The non-rational limit of D-series minimal models

As the anonymous referee, I must apologise for the late report.

This is certainly a very stimulating paper. I have spent some time thinking about it and have found it hard to come to a conclusion for reasons below.

I finds it hard to accept one of the main conclusions, which is "We will therefore have to conclude that the diagonal and mixed four-point functions cannot belong to the same CFT". If the limiting procedure results in a CFT then it is a CFT; if it does not, then what is it?

The main problem appears to be the non-analyticity in $\beta^2$ arising from the shift $\sigma(P)$ in equation (2.25). I was for some time misled by this formula which appears to suggest that $\sigma(P)$ is defined independently of $\{r_2,s_2,r_3,s_3\}$. However, since the sign has to change to accommodate changes in sign of three point functions of fields with spin, this cannot be true - for some pairs of $(r_2,s_2)$ and $(r_3,s_3)$, the three point function has to change sign and so $\sigma(P)$ does depend on $\{r_2,s_2,r_3,s_3\}$.

Unfortunately, I cannot find a formula for $\sigma(P)$, or for $C_{P_1,{r_2,s_2},{r_3,_3}}$ itself. It says in [5] that the shift equations are solved (for a sign $f_{2,3}(P)$) but I could not find that solution in that paper, which may very well just be my fault. As it is, the absence of a clear formula for $C_{P_1,{r_2,s_2},{r_3,_3}}$ makes it hard to check whether there are other ways to fix the problem of non-analyticity.

I could imagine that the sign $\sigma(P)$ might have a component (like a cocyle) that depends on $\{r_2,s_2,r_3,s_3\}$ and part that depends on $P$ in such a way that one could split the fields into two sets, one for two possible signs, and for which new fields then have structure constants that are analytic, up to some signs required because of the presence of fields with spin - but it is hard to start thinking about this in the absence of a formula for $C_{P_1,{r_2,s_2},{r_3,_3}}$.

So, to summarise, I have found the ideas very stimulating but have been unable to decide what I think of the results presented here. They seem to me outside what I regard as "likely" and I have not been able to do the sorts of tests of the ideas I would like to do because I have not been able to find the required formulae. This could just be my incompetence, for which I apologise.

If the author can provide a formula for $\sigma(P)$, either explicitly, or a clear indication where it is to be found, then I think that would be a substantial improvement and at that stage I would be very likely to recommend publication. Even if the eventual conclusions are wrong, the paper is still very interesting and stimulating.