Inverse reduction for hook-type W-algebras

Abstract

Originating in the work of A.M. Semikhatov and D. Adamović, 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.

arXiv comment27 pages, 1 figure