Reduction by stages for affine W-algebras

TL;DR

This paper establishes a method called reduction by stages for affine W-algebras associated with nilpotent orbits in simple Lie algebras, showing how one W-algebra can be derived from another through quantum Hamiltonian reduction, supported by examples and cohomology techniques.

Contribution

It introduces the reduction by stages property for affine W-algebras and demonstrates its validity using reduction techniques, examples, and BRST cohomology constructions.

Findings

Reduction by stages holds for affine W-algebras under certain conditions.

Several classical and exceptional examples are provided.

A connection between different W-algebras via BRST cohomology is established.

Abstract

Given a pair of nilpotent orbits in a simple Lie algebra, one can associate a pair of vertex algebras called affine W-algebras. Under some compatibility conditions on these orbits, we prove that one of these W-algebras can be obtained as the quantum Hamiltonian reduction of the other. This property is called reduction by stages. We provide several examples in classical and exceptional types. To prove reduction by stages for affine W-algebras, we use our previous work on reduction by stages for the Slodowy slices associated with these nilpotent orbits, these slices being the associated varieties of the W-algebras. We also prove and use the fact that each W-algebra can be defined using several equivalent BRST cohomology constructions: choosing the right BRST complexes allows us to connect the two W-algebras in a natural way.