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

TL;DR

This paper establishes a braided tensor equivalence between the Kazhdan-Lusztig categories of affine vertex algebras and their associated W-algebras for simply-laced Lie algebras at irrational levels, extending understanding of their categorical structures.

Contribution

It proves that the Kazhdan-Lusztig category of W-algebras is equivalent to that of affine vertex algebras at irrational levels for simply-laced Lie algebras, a new categorical correspondence.

Findings

Braided tensor equivalence established for irrational levels

Category equivalence holds for all nilpotent elements f

Extends known results to irrational level cases

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