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

TL;DR

This paper explores the algebraic structures of integrable hierarchies related to $W$-gravity and topological theories, revealing connections with quantum cohomology and generalizations of Drinfeld-Sokolov systems.

Contribution

It introduces a generalized framework for Drinfeld-Sokolov hierarchies linked to $W$-gravity, extending to higher representations and connecting with quantum cohomology.

Findings

Hierarchies relate to quantum cohomology of grassmannians

Lax operators expressed via topological LG superpotentials

Generalization of matrix Drinfeld-Sokolov systems to higher representations

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.