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The Kazhdan-Lusztig category of W-algebras of simply-laced Lie algebras at irrational levels

Thomas Creutzig, Gurbir Dhillon, Shigenori Nakatsuka

Abstract

Let $\mathfrak{g}$ be a simple, simply-laced Lie algebra and $f \in \mathfrak{g}$ nilpotent. The Kazhdan-Lusztig category of the W-algebra $W^κ(\mathfrak{g},f)$ associated with $(\mathfrak{g},f)$ at level $κ\in \mathbb{C}$ is obtained from the Kazhdan-Lusztig category of the affine vertex algebra $V^κ(\mathfrak{g})$ via the quantum Hamiltonian reduction associated with $f$. We show that this is a braided tensor equivalence for any $f$ and any irrational level $κ\in \mathbb{C} \backslash \mathbb{Q}$.

The Kazhdan-Lusztig category of W-algebras of simply-laced Lie algebras at irrational levels

Abstract

Let be a simple, simply-laced Lie algebra and nilpotent. The Kazhdan-Lusztig category of the W-algebra associated with at level is obtained from the Kazhdan-Lusztig category of the affine vertex algebra via the quantum Hamiltonian reduction associated with . We show that this is a braided tensor equivalence for any and any irrational level .
Paper Structure (20 sections, 23 theorems, 90 equations)

This paper contains 20 sections, 23 theorems, 90 equations.

Table of Contents

  1. Introduction
  2. Affine vertex algebras and $\mathcal{W}$-algebras
  3. Affine vertex algebras
  4. ABC-algebras
  5. Twisted reductions
  6. Equivariant ABC-algebras at irrational levels
  7. Cosets à la Goddard-Kent-Olive
  8. Braided tensor categories
  9. Intertwining Operators
  10. Commutative algebras
  11. ABC-cofinite modules
  12. Mirror equivalence
  13. Principal nilpotent case
  14. ABC-cofiniteness
  15. Centralizing property
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $\mathfrak{g}$ be of type $ADE$ and $\kappa \in \mathbb{C} \backslash \mathbb{Q}$. Then, the BRST reduction $H_f\colon \mathrm{KL}^\kappa(\mathfrak{g}) \xrightarrow{\simeq} \mathrm{KL}_f^\kappa(\mathfrak{g})$ is an equivalence of braided tensor categories.

Theorems & Definitions (36)

  • Theorem 1.1: Theorem \ref{['thm:main']}
  • Corollary 1.2: Corollary \ref{['cor:main']}
  • Theorem 2.1: AF
  • Theorem 2.2: A4
  • Theorem 2.3: ACLCNCL1
  • Corollary 2.4: ACF
  • Proposition 3.1: C1
  • Theorem 3.2: CKM
  • Example 3.3: KL
  • Theorem 3.4: H1
  • ...and 26 more