The super-Virasoro singular vectors and Jack superpolynomials relationship revisited

Extends a free field derivation of Virasoro singular vectors to the \(\mathcal{N}=1\) superconformal algebra, expressing the singular vectors in both the Neveu–Schwarz and Ramond sectors as a simple differential operator acting on a single Jack superpolynomial. This proves formulae that are more compact than those previously conjectured.

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.

Recommended citation: O. Blondeau-Fournier, P. Mathieu, D. Ridout and S. Wood, Nucl. Phys. B913, 34-63 (2016)
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