An analysis is given of the structure of a general two-dimensional Toda 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 simple root system for a superalgebra has two natural bosonic root systems associated with it which can be found very simply using Dynkin diagrams; the construction is closely related to the question of how to recover the signs of the entries of a Cartan matrix for a superalgebra from its Dynkin diagram. The significance for Toda theories is that the 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 kinds of models and their implications are investigated for features such as supersymmetry, positive kinetic energy and generalized reality conditions for the Toda fields. As a result, some new families of integrable theories with positive kinetic energy are constructed, some containing a mixture of massless and massive degrees of freedom, others being purely massive and supersymmetric, involving a number of coupled sine/sinh-Gordon theories.

We present a systematic construction of classical extended superconformal algebras from the hamiltonian reduction of a class of affine Lie superalgebras, which include an even subalgebra sl(2). In particular, we obtain the doubly extended N = 4 superconformal algebraà γ from the hamiltonian reduction of the exceptional Lie superalgebra D(2|1; γ/(1 − γ)). We also find the Miura transformation for these algebras and give the free field representation. A W -algebraic generalization is discussed.

We study the classical and quantum G extended superconformal algebras from the hamiltonian reduction of affine Lie superalgebras, with even subalgebras G ⊕ sl(2). At the classical level we obtain generic formulas for the Poisson bracket structure of the algebra. At the quantum level we get free field (Feigin-Fuchs) representations of the algebra by using the BRST formalism and the free field realization of the affine Lie superalgebra. In particular we get the free field representation of the sl(2) ⊕ sp(2N) extended superconformal algebra from the Lie superalgebra osp(4|2N). We also discuss the screening operators of the algebra and the structure of singular vectors in the free field representation.

Recently the quantum hamiltonian reduction was done in the case of general sℓ(2) embeddings into Lie algebras and superalgebras. In this paper we extend the results to the quantum hamiltonian reduction of N = 1 affine Lie superalgebras in the superspace formalism. We show that if we choose a gauge for the supersymmetry, and consider only certain equivalence classes of fields, then our quantum hamiltonian reduction reduces to quantum hamiltonian reduction of non-supersymmetric Lie superalgebras. We construct explicitly the super energy-momentum tensor, as well as all generators of spin 1 (and 1 2 ); thus we construct explicitly all generators in the superconformal, quasi-superconformal and Z Z 2 × Z Z 2 superconformal algebras.