The Virasoro Completeness Relation and Inverse Shapovalov Form

TL;DR

This paper presents an explicit formula for the inverse Shapovalov form in Virasoro Verma modules, facilitating the computation of conformal blocks and the resolution of identity in conformal field theory.

Contribution

It introduces a novel explicit expression for the inverse Shapovalov form using singular vector operators and conformal dimensions, advancing the mathematical tools for Virasoro algebra analysis.

Findings

Explicit inverse Shapovalov form formula derived

Resolution of identity for Virasoro Verma modules established

Enables improved computation of conformal blocks

Abstract

In this work, we introduce an explicit expression for the inverse of the symmetric bilinear form of Virasoro Verma modules, the so-called Shapovalov form, in terms of singular vector operators and their conformal dimensions. Our proposed expression also determines the resolution of the identity for Verma modules of the Virasoro algebra, and can be thus employed in the computation of Virasoro conformal blocks via the sewing procedure.