The super-Virasoro singular vectors and Jack superpolynomials relationship revisited

TL;DR

This paper extends a recent method to express Virasoro singular vectors using Jack polynomials into the supersymmetric case, resulting in more compact formulas and revealing special properties at specific parameters.

Contribution

It provides a new, simplified representation of super-Virasoro singular vectors in terms of Jack superpolynomials, generalizing previous results to the supersymmetric setting.

Findings

Derived a differential operator representation for super-Virasoro singular vectors.

Expressed singular vectors as linear combinations of Jack superpolynomials with a more compact form.

Identified a remarkable property of Jack superpolynomials at alpha=-3.

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 alpha=-3 is pointed out.