NU CLEAR Nuclear Physics B 368 (1992) 361—376 P H Y S I CS B North-Holland _________________ Classical N = 2 W-superalgebras and supersymmetric Gel’fand—Dickey brackets José M. Figueroa-O’Farrill * and Eduardo Ramos ** Instituut L’oor Theoretische Fysica, Unicersiteit Leuien, Celestijnenlaan 200 D, B-3001 Hererlee, Belgium Received 11 April 1991 (Revised 6 September 1991) Accepted for publication 6 September 1991 We construct an infinite series of classical N = 2 W-superalgebras as a reduction of a recently constructed supersymmetric version of the Gel’fand—Dickey brackets for the generalized super KdV hierarchies. This is achieved via a supersymmetric Miura transformation which yields, as a by-product, classical free-field realizations. 1. Introduction W-algebras [1,2] play an increasing role in two-dimensional conformal field theory and in string theory. In the former context it has been shown that they appear as the symmetry [3] algebras of statistical mechanics systems at criticality; and in the latter case, their minimal models can be used for the construction of classical vacua. Furthermore, unexpected relationships have been recently unveiled between W-algebras and non-critical strings [4] via their matrix model formulation, where the “string equation” corresponds to generalized KdV flows with specific boundary conditions [51. This observation allows one to translate computations of correlation functions in non-critical string theory to more manageable algebraic computations in terms of the associated Lax system, without having to appeal directly to the matrix model formulation. This has a variety of advantages, not the least of which is that it allows for generalizations — one very important generaliza- tion being the one to non-critical superstrings, where a direct treatment in terms of (super)matrix models is lacking [6]. W-superalgebras appear naturally in the context of supersymmetric Toda field theory [7] as the algebras of conserved currents. In this paper we present a systematic treatment of N = 2 W-superalgebras based on the formalism of super- symmetric KdV hierarchies, which seems especially well suited for the applications to superstring theory. * E-mail: jmf@iks.kuleuven.ac.be and fgbdall(ãblekulll.BITNET Present address: Physikalisches Institut der Universität Bonn, Germany. ** E-mail: fgbdaO6@blekulll.BITNET and fgbda06@cc1.kuleuven.ac.be 0550-3213/92/$05.OO © 1992 — Elsevier Science Publishers B.V. All rights reserved