Higher Dimensional Classical W-Algebras

TL;DR

This paper constructs higher-dimensional classical W-algebras by generalizing Gel'fand-Dickey brackets, leading to new Poisson structures linked to a higher-dimensional dispersionless KP hierarchy.

Contribution

It introduces a novel method to build higher-dimensional classical W-algebras via generalization of classical brackets, extending their application to multidimensional integrable systems.

Findings

Explicit two-dimensional case analysis

Identification of local diffeomorphisms as symmetry algebra

Connection to higher-dimensional dispersionless KP hierarchy

Abstract

Classical $W$-algebras in higher dimensions are constructed. This is achieved by generalizing the classical Gel'fand-Dickey brackets to the commutative limit of the ring of classical pseudodifferential operators in arbitrary dimension. These $W$-algebras are the Poisson structures associated with a higher dimensional version of the Khokhlov-Zabolotskaya hierarchy (dispersionless KP-hierarchy). The two dimensional case is worked out explicitly and it is shown that the role of Diff$S(1)$ is taken by the algebra of generators of local diffeomorphisms in two dimensions.