A Class of W-Algebras with Infinitely Generated Classical Limit

TL;DR

This paper introduces a new class of deformable W-algebras with infinitely generated classical limits, contrasting with the well-understood finitely generated cases, and explores their quantum corrections and algebraic structures.

Contribution

It identifies and illustrates a second class of deformable W-algebras with infinitely generated classical limits, expanding the understanding of their structure and quantum properties.

Findings

Classical limits are infinitely, nonfreely generated rings of differential polynomials.

Quantum corrections lead to finitely generated quantum algebras.

Invariant theory explains the relations and the origin of specific W-algebras.

Abstract

There is a relatively well understood class of deformable W-algebras, resulting from Drinfeld-Sokolov (DS) type reductions of Kac-Moody algebras, which are Poisson bracket algebras based on finitely, freely generated rings of differential polynomials in the classical limit. The purpose of this paper is to point out the existence of a second class of deformable W-algebras, which in the classical limit are Poisson bracket algebras carried by infinitely, nonfreely generated rings of differential polynomials. We present illustrative examples of coset constructions, orbifold projections, as well as first class Hamiltonian reductions of DS type W-algebras leading to reduced algebras with such infinitely generated classical limit. We also show in examples that the reduced quantum algebras are finitely generated due to quantum corrections arising upon normal ordering the relations obeyed by the classical generators. We apply invariant theory to describe the relations and to argue that classical cosets are infinitely, nonfreely generated in general. As a by-product, we also explain the origin of the previously constructed and so far unexplained deformable quantum W(2,4,6) and W(2,3,4,5) algebras.