DYNKIN DIAGRAMS AND INTEGRABLE MODELS BASED ON LIE-SUPERALGEBRAS

An analysis is given of the structure of a general two-dimensional Tod a field theory involving bosons and fermions which is defined in terms of a set of simple roots for a Lie superalgebra. It is shown that a s imple root system for a superalgebra has two natural bosonic root syst ems associated with it which can be found very simply using Dynkin dia grams; the construction is closely related to the question of how to r ecover the signs of the entries of a Cartan matrix for a superalgebra from its Dynkin diagram. The significance for Toda theories is that th e bosonic root systems correspond to the purely bosonic sector of the integrable model, knowledge of which can determine the bosonic part of the extended conformal symmetry in the theory, or its classical mass spectrum, as appropriate, These results are applied to some special ki nds of models and their implications are investigated for features suc h as supersymmetry, positive kinetic energy and generalized reality co nditions for the Toda fields. As a result, some new families of integr able theories with positive kinetic energy are constructed, some conta ining a mixture of massless and massive degrees of freedom, others bei ng purely massive and supersymmetric, involving a number of coupled si ne/sinh-Gordon theories. (C) 1997 Elsevier Science B.V.