OSp(1|2) and Sl(2) reductions in generalised super-Toda models and factorization of spin 1/2 fields

I show that the classical Toda models built on superalgebras, and obtained from a reduction with respect to an Sl(2) algebra, are ”linearly supersymmetrizable” (by adding spin 1/2 fields) if and only if the Sl(2) is the bosonic part of an OSp(1|2) algebra. In that case, the model is equivalent to the one constructed from a reduction with respect to the OSp(1|2) algebra, up to spin 1/2 fields. The corresponding W algebras are related through a factorization of spin 1/2 fields (bosons and fermions). I illustrate this factorization on an example: the superconformal algebra built on Sl(n+ 1|2). On leave from absence of Laboratoire de Physique Théorique ENSLAPP Chemin de Bellevue BP 110, F 74941 Annecy-le-Vieux Cedex, France