensions but varying $$\sigma_s$$?
 * 11) The role of Assumption 4.1 could be clarified. It could be stated just before being used (rather than long before), and its role in the proof of Theorem 1 (if any) could be stated.
 * 12) The proof of Proposition 4.3 could be streamlined. First, the idea that acting with the difference operator results in a shift of the integration variable could be stated more explicitly (and from the start). Then, the proof of $$f_1(\sigma_s)=f_2(\sigma_s)$$ does not bring much more than saying that these two trigonometric functions agree by a brute force calculation. Finally, Eqs. (4.31a) and (4.31c) might be unneeded, as they are quite obvious, and all we need is Eq. (4.33).
 * 13) In the Introduction and at the beginning of Section 5, the enumeration of the three obstacles to identifying $$F$$ and $$R_\text{ren}$$ is a bit misleading, given the simplicity of the eventual proof. It could be made clearer than the proof of Theorem 1 is rather simple using Eq. (B.4). There is little need to insist on the obstacles: the complexity of the formulas speaks for itself.
 * 14) The proof of the main result could be made even shorter, by combining the various operations into one identity of hyperbolic Barnes integrals: the operation $$\omega$$, the changes of integration variables, and the identity for $$s_b$$.