99 4 B . R . S . renormalisation of some on -

In order to study in a regularisation free manner the renormalisability of d=2 supersymmetric non-linear σ models, one has to use the algebraic BRS methods ; moreover, in the absence of an off-shell formulation, one has often to deal with open algebras. We then recall in a pedagogical and non technical manner the standard methods used to handle these questions and illustrate them on N=1 supersymmetric non-linear σ model in component fields, giving the first rigorous proof of their renormalisability. In the special case of compact homogeneous manifolds (non-linear σ model on a coset space G/H), we obtain the supersymmetric extension of the analysis done some years ago in the bosonic case. A further publication will be devoted to extended supersymmetry. PAR/LPTHE/94-10 Mars 1994 Laboratoire de Physique Théorique et des Hautes Energies, Unité associée au CNRS URA 280, Université Paris 7, 2 Place Jussieu, 75251 Paris Cedex 05. Electronic address [email protected]