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Virasoro blocks at rational central charge from Zamolodchikov recursion

Develop a rigorous procedure to compute Virasoro conformal blocks at rational central charge by taking appropriate limits of the Zamolodchikov recursion relation and proving cancellation of the divergences that arise in this limit.

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Background

At rational central charges (as in minimal models), the Zamolodchikov recursion used to compute Virasoro blocks develops apparent divergences even though the blocks themselves remain finite, implying delicate cancellations must occur.

A general, systematic method to take the rational‑c limit within the recursion and demonstrate the cancellations has not been established, limiting exact computations at rational points.

References

Of course, the Virasoro block itself does not diverge, so each divergence must cancel out nicely. Therefore, it is theoretically possible to obtain the Virasoro blocks for rational central charges by taking the appropriate limit from the Zamolodchikov recursion relation. However, due to technical difficulties, this remains an open problem.

— Modern Approach to 2D Conformal Field Theory (2412.18307 - Kusuki, 24 Dec 2024) in Section 3.1 (Minimal Models) — Conformal Bootstrap in Minimal Models, footnote to Equation (3.62) [Ising Virasoro blocks]