The Classical Limit of W-Algebras

TL;DR

This paper explicitly computes the classical limits of $W_n$ algebras as reductions of a universal algebra $w_{KP}$, revealing their structure and relation to $w_{1+ ablafty}$, and introduces a new algebraic framework for classical $W$-algebras.

Contribution

It defines and computes the classical limits of $W_n$ algebras, introduces the universal algebra $w_{KP}$, and relates these to $w_{1+ ablafty}$ as a deformation.

Findings

Classical limits of $W_n$ are explicitly computed.

All $w_n$ are reductions of a new universal algebra $w_{KP}.

The algebra $w_{1+ ablafty}$ is obtained as a deformation of $w_{KP}$.

Abstract

We define and compute explicitly the classical limit of the realizations of $W_n$ appearing as hamiltonian structures of generalized KdV hierarchies. The classical limit is obtained by taking the commutative limit of the ring of pseudodifferential operators. These algebras---denoted $w_n$---have free field realizations in which the generators are given by the elementary symmetric polynomials in the free fields. We compute the algebras explicitly and we show that they are all reductions of a new algebra $w_{\rm KP}$, which is proposed as the universal classical $W$-algebra for the $w_n$ series. As a deformation of this algebra we also obtain $w_{1+\infty}$, the classical limit of $W_{1+\infty}$.