Explicit expressions for Virasoro singular vectors

TL;DR

This paper provides explicit formulas for Virasoro algebra singular vectors, extending previous recursive methods by Benoit-Saint Aubin and Bauer et al., with practical implementation via Mathematica.

Contribution

It introduces two new explicit expressions for generic Virasoro singular vectors, generalizing earlier recursive constructions and offering computational tools.

Findings

Explicit formulas for (r,s) singular vectors derived

Two different methods for expressing vectors provided

Mathematica implementation available for practical use

Abstract

We present two explicit expressions for generic singular vectors of type $(r,s)$ of the Virasoro algebra. These results follow from the paper of Bauer et al which presented recursive methods to construct the vectors. The expressions presented here generalise the results of Benoit-Saint Aubin for the type $(1,s)$ singular vectors in two different ways: the first simply solves the recursion through the use of partitions; the second gives explicit formulae for the coefficients in a particular expansion. A Mathematica notebook is available which implements the formulae.