A One-Parameter Family of Hamiltonian Structures for the KP Hierarchy and a Continuous Deformation of the Nonlinear $\W_{\rm KP}$ Algebra

TL;DR

This paper introduces a one-parameter family of Hamiltonian structures for the KP hierarchy, leading to a continuous deformation of the $ ext{W}$-algebra that interpolates between known algebras and reveals new algebraic structures.

Contribution

It constructs a novel one-parameter family of Poisson structures for the KP hierarchy, generalizing the $ ext{W}$-algebra and connecting various known $ ext{W}$-algebras through deformation and contraction.

Findings

The family admits a central extension for generic parameters.

Special parameter values recover classical $ ext{W}_n$ algebras.

The algebra contracts to a new nonlinear $ ext{W}_ ext{infinity}$-type algebra.

Abstract

The KP hierarchy is hamiltonian relative to a one-parameter family of Poisson structures obtained from a generalized Adler map in the space of formal pseudodifferential symbols with noninteger powers. The resulting $\W$-algebra is a one-parameter deformation of $\W_{\rm KP}$ admitting a central extension for generic values of the parameter, reducing naturally to $\W_n$ for special values of the parameter, and contracting to the centrally extended $\W_{1+\infty}$, $\W_\infty$ and further truncations. In the classical limit, all algebras in the one-parameter family are equivalent and isomorphic to $\w_{\rm KP}$. The reduction induced by setting the spin-one field to zero yields a one-parameter deformation of $\widehat{\W}_\infty$ which contracts to a new nonlinear algebra of the $\W_\infty$-type.