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It is known in Liouville CFT from the crossing symmetry that the four points $s$-channel and 4t$-channel conformal blocks are related to each other via an integral transformation

$$\mathcal{F}\left[\begin{matrix}\theta_1,\theta_{t}\\ \theta_{\infty},\theta_0 \end{matrix};\sigma,t\right] = \int d\rho F\left[\begin{matrix}\theta_1,\theta_{t}\\ \theta_{\infty},\theta_0 \end{matrix};\begin{matrix}\sigma \\ \rho \end{matrix} \right] \mathcal{F}\left[\begin{matrix}\theta_0,\theta_{t}\\ \theta_{\infty},\theta_1 \end{matrix};\rho,1-t\right],$$

where $F\left[\begin{matrix}\theta_1,\theta_{t}\\ \theta_{\infty},\theta_0 \end{matrix};\begin{matrix}\sigma \\ \rho \end{matrix} \right]$ is the fusion matrix and has been constructed by Ponsot and Teschner in https://arxiv.org/abs/math/0007097.

The fusion matrix is quite complicated, and is essentially a contour integral of a product of quantum dilogarithm functions.

However, there is a special case (when one of the conformal dimensions corresponds to a degenerate field) where the conformal blocks become the hypergeometric function $_2F_1$. The integral simply becomes a sum of 2 terms, and it gives the connection formula for the hypergeometric function $_2F_1$.

I would like to understand how to get these known connection formulas from the general fusion transformation. I guess that one has to compute some residues, but in details I don't know how to do.

Qmechanic
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Gropillon
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1 Answers1

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First of all let me point out that Ponsot-Teschner has been superseded by Teschner-Vartanov https://arxiv.org/abs/1202.4698 , whose formulas are more symmetric (although not less complicated).

Then the fusion transformations is a feature of the Virasoro algebra, not specifically of the Liouville CFT.

You are right that the contour integral reduces to a product of Gamma functions if one of your four fields is degenerate at level two. I do not know a reference that does this in detail but I know two related calculations:

  • The case of the identity field (instead of a level two degenerate field) is explained in Section 4 of this article: https://arxiv.org/abs/1606.07458 This is slightly simpler than what you want, but the idea is the same.

  • The case of OPEs in Liouville theory (instead of the fusion transformation of the Virasoro algebra) is done in Section 3.1.4 of my review article https://arxiv.org/abs/1406.4290 for arbitrary degenerate fields, not just level two. This is technically simpler than what you want, but again the idea is the same.

Sylvain Ribault
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