ar X iv : s ol v - in t / 9 51 10 07 v 2 1 7 N ov 1 99 5 Algebra of Non - Local Charges in Supersymmetric Non - Linear Sigma Models

We propose a graphic method to derive the classical algebra (Dirac brackets) of non-local conserved charges in the two-dimensional supersymmetric non-linear O(N) sigma model. As in the purely bosonic theory we find a cubic Yangian algebra. We also consider the extension of graphic methods to other integrable theories. 1 Introduction Non-linear sigma models [1-3] are prototypes of a remarkable class of integrable two dimensional models which contain an infinite number of conserved local and non-local charges [4-7]. The algebraic relations obeyed by such charges are supposed to be an important ingredient in the complete solution of those models [8-11]. The local charges form an Abelian algebra. Opposing to that simplicity, the algebra of non-local charges is non-Abelian and actually non-linear [12-28]. In ref.[29] the O(N) sigma model was investigated and a particular set of non-local charges – called improved charges – was found to satisfy a cubic algebra related to a Yangian structure. In this work we intend to extend that result to the corresponding supersymmetric case [30-32]. The introduction of supersymmetry might have rendered a much more involved algebra [33]. However, it has been conjectured [29,32] that, in the sigma model, the algebra of supersymmetric non-local charges would remain the same as in the bosonic theory and we shall present results that confirm such conjecture. This paper is organized as follows. In Sect.2 we briefly review the results from the purely bosonic theory. A graphic technique to compute charges and their algebra is introduced in Sect.3. In Sect.4 we discuss the supersymmetric model and the main results of the paper. Another application of graphic rules is shown in Sect. 5 concerning the O(N) Gross-Neveu model. Sect.6 is left for conclusions while an appendix contains examples of the graphic technique.