AN EXPLICIT FORMULA FOR SOME SINGULAR VECTORS OF THE NEVEU–SCHWARZ ALGEBRA

Abstract

Similarly to the Virasoro algebra, the Neveu–Schwarz algebra has a discrete series of unitary irreducible highest weight representations. These are labeled by the values of [Formula: see text] (the central charge) and of the highest weight hpq = [(p (m + 2) − qm)2 − 4]/(8m (m + 2)) where m, p, q are some integers. The Verma modules constructed with these values (c, h) are not irreducible, however, as they contain two Verma submodules, each generated by a singular vector ψp,q (of weight hpq + pq/2) and ψm−p, m+2−q (of weight hpq + (m−p)(m+2−q)/2), respectively. We give an explicit expression for these singular vectors whenever one of its indices is 1.