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

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 (the central charge) and of the highest weight h_pq = [(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 ψ_p,q (of weight h_pq + pq/2) and ψ_{m−p, m+2−q} (of weight h_pq + (m−p)(m+2−q)/2), respectively. We give an explicit expression for these singular vectors whenever one of its indices is 1.