Casimirs of the Virasoro Algebra

TL;DR

This paper explicitly derives the Casimir operators of the Virasoro algebra by solving a recurrence relation and expressing them through the inverse of the Shapovalov form, linking singular vectors and algebra invariants.

Contribution

It provides a new explicit formula for Virasoro Casimirs using the inverse Shapovalov form, advancing understanding of algebraic invariants in conformal field theory.

Findings

Derived explicit formulas for Virasoro Casimirs.

Connected Casimirs with singular vectors and the inverse Shapovalov form.

Enhanced computational methods for algebraic invariants.

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.