Orbifolds and Cosets of Minimal ${\mathcal{W}}$-Algebras

Abstract

Let ${\mathfrak{g}}$ be a simple, finite-dimensional Lie (super)algebra equipped with an embedding of ${\mathfrak{s}\mathfrak{l}_2}$ inducing the minimal gradation on ${\mathfrak{g}}$. The corresponding minimal ${\mathcal{W}}$-algebra ${\mathcal{W}^k(\mathfrak{g}, e_{-\theta})}$ introduced by Kac and Wakimoto has strong generators in weights ${1,2,3/2}$, and all operator product expansions are known explicitly. The weight one subspace generates an affine vertex (super)algebra ${V^{k'}(\mathfrak{g}^{\natural})}$, where ${\mathfrak{g}^{\natural} \subset \mathfrak{g}}$ denotes the centralizer of ${\mathfrak{s}\mathfrak{l}_2}$. Therefore, ${\mathcal{W}^k(\mathfrak{g}, e_{-\theta})}$ has an action of a connected Lie group ${G^{\natural}_0}$ with Lie algebra ${\mathfrak{g}^{\natural}_0}$, where ${\mathfrak{g}^{\natural}_0}$ denotes the even part of ${\mathfrak{g}^{\natural}}$. We show that for any reductive subgroup ${G \subset G^{\natural}_0}$, and for any reductive Lie algebra ${\mathfrak{g}' \subset \mathfrak{g}^{\natural}}$, the orbifold ${\mathcal{O}^k = \mathcal{W}^k(\mathfrak{g}, e_{-\theta})^{G}}$ and the coset ${\mathcal{C}^k = {\rm Com}(V(\mathfrak{g}'),\mathcal{W}^k(\mathfrak{g}, e_{-\theta}))}$ are strongly finitely generated for generic values of ${k}$. Here ${V(\mathfrak{g}')}$ denotes the affine vertex algebra associated to ${\mathfrak{g}'}$. We find explicit minimal strong generating sets for ${\mathcal{C}^k}$ when ${\mathfrak{g}' = \mathfrak{g}^{\natural}}$ and ${\mathfrak{g}}$ is either ${\mathfrak{s}\mathfrak{l}_n}$, ${\mathfrak{s}\mathfrak{p}_{2n}}$, ${\mathfrak{s}\mathfrak{l}(2|n)}$ for ${n \neq 2}$, ${\mathfrak{p}\mathfrak{s}\mathfrak{l}(2|2)}$, or ${\mathfrak{o}\mathfrak{s}\mathfrak{p}(1|4)}$. Finally, we conjecture some surprising coincidences among families of cosets ${\mathcal{C}_k}$ which are the simple quotients of ${\mathcal{C}^k}$, and we prove several cases of our conjecture.