Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Reduction by stages for affine W-algebras

Naoki Genra, Thibault Juillard

TL;DR

The paper develops a general framework for reduction by stages of affine W-algebras, proving that under explicit grading and nilpotency conditions, an affine W-algebra $\\mathcal{W}^k(\\mathfrak{g}, f_2)$ can be realized as the BRST reduction of $\\mathcal{W}^k(\\mathfrak{g}, f_1)$. Central to the approach are reductions on Slodowy slices and arc spaces, together with a new BRST construction that reconstructs $\\mathcal{W}^k(\\mathfrak{g}, f_2)$ from $\\mathcal{W}^k(\\mathfrak{g}, f_1)$, plus vanishing theorems that ensure cohomology concentrates in degree 0. The results unify and extend previous work, give explicit sufficient conditions for reduction by stages, and provide numerous examples across types, including type A and other classical types. The work catalyzes potential embeddings and isomorphisms among W-algebras and informs future study of modules in Kazhdan–Lusztig categories, highlighting the geometric underpinnings via Slodowy slices and arc-space reductions. Overall, it advances the structural understanding of affine W-algebras and their interrelations through a robust BRST- and geometry-driven methodology.

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.

Reduction by stages for affine W-algebras

TL;DR

The paper develops a general framework for reduction by stages of affine W-algebras, proving that under explicit grading and nilpotency conditions, an affine W-algebra can be realized as the BRST reduction of . Central to the approach are reductions on Slodowy slices and arc spaces, together with a new BRST construction that reconstructs from , plus vanishing theorems that ensure cohomology concentrates in degree 0. The results unify and extend previous work, give explicit sufficient conditions for reduction by stages, and provide numerous examples across types, including type A and other classical types. The work catalyzes potential embeddings and isomorphisms among W-algebras and informs future study of modules in Kazhdan–Lusztig categories, highlighting the geometric underpinnings via Slodowy slices and arc-space reductions. Overall, it advances the structural understanding of affine W-algebras and their interrelations through a robust BRST- and geometry-driven methodology.

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.
Paper Structure (41 sections, 52 theorems, 271 equations, 1 figure, 1 table)

This paper contains 41 sections, 52 theorems, 271 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Context
  3. Reduction by stages
  4. Main results
  5. Future work and applications
  6. Outline
  7. Notations and conventions
  8. Acknowledgment
  9. Vertex algebra and associated Poisson geometry
  10. Graded vertex superalgebras
  11. Graded Poisson vertex algebra associated with the Li filtration
  12. Arc spaces of Poisson varieties
  13. Examples of vertex superalgebras
  14. Affine vertex algebras
  15. Clifford vertex superalgebras
  16. ...and 26 more sections

Key Result

Theorem 1

If the conditions conditions hold, there is a BRST cochain complex $\mathcal{C}^\bullet_{f_0}(\mathcal{W}^k(\mathfrak{g}, f_1))$ whose cohomology is isomorphic to $\mathcal{W}^k(\mathfrak{g}, f_2)$ as vertex algebras:

Figures (1)

  • Figure 1: Examples of pyramids

Theorems & Definitions (107)

  • Theorem 1: \ref{['theorem:reduction-by-stages-w-algebras']}
  • Theorem 2: Theorem \ref{['theorem:new-construction-w-algebra']}
  • Theorem 3: Theorem \ref{['theorem:vanishing-vertex']}
  • Conjecture 4
  • Definition 2.1.1
  • Lemma 3.1.1
  • proof
  • Theorem 3.2.1
  • Theorem 3.4.1
  • Remark 3.4.2
  • ...and 97 more