What question did this study set out to answer?

To derive expressions for the Virasoro OPE and four-point conformal blocks using the resolution of identity.

May 24, 2026Open Access

Virasoro OPE and conformal blocks from the inverse Shapovalov form

Key Points

To derive expressions for the Virasoro OPE and four-point conformal blocks using the resolution of identity.
Utilized the resolution of identity based on Virasoro singular vectors.
Derived expressions independent of the precise form of singular vectors, focusing solely on conformal weights.
Provided a Mathematica notebook for the implementation of the derived expressions.
Verified compatibility with Zamolodchikov's h-recursion relation, noting differences in expressions.
Examined the impact of various limits for large central charge on the derived formulas.

Abstract

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-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.

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