Virasoro OPE and conformal blocks from the inverse Shapovalov form

We derive expressions for the Virasoro OPE and four-point conformal blocks on the sphere via the resolution of identity recently determined in [J.-F. Fortin et al., Phys. Rev. D 111, 085010 (2025)]. Even though the resolution of the identity depends on Virasoro singular vectors, our expression for the blocks does not depend on their precise form, but just on their well-known conformal weights. We verify that our expression is compatible with—but differs from—Zamolodchikov’s h h -recursion relation and we also examine the impact of various large central charge limits in our formula. A Mathematica notebook with a simple implementation of our expression for the Virasoro conformal blocks is provided as an ancillary file.