ON SW MINIMAL MODELS AND N = 1 SUPERSYMMETRIC QUANTUM TODA FIELD THEORIES

Abstract

Integrable N = 1 supersymmetric Toda field theories are determined by a contragredient simple super-Lie-algebra (SSLA) with purely fermionic lowering and raising operators. For the SSLA's Osp(3|2) and D(2|1; α) we construct explicitly the higher spin conserved currents and obtain free field representations of the super-W-algebras SW(3/2, 2) and SW(3/2, 3/2, 2). In constructing the corresponding series of minimal models using covariant vertex operators, we find a necessary restriction on the Cartan matrix of the SSLA, also for the general case. Within this framework, the restriction claims that there should be a minimum of one nonvanishing element on the diagonal of the Cartan matrix. This condition is without parallel in bosonic conformal field theory. As a consequence only two series of SSLA's yield minimal models, namely Osp(2n|2n−1) and Osp(2n|2n+1). Subsequently some general aspects of degenerate representations of SW algebras, notably the fusion rules, are investigated. As an application we discuss minimal models of SW(3/2, 2), which were constructed by independent methods, in this framework. Covariant formulation is used throughout this paper.