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arXiv · hep-th/9312188

Generalized Drinfeld-Sokolov Hierarchies, Quantum Rings, and W-Gravity

W. Lerche
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Abstract

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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W. Lerche. 1994-09-30. Generalized Drinfeld-Sokolov Hierarchies, Quantum Rings, and W-Gravity. https://doi.org/10.1016/0550-3213(94)00438-k

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