Physics – High Energy Physics – High Energy Physics - Theory
Scientific paper
1993-12-23
Nucl.Phys. B434 (1995) 445-474
Physics
High Energy Physics
High Energy Physics - Theory
34p, harvmac/lanlmac with hyperlinks, 2 uuencoded PostScript figures appended, CERN-TH.6988/93. (streamlined version that inco
Scientific paper
10.1016/0550-3213(94)00438-K
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 $s\ell(2)$ 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 $s\ell(n)$. 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.
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