Generalized Drinfel'd-Sokolov hierarchies, quantum rings, and W-gravity

We investigate the algebraic structure of integrable hierarchies that, we propose, underlie models of W-gravity coupled to matter. More precisely, we concentrate on the dispersionless limit of the topological subclass of such theories, by making use of a correspondence between Drinfel'd-Sokolov systems, principal sl(2) embeddings and certain chiral rings. We find that the integrable hierarchies can be viewed as generalizations of the usual matrix Drinfel'd-Sokolov systems to higher fundamental representations of sl(n). Accordingly, there are additional commuting flows as compared to the usual generalized KdV hierarchy. These are associated with the enveloping algebra and account for degeneracies of physical operators. The underlying Heisenberg algebras are nothing but specifically perturbed chiral rings of certain Kazama-Suzuki models, and have an intimate connection with the quantum cohomology of grassmannians. Correspondingly, the Lax operators are directly given in terms of multi-field superpotentials of the associated topological LG theories. We view our construction as a prototype for a multi-variable system and suspect that it might be useful also for a class of related problems. ((orig.))