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 [Phys.

Rev.

D 111, 085010 (2025), arXiv:2409.12224].

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 $h$-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.