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 Drinfeld-Sokolov systems,
principal sâ„“(2) embeddings and certain chiral rings. We find that the
integrable hierarchies can be viewed as generalizations of the usual matrix
Drinfeld-Sokolov systems to higher fundamental representations of sâ„“(n).
The underlying Heisenberg algebras have an intimate connection with the quantum
cohomology of grassmannians. 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.Comment: 34p, harvmac/lanlmac with hyperlinks, 2 uuencoded PostScript figures
appended, CERN-TH.6988/93. (streamlined version that incorporates recent
results on W-ground ring spectra