Casimirs of the Virasoro Algebra

TL;DR

This work resolves the long-standing problem of obtaining all-level Casimir operators for the Virasoro algebra by refining the Feigin–Fuchs recurrence and expressing the Casimirs in terms of the inverse Shapovalov form and singular-vector data. By combining this improved recurrence with a prior construction of the inverse Shapovalov form, the authors derive an explicit all-level formula for generic Virasoro Casimirs, encapsulating contributions from singular vectors and their conformal dimensions through a structured product over Virasoro generators. They identify distinct global and local constructions: globally, Casimirs can be written as $\mathcal{C}^g(M_0)$ with $M_0$ tied to the quadratic Casimir $\mathcal{Q}$ via $M_0=\tfrac12\bigl(1-\sqrt{1+4\mathcal{Q}}\bigr)$, while locally they reduce to functions of the Heisenberg zero mode $a_0$ and are captured by $\mathcal{C}(M_0)=M_0\bigl(\tfrac12 a_0^2 - \tfrac{1}{\sqrt{2}}(\sqrt{t}-1/\sqrt{t})a_0,\hat c\bigr)$. The results illuminate the pole structure at singular-vector dimensions and offer a natural route to expressing Virasoro Casimirs via singular vectors, with potential applications to Virasoro conformal blocks and extensions to arbitrary algebras.

Abstract

We explicitly solve a recurrence relation due to Feigin and Fuchs to obtain the Casimirs of the Virasoro algebra in terms of the inverse of the Shapovalov form. Combined with our recent result for the inverse Shapovalov form, this allows us to write the Casimir operators as linear combinations of products of singular vectors.