Inverse reduction for hook-type W-algebras

TL;DR

This paper demonstrates the existence of inverse reduction embeddings between affine rak{sl}_{n+1} vertex operator algebras and hook-type W-algebras, generalizing previous specific cases using free-field realizations.

Contribution

It establishes the existence of inverse reduction embeddings for all hook-type rak{sl}_{n+1} W-algebras, extending prior results for specific cases.

Findings

Inverse reduction embeddings exist between affine rak{sl}_{n+1} VOAs and hook-type W-algebras.

The results generalize previous specific cases to all hook-type rak{sl}_{n+1} W-algebras.

Uses free-field realizations and screening operators to construct embeddings.

Abstract

Originating in the work of A.M. Semikhatov and D. Adamovi\'c, inverse reductions are embeddings involving W-algebras corresponding to the same Lie algebra but different nilpotent orbits. Here, we show that an inverse reduction embedding between the affine $\mathfrak{sl}_{n+1}$ vertex operator algebra and the minimal $\mathfrak{sl}_{n+1}$ W-algebra exists. This generalises the realisations for $n=1,2$ in [arXiv:1711.11342, arXiv:2110.15203]. A similar argument is then used to show that inverse reduction embeddings exists between all hook-type $\mathfrak{sl}_{n+1}$ W-algebras, which includes the principal/regular, subregular, minimal $\mathfrak{sl}_{n+1}$ W-algebras, and the affine $\mathfrak{sl}_{n+1}$ vertex operator algebra. This generalises the regular-to-subregular inverse reduction of [arXiv:2111.05536], and similarly uses free-field realisations and their associated screening operators.