The super-Virasoro singular vectors and Jack superpolynomials relationship revisited

O Blondeau-Fournier,P Mathieu,D Ridout,S Wood

doi:10.1016/j.nuclphysb.2016.09.003

Abstract

A recent novel derivation of the representation of Virasoro singular vectors in terms of Jack polynomials is extended to the supersymmetric case. The resulting expression of a generic super-Virasoro singular vector is given in terms of a simple differential operator (whose form is characteristic of the sector, Neveu–Schwarz or Ramond) acting on a Jack superpolynomial. The latter is indexed by a superpartition depending upon the two integers r,s that specify the reducible module under consideration. The corresponding singular vector (at grade rs/2), when expanded as a linear combination of Jack superpolynomials, results in an expression that (in addition to being proved) turns out to be more compact than those that have been previously conjectured. As an aside, in relation with the differential operator alluded to above, a remarkable property of the Jack superpolynomials at α=−3 is pointed out.

Highlights

A recent novel derivation of the representation of Virasoro singular vectors in terms of Jack polynomials is extended to the supersymmetric case
The resulting expression of a generic super-Virasoro singular vector is given in terms of a simple differential operator acting on a Jack superpolynomial
The first one [10] relies on the extension of the Jack polynomials to superpolynomials, which are polynomials that depend upon additional anticommuting variables

Summary

Introduction

The first one [10] relies on the extension of the Jack polynomials to superpolynomials, which are polynomials that depend upon additional anticommuting variables In this context, the simple link which exists in the Virasoro case, namely that a singular vector is represented by a single Jack indexed by a rectangular partition, becomes somewhat more complicated. These analyses rely on the theory of Jack superpolynomials, which is summarized in Appendix A. In Appendix B, which is somewhat off the main theme of the article, we display a remarkable formula for the action of the operator B, in the NS sector, on the sJacks PΓ(−3)

Super-Virasoro singular vectors from screening charges

The Ramond algebra

They decompose triangularly in the monomial basis