The Drinfeld--Sokolov reduction of admissible representations of affine Lie algebras

TL;DR

This paper proves a key conjecture relating admissible affine Lie algebra modules to minimal series W-algebra modules via Drinfeld--Sokolov reduction, extending previous results to spectral flow cases.

Contribution

It establishes the conjecture that Drinfeld--Sokolov reduction maps admissible modules to minimal series modules, including spectral flow cases, generalizing Arakawa's earlier work.

Findings

Confirmed the conjecture for all admissible modules

Extended results to spectral flow Drinfeld--Sokolov reduction

Unified understanding of module transformations under reduction

Abstract

Fix an affine Lie algebra $\widehat{\mathfrak{g}}_\kappa$ with associated principal affine W-algebra $\mathcal{W}_\kappa$. A basic conjecture of Frenkel--Kac--Wakimoto asserts that Drinfeld--Sokolov reduction sends admissible $\widehat{\mathfrak{g}}_\kappa$-modules to zero or cohomological shifts of minimal series $\mathcal{W}_\kappa$-modules. We prove this conjecture and a natural generalization to the spectrally flowed Drinfeld--Sokolov reduction functors. This extends the previous results of Arakawa for the minus reduction.