Minimal W-algebras of $\mathfrak{so}_N$ at level minus one

TL;DR

This paper establishes an isomorphism between certain minimal W-algebras of so_N at level -1 and tensor products involving affine vertex superalgebras and free fermions, confirming their rationality for even N.

Contribution

It explicitly describes the structure of minimal W-algebras of so_N at level -1 and proves their strong rationality when N is even.

Findings

Isomorphism with tensor products of affine vertex superalgebras and free fermions

Confirmation of strong rationality for even N

Explicit description of minimal W-algebras of so_N

Abstract

For $N \in\mathbb Z_{\geq 7}$ we show that the simple minimal $\mathcal{W}$-algebra of $\mathfrak{so}_N$ at level minus one is isomorphic to the even subalgebra of the tensor product of the simple affine vertex superalgebra of $\mathfrak{osp}_{1|2}$ at level $\frac{N-6}{2}$ with $N-4$ free fermions. In particular when $N$ is even this minimal $\mathcal{W}$-algebra is strongly rational as conjectured by Arakawa-Moreau.