The Virasoro Completeness Relation and Inverse Shapovalov Form

TL;DR

This work tackles the lack of a closed-form inverse Shapovalov form for Virasoro Verma modules at generic central charge by deriving an explicit index-free expression $\mathbf{S}_\ell^{-1}(L_0,\hat{c})$ written in terms of singular vectors and their conformal dimensions. The authors formulate a level-by-level sum over singular-vector operators, with coefficients $q_{\langle r,s\rangle}$ built from Zamolodchikov’s regularized norms, and show how this structure yields a complete resolution of the identity $\mathds{1}(h,c)$ on Verma modules, thereby enabling direct computation of Virasoro conformal blocks via sewing. The paper proves the completeness relation and provides a detailed inductive proof that the proposed inverse Shapovalov form acts as the identity on all level-$\ell$ descendants, while clarifying the pole structure (simple poles at $h=h_{\langle r,s\rangle}$) and the role of singular vectors in the construction. These results offer a practical, OPE-compatible route to compute conformal blocks and pave the way for applications to the AGT correspondence and conformal bootstrap, as well as potential generalizations to related algebras and generalized Casimir operators.

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.

Theorems & Definitions (4)

• Lemma 1
• proof
• Lemma 2
• proof